← #27299455
Subgroups — #27299455
Order 13568.
326 conjugacy classes of subgroups.
The subgroup lattice has 240 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16810443 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20276314 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20276312 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#16810398 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #20274595 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #20272803 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C53×#1379 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#13133152 | C4 | 1 | 1 | yes | characteristic |
| 11 | #16810659 | C4 | 1 | 1 | yes | characteristic |
| 12 | #16810578 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C8×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16810443 | C4 | 1 | 4 | yes | complemented |
| 15 | #16810399 | — | 2 | 1 | no | — |
| 16 | #16810398 | — | 2 | 1 | no | — |
| 17 | #16810398 | C4 | 1 | 4 | yes | complemented |
| 18 | C4×#11719908 | — | 2 | 1 | no | — |
| 19 | C16×Dic53 | — | 2 | 1 | no | — |
| 20 | #16810384 | C4 | 1 | 1 | yes | characteristic |
| 21 | #16810382 | C4 | 1 | 1 | yes | characteristic |
| 22 | C2×M32×C53 | C22 | 1 | 1 | yes | characteristic |
| 23 | C22×C16×C53 | C22 | 1 | 1 | yes | characteristic |
| 24 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#11719917 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C22×#88322 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | #13133176 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C2×C4×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #13133157 | — | 2 | 1 | no | — |
| 30 | #13133157 | C2×C4 | 1 | 2 | yes | — |
| 31 | C2×#11719908 | — | 2 | 1 | no | — |
| 32 | C2×#11719908 | C2×C4 | 1 | 2 | yes | — |
| 33 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 35 | C2×#11719908 | Q8 | 1 | 2 | yes | — |
| 36 | #13133152 | C8 | 1 | 4 | yes | — |
| 37 | #13133150 | C8 | 1 | 2 | yes | — |
| 38 | #13133148 | C8 | 1 | 2 | yes | — |
| 39 | #13133127 | — | 2 | 1 | no | — |
| 40 | C8×Dic53 | — | 2 | 1 | no | — |
| 41 | C8×Dic53 | C2×C4 | 1 | 2 | yes | — |
| 42 | C8×Dic53 | C2×C4 | 1 | 2 | yes | — |
| 43 | C2×M16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 45 | #117×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | M32×C53 | — | 2 | 1 | no | — |
| 47 | M32×C53 | C2×C4 | 1 | 2 | yes | — |
| 48 | C2×C16×C53 | — | 2 | 1 | no | — |
| 49 | C2×C16×C53 | — | 2 | 1 | no | — |
| 50 | C2×C16×C53 | — | 2 | 1 | no | — |
| 51 | C2×C16×C53 | C2×C4 | 1 | 2 | yes | — |
| 52 | #97×C53 | — | 2 | 1 | no | — |
| 53 | C4×C8×C53 | — | 2 | 1 | no | — |
| 54 | C22×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 55 | C22×Dic53 | — | 2 | 1 | no | — |
| 56 | M16×C53 | — | 2 | 1 | no | — |
| 57 | M16×C53 | C2×C8 | 1 | 2 | yes | — |
| 58 | C2×C8×C53 | — | 2 | 1 | no | — |
| 59 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 60 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 61 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 63 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 64 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 65 | C4⋊C4×C53 | C2×C8 | 1 | 2 | yes | — |
| 66 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 67 | C42×C53 | — | 2 | 1 | no | — |
| 68 | C22⋊Dic53 | — | 2 | 1 | no | — |
| 69 | C4⋊Dic53 | C2×C8 | 1 | 2 | yes | — |
| 70 | C4×Dic53 | — | 2 | 1 | no | — |
| 71 | C4×Dic53 | — | 2 | 1 | no | — |
| 72 | C4×Dic53 | C42 | 1 | 2 | yes | — |
| 73 | #11719917 | — | 2 | 1 | no | — |
| 74 | #11719917 | C2×C8 | 1 | 2 | yes | — |
| 75 | C2×#88322 | — | 2 | 1 | no | — |
| 76 | C2×#88322 | C42 | 1 | 2 | yes | — |
| 77 | C2×#88322 | C2×C8 | 1 | 1 | yes | characteristic |
| 78 | C2×#88322 | C2×C8 | 1 | 1 | yes | characteristic |
| 79 | C2×#88322 | C2×C8 | 1 | 2 | yes | — |
| 80 | C16×C53 | — | 2 | 2 | no | — |
| 81 | C16×C53 | — | 2 | 2 | no | — |
| 82 | #11719908 | — | 2 | 2 | no | — |
| 83 | #11719908 | C4⋊C4 | 1 | 2 | yes | — |
| 84 | #11719908 | C4⋊C4 | 1 | 2 | yes | — |
| 85 | C23×C53 | #131 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C53 | — | 2 | 2 | no | — |
| 87 | C2×C4×C53 | C2×C42 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C53 | C4×Q8 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×C53 | #131 | 1 | 1 | yes | characteristic |
| 93 | C2×Dic53 | — | 2 | 2 | no | — |
| 94 | C2×Dic53 | — | 2 | 2 | no | — |
| 95 | C2×Dic53 | — | 4 | 2 | no | — |
| 96 | C8×C53 | — | 2 | 2 | no | — |
| 97 | C8×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 98 | C8×C53 | #117 | 1 | 1 | yes | characteristic |
| 99 | C8×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 100 | C8×C53 | C4×Q8 | 1 | 1 | yes | characteristic |
| 101 | #88322 | — | 2 | 2 | no | — |
| 102 | #88322 | C4×C8 | 1 | 2 | yes | — |
| 103 | #88322 | C4×C8 | 1 | 2 | yes | — |
| 104 | #12665 | — | 53 | 1 | no | maximal, Sylow |
| 105 | C22×C53 | — | 2 | 2 | no | — |
| 106 | C22×C53 | #405 | 1 | 1 | yes | characteristic |
| 107 | C22×C53 | C8×D4 | 1 | 1 | yes | characteristic |
| 108 | C22×C53 | C8×Q8 | 1 | 1 | yes | characteristic |
| 109 | C4×C53 | — | 4 | 2 | no | — |
| 110 | C4×C53 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 111 | C4×C53 | C2×C4×C8 | 1 | 1 | yes | characteristic |
| 112 | C4×C53 | C8×D4 | 1 | 1 | yes | characteristic |
| 113 | C4×C53 | C8×Q8 | 1 | 1 | yes | characteristic |
| 114 | Dic53 | — | 4 | 2 | no | — |
| 115 | Dic53 | — | 4 | 2 | no | — |
| 116 | C2×#364 | — | 53 | 1 | no | — |
| 117 | #2108 | — | 53 | 1 | no | — |
| 118 | #2108 | — | 53 | 1 | no | — |
| 119 | C2×#346 | — | 53 | 1 | no | — |
| 120 | #1772 | — | 53 | 1 | no | — |
| 121 | #1379 | — | 53 | 1 | no | — |
| 122 | #1379 | D53 | 1 | 1 | yes | characteristic |
| 123 | C2×C53 | — | 2 | 1 | no | — |
| 124 | C2×C53 | — | 2 | 1 | no | — |
| 125 | C2×C53 | C8×C4⋊C4 | 1 | 1 | yes | characteristic |
| 126 | C2×C53 | #2171 | 1 | 2 | yes | — |
| 127 | C2×M32 | — | 53 | 1 | no | — |
| 128 | C2×M32 | C2×D53 | 1 | 1 | yes | characteristic |
| 129 | C22×C16 | — | 53 | 1 | no | — |
| 130 | C22×C16 | C2×D53 | 1 | 1 | yes | characteristic |
| 131 | C2×#108 | — | 53 | 1 | no | — |
| 132 | #427 | — | 53 | 1 | no | — |
| 133 | #405 | — | 53 | 1 | no | — |
| 134 | #405 | C2×D53 | 1 | 1 | yes | characteristic |
| 135 | C2×C4×C8 | — | 53 | 1 | no | — |
| 136 | #364 | — | 53 | 4 | no | — |
| 137 | #346 | — | 53 | 4 | no | — |
| 138 | #346 | — | 106 | 1 | no | — |
| 139 | #346 | — | 106 | 1 | no | — |
| 140 | C4×C16 | — | 106 | 1 | no | — |
| 141 | C4×C16 | — | 106 | 1 | no | — |
| 142 | #339 | — | 53 | 1 | no | — |
| 143 | #339 | — | 53 | 1 | no | — |
| 144 | C53 | #12665 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 145 | C2×M16 | — | 53 | 1 | no | — |
| 146 | C2×M16 | C4×D53 | 1 | 1 | yes | characteristic |
| 147 | C22×C8 | — | 53 | 1 | no | — |
| 148 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 149 | #117 | — | 53 | 1 | no | — |
| 150 | #117 | C4×D53 | 1 | 1 | yes | characteristic |
| 151 | C2×C42 | — | 53 | 1 | no | — |
| 152 | M32 | — | 2 | 1 | no | — |
| 153 | M32 | — | 53 | 2 | no | — |
| 154 | M32 | — | 106 | 1 | no | — |
| 155 | M32 | C4×D53 | 1 | 2 | yes | — |
| 156 | C2×C16 | — | 2 | 1 | no | — |
| 157 | C2×C16 | — | 2 | 1 | no | — |
| 158 | C2×C16 | — | 2 | 1 | no | — |
| 159 | C2×C16 | — | 53 | 2 | no | — |
| 160 | C2×C16 | — | 53 | 1 | no | — |
| 161 | C2×C16 | — | 53 | 1 | no | — |
| 162 | C2×C16 | — | 53 | 2 | no | — |
| 163 | C2×C16 | — | 106 | 1 | no | — |
| 164 | C2×C16 | C4×D53 | 1 | 2 | yes | — |
| 165 | #108 | — | 53 | 4 | no | — |
| 166 | #107 | — | 53 | 2 | no | — |
| 167 | #106 | — | 53 | 2 | no | — |
| 168 | #97 | — | 2 | 1 | no | — |
| 169 | #97 | — | 106 | 1 | no | — |
| 170 | C4×C8 | — | 2 | 1 | no | — |
| 171 | C4×C8 | — | 53 | 2 | no | — |
| 172 | C4×C8 | — | 53 | 2 | no | — |
| 173 | C4×C8 | — | 106 | 1 | no | — |
| 174 | C22×C4 | — | 106 | 1 | no | — |
| 175 | C22×C4 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 176 | M16 | — | 2 | 1 | no | — |
| 177 | M16 | — | 53 | 2 | no | — |
| 178 | M16 | — | 106 | 1 | no | — |
| 179 | M16 | C8×D53 | 1 | 2 | yes | — |
| 180 | C2×C8 | — | 2 | 1 | no | — |
| 181 | C2×C8 | — | 53 | 2 | no | — |
| 182 | C2×C8 | — | 53 | 1 | no | — |
| 183 | C2×C8 | — | 53 | 1 | no | — |
| 184 | C2×C8 | — | 53 | 2 | no | — |
| 185 | C2×C8 | — | 106 | 1 | no | — |
| 186 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 187 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 188 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 189 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 190 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 191 | C2×C8 | Q8×D53 | 1 | 1 | yes | characteristic |
| 192 | C4⋊C4 | — | 53 | 2 | no | — |
| 193 | C4⋊C4 | C8×D53 | 1 | 2 | yes | — |
| 194 | C22⋊C4 | — | 2 | 1 | no | — |
| 195 | C22⋊C4 | — | 106 | 1 | no | — |
| 196 | C42 | — | 2 | 1 | no | — |
| 197 | C42 | — | 53 | 2 | no | — |
| 198 | C42 | — | 106 | 1 | no | — |
| 199 | C42 | — | 106 | 1 | no | — |
| 200 | C16 | — | 2 | 2 | no | — |
| 201 | C16 | — | 2 | 2 | no | — |
| 202 | C16 | — | 53 | 2 | no | — |
| 203 | C16 | — | 53 | 2 | no | — |
| 204 | C16 | — | 106 | 2 | no | — |
| 205 | C23 | #13133225 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | — | 2 | 2 | no | — |
| 207 | C2×C4 | — | 106 | 2 | no | — |
| 208 | C2×C4 | — | 106 | 2 | no | — |
| 209 | C2×C4 | — | 212 | 2 | no | — |
| 210 | C2×C4 | C42×D53 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | #13133177 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | #13133182 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 215 | C2×C4 | #13133225 | 1 | 1 | yes | characteristic |
| 216 | C8 | — | 2 | 2 | no | — |
| 217 | C8 | — | 53 | 2 | no | — |
| 218 | C8 | — | 53 | 2 | no | — |
| 219 | C8 | — | 106 | 2 | no | — |
| 220 | C8 | C4⋊C4×D53 | 1 | 1 | yes | characteristic |
| 221 | C8 | #13133173 | 1 | 1 | yes | characteristic |
| 222 | C8 | #13133177 | 1 | 1 | yes | characteristic |
| 223 | C8 | #13133182 | 1 | 1 | yes | characteristic |
| 224 | C22 | — | 2 | 2 | no | — |
| 225 | C22 | #16810580 | 1 | 1 | yes | characteristic |
| 226 | C22 | #16810686 | 1 | 1 | yes | characteristic |
| 227 | C22 | #16810747 | 1 | 1 | yes | characteristic |
| 228 | C4 | — | 4 | 2 | no | — |
| 229 | C4 | — | 212 | 2 | no | — |
| 230 | C4 | — | 212 | 2 | no | — |
| 231 | C4 | #16810474 | 1 | 1 | yes | characteristic |
| 232 | C4 | C4×C8×D53 | 1 | 1 | yes | characteristic |
| 233 | C4 | #16810686 | 1 | 1 | yes | characteristic |
| 234 | C4 | #16810747 | 1 | 1 | yes | characteristic |
| 235 | C2 | — | 2 | 1 | no | — |
| 236 | C2 | — | 2 | 1 | no | — |
| 237 | C2 | #20274576 | 1 | 1 | yes | characteristic |
| 238 | C2 | #20276554 | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta