Editing Monster group (section)

== Maximal subgroups ==

The monster has 46 conjugacy classes of maximal [[subgroups]].{{sfn|Dietrich|Lee|Popiel|2025|}} Non-abelian simple groups of some 60 [[isomorphism]] types are found as subgroups or as quotients of subgroups. The largest [[alternating group]] represented is A<sub>12</sub>.

The 46 classes of maximal subgroups of the monster are given by the following table. Previous unpublished work of Wilson et. al had purported to rule out any almost simple subgroups with non-abelian simple [[Socle of a group|socles]] of the form U<sub>3</sub>(4), L<sub>2</sub>(8), and L<sub>2</sub>(16).{{sfn|Wilson|2010|pp=393–403}}{{sfn|Norton|Wilson|2013|pp=943–962}}{{sfn|Wilson|2016|pp=355–364}} However, the latter was contradicted by Dietrich et al., who found a new maximal subgroup of the form U<sub>3</sub>(4). The same authors had previously found a new maximal subgroup of the form L<sub>2</sub>(13) and confirmed that there are no maximal subgroups with socle L<sub>2</sub>(8) or L<sub>2</sub>(16), thus completing the classification in the literature.{{sfn|Dietrich|Lee|Popiel|2025|}}

{| class="wikitable"
|+ Maximal subgroups of the Monster
|-
! No. !! Structure !! Order !! Comments
|-
| 1||2<sup>&thinsp;·&thinsp;</sup>[[Baby Monster group|B]]||style="text-align:right;"|8,309,562,962,452,852,382,355,161,088,000,000<br />= 2<sup>42</sup>·3<sup>13</sup>·5<sup>6</sup>·7<sup>2</sup>·11·13·17·19·23·31·47||centralizer of an involution of class 2A; contains the normalizer (47:23) × 2 of a Sylow 47-subgroup
|- 
| 2||2{{su|a=l|b=+|p=1+24}}<sup>&thinsp;·&thinsp;</sup>[[Conway group Co1|Co<sub>1</sub>]]||style="text-align:right;"|139,511,839,126,336,328,171,520,000<br />= 2<sup>46</sup>·3<sup>9</sup>·5<sup>4</sup>·7<sup>2</sup>·11·13·23||centralizer of an involution of class 2B
|-
| 3||3<sup>&thinsp;·&thinsp;</sup>[[Fischer group Fi24|Fi<sub>24</sub>]]||style="text-align:right;"|7,531,234,255,143,970,327,756,800<br />= 2<sup>22</sup>·3<sup>17</sup>·5<sup>2</sup>·7<sup>3</sup>·11·13·17·23·29||normalizer of a subgroup of order 3 (class 3A); contains the normalizer ((29:14) × 3).2 of a Sylow 29-subgroup
|-
| 4||2<sup>2&thinsp;·&thinsp;2</sup>E<sub>6</sub>(2):S<sub>3</sub>||style="text-align:right;"|1,836,779,512,410,596,494,540,800<br />= 2<sup>39</sup>·3<sup>10</sup>·5<sup>2</sup>·7<sup>2</sup>·11·13·17·19||normalizer of a Klein 4-group of type 2A<sup>2</sup>
|- 
| 5||{{nowrap|2<sup>10+16 ·&thinsp;</sup>O{{su|a=c|b=10|p=+}}(2)}}||style="text-align:right;"|1,577,011,055,923,770,163,200<br />= 2<sup>46</sup>·3<sup>5</sup>·5<sup>2</sup>·7·17·31||
|-
| 6||2<sup>2+11+22</sup>.(S<sub>3</sub> × [[Mathieu group M24|M<sub>24</sub>]])||style="text-align:right;"|50,472,333,605,150,392,320<br />= 2<sup>46</sup>·3<sup>4</sup>·5·7·11·23||normalizer of a Klein 4-group; contains the normalizer (23:11) × S<sub>4</sub> of a Sylow 23-subgroup
|-
| 7||3{{su|a=l|b=+|p=1+12}}.2[[Suzuki group (mathematics)|Suz]].2||style="text-align:right;"|2,859,230,155,080,499,200<br />= 2<sup>15</sup>·3<sup>20</sup>·5<sup>2</sup>·7·11·13||normalizer of a subgroup of order 3 (class 3B)
|-
| 8||2<sup>5+10+20</sup>.(S<sub>3</sub> × L<sub>5</sub>(2))||style="text-align:right;"|2,061,452,360,684,666,880<br />= 2<sup>46</sup>·3<sup>3</sup>·5·7·31||
|- 
| 9||S<sub>3</sub> × [[Thompson group (mathematics)|Th]]||style="text-align:right;"|544,475,663,327,232,000<br />= 2<sup>16</sup>·3<sup>11</sup>·5<sup>3</sup>·7<sup>2</sup>·13·19·31||normalizer of a subgroup of order 3 (class 3C); contains the normalizer (31:15) × S<sub>3</sub> of a Sylow 31-subgroup
|-
|10||2<sup>3+6+12+18</sup>.(L<sub>3</sub>(2) × 3S<sub>6</sub>)||style="text-align:right;"|199,495,389,743,677,440<br />= 2<sup>46</sup>·3<sup>4</sup>·5·7||
|-
|11||3<sup>8&thinsp;·&thinsp;</sup>O{{su|a=c|b=8|p=&minus;}}(3)<sup>&thinsp;·&thinsp;</sup>2<sub>3</sub>||style="text-align:right;"|133,214,132,225,341,440<br />= 2<sup>11</sup>·3<sup>20</sup>·5·7·13·41||
|-
|12||(D<sub>10</sub> × [[Harada-Norton group|HN]]).2||style="text-align:right;"|5,460,618,240,000,000<br />= 2<sup>16</sup>·3<sup>6</sup>·5<sup>7</sup>·7·11·19||normalizer of a subgroup of order 5 (class 5A)
|-
|13||(3<sup>2</sup>:2 × {{nowrap|O{{su|a=c|b=8|p=+}}(3)}}).S<sub>4</sub>||style="text-align:right;"|2,139,341,679,820,800<br />= 2<sup>16</sup>·3<sup>15</sup>·5<sup>2</sup>·7·13||
|-
|14||3<sup>2+5+10</sup>.([[Mathieu group M11|M<sub>11</sub>]] × 2S<sub>4</sub>)||style="text-align:right;"|49,093,924,366,080<br />= 2<sup>8</sup>·3<sup>20</sup>·5·11||
|-
|15||3<sup>3+2+6+6</sup>:(L<sub>3</sub>(3) × SD<sub>16</sub>)||style="text-align:right;"|11,604,018,486,528<br />= 2<sup>8</sup>·3<sup>20</sup>·13||
|-
|16||5{{su|a=l|b=+|p=1+6}}:2[[Janko group J2|J<sub>2</sub>]]:4||style="text-align:right;"|378,000,000,000<br />= 2<sup>10</sup>·3<sup>3</sup>·5<sup>9</sup>·7||normalizer of a subgroup of order 5 (class 5B)
|- 
|17||(7:3 × [[Held group|He]]):2||style="text-align:right;"|169,276,262,400<br />= 2<sup>11</sup>·3<sup>4</sup>·5<sup>2</sup>·7<sup>4</sup>·17||normalizer of a subgroup of order 7 (class 7A)
|-
|18||(A<sub>5</sub> × A<sub>12</sub>):2||style="text-align:right;"|28,740,096,000<br />= 2<sup>12</sup>·3<sup>6</sup>·5<sup>3</sup>·7·11||
|-
|19||5<sup>3+3</sup>.(2 × L<sub>3</sub>(5))||style="text-align:right;"|11,625,000,000<br />= 2<sup>6</sup>·3·5<sup>9</sup>·31||
|-
|20||(A<sub>6</sub> × A<sub>6</sub> × A<sub>6</sub>).(2 × S<sub>4</sub>)||style="text-align:right;"|2,239,488,000<br />= 2<sup>13</sup>·3<sup>7</sup>·5<sup>3</sup>||
|-
|21||(A<sub>5</sub> × U<sub>3</sub>(8):3<sub>1</sub>):2||style="text-align:right;"|1,985,679,360<br />= 2<sup>12</sup>·3<sup>6</sup>·5·7·19||contains the normalizer ((19:9) × A<sub>5</sub>):2 of a Sylow 19-subgroup
|-
|22||5<sup>2+2+4</sup>:(S<sub>3</sub> × GL<sub>2</sub>(5))||style="text-align:right;"|1,125,000,000<br />= 2<sup>6</sup>·3<sup>2</sup>·5<sup>9</sup>||
|-
|23||(L<sub>3</sub>(2) × S<sub>4</sub>(4):2).2||style="text-align:right;"|658,022,400<br />= 2<sup>13</sup>·3<sup>3</sup>·5<sup>2</sup>·7·17||contains the normalizer ((17:8) × L<sub>3</sub>(2)).2 of a Sylow 17-subgroup
|-
|24||7{{su|a=l|b=+|p=1+4}}:(3 × 2S<sub>7</sub>)||style="text-align:right;"|508,243,680<br />= 2<sup>5</sup>·3<sup>3</sup>·5·7<sup>6</sup>||normalizer of a subgroup of order 7 (class 7B)
|-
|25||(5<sup>2</sup>:4.2<sup>2</sup> × U<sub>3</sub>(5)).S<sub>3</sub>||style="text-align:right;"|302,400,000<br />= 2<sup>9</sup>·3<sup>3</sup>·5<sup>5</sup>·7||
|-
|26||(L<sub>2</sub>(11) × [[Mathieu group M12|M<sub>12</sub>]]):2||style="text-align:right;"|125,452,800<br />= 2<sup>9</sup>·3<sup>4</sup>·5<sup>2</sup>·11<sup>2</sup>||contains the normalizer (11:5 × M<sub>12</sub>):2 of a subgroup of order 11
|-
|27||(A<sub>7</sub> × (A<sub>5</sub> × A<sub>5</sub>):2<sup>2</sup>):2||style="text-align:right;"|72,576,000<br />= 2<sup>10</sup>·3<sup>4</sup>·5<sup>3</sup>·7||
|-
|28||5<sup>4</sup>:(3 × 2L<sub>2</sub>(25)):2<sub>2</sub>||style="text-align:right;"|58,500,000<br />= 2<sup>5</sup>·3<sup>2</sup>·5<sup>6</sup>·13||
|-
|29||7<sup>2+1+2</sup>:GL<sub>2</sub>(7)||style="text-align:right;"|33,882,912<br />= 2<sup>5</sup>·3<sup>2</sup>·7<sup>6</sup>||
|-
|30||[[Mathieu group M11|M<sub>11</sub>]] × A<sub>6</sub>.2<sup>2</sup>||style="text-align:right;"|11,404,800<br />= 2<sup>9</sup>·3<sup>4</sup>·5<sup>2</sup>·11||
|-
|31||(S<sub>5</sub> × S<sub>5</sub> × S<sub>5</sub>):S<sub>3</sub>||style="text-align:right;"|10,368,000<br />= 2<sup>10</sup>·3<sup>4</sup>·5<sup>3</sup>||
|-
|32||(L<sub>2</sub>(11) × L<sub>2</sub>(11)):4||style="text-align:right;"|1,742,400<br />= 2<sup>6</sup>·3<sup>2</sup>·5<sup>2</sup>·11<sup>2</sup>||
|- 
|33||13<sup>2</sup>:2L<sub>2</sub>(13).4||style="text-align:right;"|1,476,384<br />= 2<sup>5</sup>·3·7·13<sup>3</sup>||
|-
|34||(7<sup>2</sup>:(3 × 2A<sub>4</sub>) × L<sub>2</sub>(7)):2||style="text-align:right;"|1,185,408<br />= 2<sup>7</sup>·3<sup>3</sup>·7<sup>3</sup>||
|-
|35||(13:6 × L<sub>3</sub>(3)).2||style="text-align:right;"|876,096<br />= 2<sup>6</sup>·3<sup>4</sup>·13<sup>2</sup>||normalizer of a subgroup of order 13 (class 13A)
|-
|36||13{{su|a=l|b=+|p=1+2}}:(3 × 4S<sub>4</sub>)||style="text-align:right;"|632,736<br />= 2<sup>5</sup>·3<sup>2</sup>·13<sup>3</sup>||normalizer of a subgroup of order 13 (class 13B); normalizer of a Sylow 13-subgroup
|-
|37||U<sub>3</sub>(4):4||style="text-align:right;"|249,600<br />= 2<sup>8</sup>·3·5<sup>2</sup>·13||{{sfn|Dietrich|Lee|Popiel|2025|}}
|-
|38||L<sub>2</sub>(71)||style="text-align:right;"|178,920<br />= 2<sup>3</sup>·3<sup>2</sup>·5·7·71||contains the normalizer 71:35 of a Sylow 71-subgroup{{sfn|Holmes|Wilson|2008|pp=2653–2667}}
|-
|39||11<sup>2</sup>:(5 × 2A<sub>5</sub>)||style="text-align:right;"|72,600<br />= 2<sup>3</sup>·3·5<sup>2</sup>·11<sup>2</sup>||normalizer of a Sylow 11-subgroup.
|-
|40||L<sub>2</sub>(41)||style="text-align:right;"|34,440<br />= 2<sup>3</sup>·3·5·7·41||Norton and Wilson found a maximal subgroup of this form; due to a subtle error pointed out by Zavarnitsine some previous lists and papers stated that no such maximal subgroup existed{{sfn|Norton|Wilson|2013|pp=943–962}}
|-
|41||L<sub>2</sub>(29):2||style="text-align:right;"|24,360<br />= 2<sup>3</sup>·3·5·7·29||{{sfn|Holmes|Wilson|2002|pp=435–447}}
|-
|42||7<sup>2</sup>:SL<sub>2</sub>(7)||style="text-align:right;"|16,464<br />=2<sup>4</sup>·3·7<sup>3</sup>||this was accidentally omitted from some previous lists of 7-local subgroups
|-
|43||L<sub>2</sub>(19):2||style="text-align:right;"|6,840<br />= 2<sup>3</sup>·3<sup>2</sup>·5·19||{{sfn|Holmes|Wilson|2008|pp=2653–2667}}
|-
|44||L<sub>2</sub>(13):2||style="text-align:right;"|2,184<br />= 2<sup>3</sup>·3·7·13||{{sfn|Dietrich|Lee|Popiel|2025|}}
|-
|45||59:29||style="text-align:right;"|1,711<br />= 29·59||previously thought to be L<sub>2</sub>(59);{{sfn|Dietrich|Lee|Popiel|2025|}} normalizer of a Sylow 59-subgroup
|-
|46||41:40||style="text-align:right;"|1,640<br />= 2<sup>3</sup>·5·41||normalizer of a Sylow 41-subgroup
|}

Note that tables of maximal subgroups have often been found to contain subtle errors, and in particular at least two of the subgroups in this table were incorrectly omitted from some previous lists.