## Orders of non-abelian simple groups

This list contains finite non-abelian simple groups with unusual properties, such as:

1. Its order has at most four distinct prime factors, or is a powerful number;
2. the p-Sylow group (where p = 2 for alternating groups) is not the largest Sylow subgroup; and/or
3. there is an exceptional isomorphism, outer automorphism group, or Schur multiplier.

Sporadic groups have their own section.

Group(s) Order Factorization Remarks
A5A1(4) ≃ A1(5) 60 22 × 3 × 5 Exceptional Schur multiplier (for A1(4)), and the 5-Sylow group is the largest Sylow subgroup.
A1(7) ≃ A2(2) 168 23 × 3 × 7 Exceptional Schur multiplier (for A2(2)), and the 2-Sylow group is the largest Sylow subgroup.
A6A1(9) ≃ B2(2)′ 360 23 × 32 × 5 Exceptional outer automorphism group (for A6) and Schur multiplier, and the 3-Sylow group is the largest Sylow subgroup.
A1(8) ≃ 2G2(3)′ 504 23 × 32 × 7 The 3-Sylow group is the largest Sylow subgroup.
It is also the number of possible queen moves in starchess.
A1(11) 660 22 × 3 × 5 × 11 It is also the number of feet in a furlong.
And the engine displacement of kei cars is limited to 660 cm3.
A1(13) 1,092 22 × 3 × 7 × 13 It is also the number of pips in a double-12 domino set.
A1(17) 2,448 24 × 32 × 17
A7 2,520 23 × 32 × 5 × 7 Exceptional Schur multiplier, and the 2-Sylow group is not the largest Sylow subgroup.
It is also the maximum possible cycle length of any given algorithm on the Rubik's cube (one such algorithm is "RL2U'F'd").
A1(19) 3,420 22 × 32 × 5 × 19 It is also the number of pips in a double-18 domino set.
A1(16) 4,080 24 × 3 × 5 × 17 The 2-Sylow group is not the largest Sylow subgroup.
A2(3) 5,616 24 × 33 × 13
G2(2)′ ≃ 2A2(32) 6,048 25 × 33 × 7 The 2-Sylow group is the largest Sylow subgroup.
A8A3(2); A2(4) 20,160 26 × 32 × 5 × 7 Smallest order with more than one simple group.
Exceptional Schur multiplier (for A3(2) and A2(4)).
It is also the number of minutes in a fortnight.
B2(3) ≃ 2A3(22) 25,920 26 × 34 × 5 Exceptional Schur multiplier (for 2A3(22)), and the 3-Sylow group is the largest Sylow subgroup.
It is also the number of halakim in a day.
A9 181,440 26 × 34 × 5 × 7 Largest alternating group, for which the 2-Sylow group is not the largest Sylow subgroup.
It is also the number of halakim in a week.
D4(2) 174,182,400 212 × 35 × 52 × 7 Exceptional Schur multiplier.
G2(4) 251,596,800 212 × 33 × 52 × 7 × 13 Exceptional Schur multiplier.
2A5(22) 9,196,830,720 215 × 36 × 5 × 7 × 11 Exceptional Schur multiplier.
F4(2)   3,311,126,
603,366,400
224 × 36 × 52 × 72 × 13 × 17 Exceptional Schur multiplier.
2E6(22)      76,532,
479,683,774,
853,939,200
236 × 39 × 52 × 72 × 11 × 13 × 17 × 19 Exceptional Schur multiplier.

Group Order Factorization Number of subgroups Factorization
Mathieu group M11 7,920 24 × 32 × 5 × 11 8,651 41 × 211
Mathieu group M12 95,040 26 × 33 × 5 × 11 214,871 19 × 43 × 263
Janko group J1 175,560 23 × 3 × 5 × 7 × 11 × 19 158,485 5 × 29 × 1093
Mathieu group M22 443,520 27 × 32 × 5 × 7 × 11 941,627 73 × 12,899
Janko group J2 604,800 27 × 33 × 52 × 7 1,104,344 23 × 31 × 61 × 73
Mathieu group M23 10,200,960 27 × 32 × 5 × 7 × 11 × 23 17,318,406 2 × 3 × 7 × 412,343
Tits group 17,971,200 211 × 33 × 52 × 13 50,285,950 2 × 52 × 11 × 132 × 541
Higman–Sims group 44,352,000 29 × 32 × 53 × 7 × 11 149,985,646 2 × 3,929 × 19,087
Janko group J3 50,232,960 27 × 35 × 5 × 17 × 19 71,564,248 23 × 7 × 239 × 5,347
Mathieu group M24 244,823,040 210 × 33 × 5 × 7 × 11 × 23 1,363,957,253 Prime
McLaughlin group 898,128,000 27 × 36 × 53 × 7 × 11 1,719,739,392 210 × 3 × 7 × 79,973
Held group 4,030,387,200 210 × 33 × 52 × 73 × 17 22,303,017,686 2 × 17 × 211 × 310,889
Rudvalis group 145,926,144,000 214 × 33 × 53 × 7 × 13 × 29 963,226,363,401 32 × 1,549 × 69,093,061
Suzuki group 448,345,497,600 213 × 37 × 52 × 7 × 11 × 13 4,057,939,316,149 7 × 19 × 127 × 27,111,439
O'Nan group 460,815,505,920 29 × 34 × 5 × 73 × 11 × 19 × 31 1,169,254,703,685 3 × 5 × 1,109 × 7,681 × 9,151
Conway group Co3 495,766,656,000 210 × 37 × 53 × 7 × 11 × 23 2,547,911,497,738 2 × 1,273,955,748,869
Group Order Factorization
Conway group Co2 42,305,421,312,000 218 × 36 × 53 × 7 × 11 × 23
Fischer group Fi22 64,561,751,654,400 217 × 39 × 52 × 7 × 11 × 13
Harada–Norton group 273,030,912,000,000 214 × 36 × 56 × 7 × 11 × 19
Lyons group 51,765,179,004,000,000 28 × 37 × 56 × 7 × 11 × 31 × 37 × 67
Thompson sporadic group 90,745,943,887,872,000 215 × 310 × 53 × 72 × 13 × 19 × 31
Fischer group Fi23 4,089,470,473,293,004,800 218 × 313 × 52 × 7 × 11 × 13 × 17 × 23
Conway group Co1 4,157,776,806,543,360,000 221 × 39 × 54 × 72 × 11 × 13 × 23
Janko group J4 86,775,571,046,077,562,880 221 × 33 × 5 × 7 × 113 × 23 × 29 × 31 × 37 × 43
Fischer group Fi24 1,255,205,709,190,661,721,292,800 221 × 316 × 52 × 73 × 11 × 13 × 17 × 23 × 29
Baby monster group   4,154,781,481,226,426,
191,177,580,544,000,000
241 × 313 × 56 × 72 × 11 × 13 × 17 × 19 × 23 × 31 × 47
Monster group 808,017,424,794,512,875,886,459,904,
961,710,757,005,754,368,000,000,000
246 × 320 × 59 × 76 × 112 × 133 × 17 × 19 × 23 × 29 × 31 × 41 × 47 × 59 × 71