Editing List of small groups (section)

== List of small abelian groups ==
The finite abelian groups are either cyclic groups, or direct products thereof; see [[Abelian group]]. The numbers of nonisomorphic abelian groups of orders ''n'' = 1, 2, ... are
: 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, ... {{OEIS|id=A000688}}
For labeled abelian groups, see {{oeis|A034382}}.

{| class="wikitable"
|+ List of all abelian groups up to order 31
|-
! Order
! Id.{{efn|name=id|Identifier when groups are numbered by order, ''o'', then by index, ''i'',  from the small groups library, starting at 1.}}
! G<sub>''o''</sub><sup>''i''</sup>
! Group
! Non-trivial proper subgroups{{r|Dockchitser}}
! [[Cycle graph (algebra)|Cycle<br />graph]]
! Properties
|-
! 1
! 1
! G<sub>1</sub><sup>1</sup>
| Z<sub>1</sub> = S<sub>1</sub> = A<sub>2</sub>
| –
| align=center|[[Image:GroupDiagramMiniC1.svg|40px]]
| [[Trivial group|Trivial]]. Cyclic. Alternating. Symmetric. [[Elementary abelian group|Elementary]].
|-
! 2
! 2
! G<sub>2</sub><sup>1</sup>
| Z<sub>2</sub> = S<sub>2</sub> = D<sub>2</sub>
| –
| align=center|[[Image:GroupDiagramMiniC2.svg|40px]]
| Simple. Symmetric. Cyclic. Elementary. (Smallest non-trivial group.)
|-
! 3
! 3
! G<sub>3</sub><sup>1</sup>
| Z<sub>3</sub> = A<sub>3</sub>
| –
| align=center|[[Image:GroupDiagramMiniC3.svg|40px]]
| Simple. Alternating. Cyclic. Elementary. 
|-
! rowspan="2" | 4
! 4
! G<sub>4</sub><sup>1</sup>
| Z<sub>4</sub> = Q<sub>4</sub>
| Z<sub>2</sub>
| align=center|[[Image:GroupDiagramMiniC4.svg|40px]]
| Cyclic.
|-
! 5
! G<sub>4</sub><sup>2</sup>
| Z<sub>2</sub><sup>2</sup> = K<sub>4</sub> = D<sub>4</sub>
| Z<sub>2</sub> (3)
| align=center|[[Image:GroupDiagramMiniD4.svg|40px]]
| Elementary. [[Direct product of groups|Product]]. ([[Klein four-group]]. The smallest non-cyclic group.)
|-
! 5
! 6
! G<sub>5</sub><sup>1</sup>
| Z<sub>5</sub>
| –
| align=center|[[Image:GroupDiagramMiniC5.svg|40px]]
| Simple. Cyclic. Elementary.
|-
! 6
! 8
! G<sub>6</sub><sup>2</sup>
| Z<sub>6</sub> = Z<sub>3</sub> × Z<sub>2</sub><ref>See a worked [[Isomorphism#Integers modulo 6|example showing the isomorphism Z<sub>6</sub> = Z<sub>3</sub> × Z<sub>2</sub>]].</ref>
| Z<sub>3</sub>, Z<sub>2</sub>
| align=center|[[Image:GroupDiagramMiniC6.svg|40px]]
| Cyclic. Product.
|-
! 7
! 9
! G<sub>7</sub><sup>1</sup>
| Z<sub>7</sub>
| –
| align=center|[[Image:GroupDiagramMiniC7.svg|40px]]
| Simple. Cyclic. Elementary. 
|-
! rowspan="3" | 8
! 10
! G<sub name=g8>8</sub><sup>1</sup>
| Z<sub>8</sub>
| Z<sub>4</sub>, Z<sub>2</sub>
| align=center|[[Image:GroupDiagramMiniC8.svg|40px]]
| Cyclic.
|-
! 11
! G<sub>8</sub><sup>2</sup>
| Z<sub>4</sub> × Z<sub>2</sub>
| Z<sub>2</sub><sup>2</sup>, Z<sub>4</sub> (2), Z<sub>2</sub> (3)
| align=center|[[Image:GroupDiagramMiniC2C4.svg|40px]]
| Product.
|-
! 14
! G<sub>8</sub><sup>5</sup>
| Z<sub>2</sub><sup>3</sup>
| Z<sub>2</sub><sup>2</sup> (7), Z<sub>2</sub> (7)
| align=center|[[Image:GroupDiagramMiniC2x3.svg|40px]]
| Product. Elementary. (The non-identity elements correspond to the points in the [[Fano plane]], the {{nowrap|Z<sub>2</sub> × Z<sub>2</sub>}} subgroups to the lines.)
|-
! rowspan="2" | 9
! 15
! G<sub>9</sub><sup>1</sup>
| Z<sub>9</sub>
| Z<sub>3</sub> 
| align=center|[[Image:GroupDiagramMiniC9.svg|40px]]
| Cyclic.
|-
! 16
! G<sub>9</sub><sup>2</sup>
| Z<sub>3</sub><sup>2</sup>
| Z<sub>3</sub> (4) 
|align=center| [[Image:GroupDiagramMiniC3x2.svg|40px]]
| Elementary. Product.
|-
! 10
! 18
! G<sub>10</sub><sup>2</sup>
| Z<sub>10</sub> = Z<sub>5</sub> × Z<sub>2</sub>
| Z<sub>5</sub>, Z<sub>2</sub>
| align=center|[[Image:GroupDiagramMiniC10.svg|40px]]
| Cyclic. Product.
|-
! 11
! 19
! G<sub>11</sub><sup>1</sup>
| Z<sub>11</sub>
| –
| align=center|[[Image:GroupDiagramMiniC11.svg|40px]]
| Simple. Cyclic. Elementary.
|-
! rowspan="2" | 12
! 21
! G<sub>12</sub><sup>2</sup>
| Z<sub>12</sub> = Z<sub>4</sub> × Z<sub>3</sub>
| Z<sub>6</sub>, Z<sub>4</sub>, Z<sub>3</sub>, Z<sub>2</sub>
| align=center|[[Image:GroupDiagramMiniC12.svg|40px]]
| Cyclic. Product. 
|-
! 24
! G<sub>12</sub><sup>5</sup>
| Z<sub>6</sub> × Z<sub>2</sub> = Z<sub>3</sub> × Z<sub>2</sub><sup>2</sup>
| Z<sub>6</sub> (3), Z<sub>3</sub>, Z<sub>2</sub> (3), Z<sub>2</sub><sup>2</sup>
| align=center|[[Image:GroupDiagramMiniC2C6.svg|40px]]
| Product.
|-
! 13
! 25
! G<sub>13</sub><sup>1</sup>
| Z<sub>13</sub>
| –
| align=center|[[Image:GroupDiagramMiniC13.svg|40px]]
| Simple. Cyclic. Elementary. 
|-
! 14
! 27
! G<sub>14</sub><sup>2</sup>
| Z<sub>14</sub> = Z<sub>7</sub> × Z<sub>2</sub>
| Z<sub>7</sub>, Z<sub>2</sub>
|align=center| [[Image:GroupDiagramMiniC14.svg|40px]]
| Cyclic. Product.
|-
! 15
! 28
! G<sub>15</sub><sup>1</sup>
| Z<sub>15</sub> = Z<sub>5</sub> × Z<sub>3</sub>
| Z<sub>5</sub>, Z<sub>3</sub>
| align=center|[[Image:GroupDiagramMiniC15.svg|40px]]
| Cyclic. Product. 
|-
! rowspan="5" | 16
! 29
! G<sub>16</sub><sup>1</sup>
| Z<sub>16</sub>
| Z<sub>8</sub>, Z<sub>4</sub>, Z<sub>2</sub>
| align=center|[[Image:GroupDiagramMiniC16.svg|40px]]
| Cyclic. 
|-
! 30
! G<sub>16</sub><sup>2</sup>
| Z<sub>4</sub><sup>2</sup>
| Z<sub>2</sub> (3), Z<sub>4</sub> (6), Z<sub>2</sub><sup>2</sup>, {{nowrap|Z<sub>4</sub> × Z<sub>2</sub>}} (3)</td> 
| align=center|[[Image:GroupDiagramMiniC4x2.svg|40px]]
| Product.
|-
! 33
! G<sub>16</sub><sup>5</sup>
| Z<sub>8</sub> × Z<sub>2</sub>
| Z<sub>2</sub> (3), Z<sub>4</sub> (2), Z<sub>2</sub><sup>2</sup>, Z<sub>8</sub> (2), {{nowrap|Z<sub>4</sub> × Z<sub>2</sub>}}
| align=center|[[File:GroupDiagramC2C8.svg|40px]]
| Product. 
|-
! 38
! G<sub>16</sub><sup>10</sup>
| Z<sub>4</sub> × Z<sub>2</sub><sup>2</sup>
| Z<sub>2</sub> (7), Z<sub>4</sub> (4), Z<sub>2</sub><sup>2</sup> (7), Z<sub>2</sub><sup>3</sup>, {{nowrap|Z<sub>4</sub> × Z<sub>2</sub>}} (6)
| align=center|[[Image:GroupDiagramMiniC2x2C4.svg|40px]]
| Product.
|-
! 42
! G<sub>16</sub><sup>14</sup>
| Z<sub>2</sub><sup>4</sup> = K<sub>4</sub><sup>2</sup>
| Z<sub>2</sub> (15), Z<sub>2</sub><sup>2</sup> (35), Z<sub>2</sub><sup>3</sup> (15)</td> 
| align=center|[[Image:GroupDiagramMiniC2x4.svg|40px]]
| Product. Elementary.
|-
! 17
! 43
! G<sub>17</sub><sup>1</sup>
| Z<sub>17</sub>
| –
| align=center|[[Image:GroupDiagramMiniC17.svg|40px]]
| Simple. Cyclic. Elementary.
|-
! rowspan="2" | 18
! 45
! G<sub>18</sub><sup>2</sup>
| Z<sub>18</sub> = Z<sub>9</sub> × Z<sub>2</sub>
| Z<sub>9</sub>, Z<sub>6</sub>, Z<sub>3</sub>, Z<sub>2</sub>
| align=center|[[Image:GroupDiagramMiniC18.svg|40px]]
| Cyclic. Product.
|-
! 48
! G<sub>18</sub><sup>5</sup>
| Z<sub>6</sub> × Z<sub>3</sub> = Z<sub>3</sub><sup>2</sup> × Z<sub>2</sub> || Z<sub>2</sub>, Z<sub>3</sub> (4), Z<sub>6</sub> (4), Z<sub>3</sub><sup>2</sup> ||[[File:GroupDiagramMiniC3C6.png|50px]] || Product.
|-
! 19
! 49
! G<sub>19</sub><sup>1</sup>
| Z<sub>19</sub>
| –
| align=center|[[Image:GroupDiagramMiniC19.svg|40px]]
| Simple. Cyclic. Elementary.
|-
! rowspan="2" | 20
! 51
! G<sub>20</sub><sup>2</sup>
| Z<sub>20</sub> = Z<sub>5</sub> × Z<sub>4</sub>
| Z<sub>10</sub>, Z<sub>5</sub>, Z<sub>4</sub>, Z<sub>2</sub>
| align=center|[[Image:GroupDiagramMiniC20.svg|40px]]
| Cyclic. Product. 
|-
! 54
! G<sub>20</sub><sup>5</sup>
| Z<sub>10</sub> × Z<sub>2</sub> = Z<sub>5</sub> × Z<sub>2</sub><sup>2</sup> ||Z<sub>2</sub> (3), K<sub>4</sub>, Z<sub>5</sub>, Z<sub>10</sub> (3)
| align=center|[[File:GroupDiagramMiniC2C10.png|40px]]
| Product. 
|-
! 21
! 56
! G<sub>21</sub><sup>2</sup>
| Z<sub>21</sub> = Z<sub>7</sub> × Z<sub>3</sub>
| Z<sub>7</sub>, Z<sub>3</sub>
| align=center|[[Image:GroupDiagramMiniC21.svg|40px]]
| Cyclic. Product. 
|-
! 22
! 58
! G<sub>22</sub><sup>2</sup>
| Z<sub>22</sub> = Z<sub>11</sub> × Z<sub>2</sub>
| Z<sub>11</sub>, Z<sub>2</sub>
| align=center|[[Image:GroupDiagramMiniC22.svg|40px]]
| Cyclic. Product. 
|-
! 23
! 59
! G<sub>23</sub><sup>1</sup>
| Z<sub>23</sub>
| –
| align=center|[[Image:GroupDiagramMiniC23.svg|40px]]
| Simple. Cyclic. Elementary. 
|-
! rowspan=3|24
! 61
! G<sub>24</sub><sup>2</sup>
| Z<sub>24</sub> = Z<sub>8</sub> × Z<sub>3</sub>
| Z<sub>12</sub>, Z<sub>8</sub>, Z<sub>6</sub>, Z<sub>4</sub>, Z<sub>3</sub>, Z<sub>2</sub>
| align=center|[[Image:GroupDiagramMiniC24.svg|40px]]
| Cyclic. Product. 
|-
! 68
! G<sub>24</sub><sup>9</sup>
| Z<sub>12</sub> × Z<sub>2</sub> = Z<sub>6</sub> × Z<sub>4</sub> = <br />Z<sub>4</sub> × Z<sub>3</sub> × Z<sub>2</sub>
| Z<sub>12</sub>, Z<sub>6</sub>, Z<sub>4</sub>, Z<sub>3</sub>, Z<sub>2</sub>
|
| Product. 
|-
! 74
! G<sub>24</sub><sup>15</sup>
| Z<sub>6</sub> × Z<sub>2</sub><sup>2</sup> = Z<sub>3</sub> × Z<sub>2</sub><sup>3</sup>
| Z<sub>6</sub>, Z<sub>3</sub>, Z<sub>2</sub>
|
| Product.
|-
! rowspan=2|25
! 75
! G<sub>25</sub><sup>1</sup>
| Z<sub>25</sub>
| Z<sub>5</sub>
|
| Cyclic. 
|-
! 76
! G<sub>25</sub><sup>2</sup>
| Z<sub>5</sub><sup>2</sup>
| Z<sub>5</sub> (6)
|
| Product. Elementary. 
|-
! 26
! 78
! G<sub>26</sub><sup>2</sup>
| Z<sub>26</sub> = Z<sub>13</sub> × Z<sub>2</sub>
| Z<sub>13</sub>, Z<sub>2</sub>
|
| Cyclic. Product.
|-
! rowspan=3|27
! 79
! G<sub>27</sub><sup>1</sup>
| Z<sub>27</sub> ||Z<sub>9</sub>, Z<sub>3</sub>
|
| Cyclic.
|-
! 80
! G<sub>27</sub><sup>2</sup>
| Z<sub>9</sub> × Z<sub>3</sub>
| Z<sub>9</sub>, Z<sub>3</sub>
|
| Product. 
|-
! 83
! G<sub>27</sub><sup>5</sup>
| Z<sub>3</sub><sup>3</sup> || Z<sub>3</sub> || || Product. Elementary. 
|-
! rowspan=2|28
! 85
! G<sub>28</sub><sup>2</sup>
| Z<sub>28</sub> = Z<sub>7</sub> × Z<sub>4</sub> || Z<sub>14</sub>, Z<sub>7</sub>, Z<sub>4</sub>, Z<sub>2</sub> || || Cyclic. Product. 
|-
! 87
! G<sub>28</sub><sup>4</sup>
| Z<sub>14</sub> × Z<sub>2</sub> = Z<sub>7</sub> × Z<sub>2</sub><sup>2</sup> || Z<sub>14</sub>, Z<sub>7</sub>, Z<sub>4</sub>, Z<sub>2</sub>
|
| Product. 
|-
! 29
! 88
! G<sub>29</sub><sup>1</sup>
| Z<sub>29</sub>
| –
|
| Simple. Cyclic. Elementary.
|-
! 30
! 92
! G<sub>30</sub><sup>4</sup>
| style="white-space:nowrap;" | Z<sub>30</sub> = Z<sub>15</sub> × Z<sub>2</sub> = Z<sub>10</sub> × Z<sub>3</sub> = <br />Z<sub>6</sub> × Z<sub>5</sub> = Z<sub>5</sub> × Z<sub>3</sub> × Z<sub>2</sub>
| Z<sub>15</sub>, Z<sub>10</sub>, Z<sub>6</sub>, Z<sub>5</sub>, Z<sub>3</sub>, Z<sub>2</sub>
|
| Cyclic. Product.
|-
! 31
! 93
! G<sub>31</sub><sup>1</sup>
| Z<sub>31</sub>
| –
|
| Simple. Cyclic. Elementary.
|}