← #27484883
Subgroups — #27484883
Order 13568.
408 conjugacy classes of subgroups.
The subgroup lattice has 316 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20276292 | C2 | 1 | 2 | yes | maximal |
| 2 | #20276289 | C2 | 1 | 2 | yes | maximal |
| 3 | #20275503 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20275502 | C2 | 1 | 2 | yes | maximal |
| 5 | #20274885 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #20274840 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #20274839 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #20274836 | C2 | 1 | 2 | yes | maximal |
| 9 | #20274597 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #20274595 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | C53×#1953 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #16810671 | C22 | 1 | 2 | yes | — |
| 13 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#13133152 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#13133152 | C22 | 1 | 2 | yes | — |
| 16 | #16810661 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16810659 | C22 | 1 | 1 | yes | characteristic |
| 18 | #16810646 | C22 | 1 | 2 | yes | — |
| 19 | #16810581 | C22 | 1 | 2 | yes | — |
| 20 | #16810578 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×C8×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 22 | #16810395 | C22 | 1 | 2 | yes | — |
| 23 | #16810394 | C22 | 1 | 2 | yes | — |
| 24 | #16810394 | C22 | 1 | 2 | yes | — |
| 25 | #16810384 | C22 | 1 | 1 | yes | characteristic |
| 26 | #16810384 | C22 | 1 | 2 | yes | — |
| 27 | #16810382 | C22 | 1 | 1 | yes | characteristic |
| 28 | #16810380 | C22 | 1 | 2 | yes | — |
| 29 | #16810379 | C22 | 1 | 2 | yes | — |
| 30 | C53×#430 | C22 | 1 | 2 | yes | — |
| 31 | C2×#108×C53 | C22 | 1 | 1 | yes | characteristic |
| 32 | C2×#108×C53 | C22 | 1 | 1 | yes | characteristic |
| 33 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 34 | C53×#344 | C22 | 1 | 2 | yes | — |
| 35 | C2×#11719917 | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×#11719917 | C23 | 1 | 2 | yes | — |
| 37 | C22×#88322 | C23 | 1 | 1 | yes | characteristic |
| 38 | #13133176 | C23 | 1 | 1 | yes | characteristic |
| 39 | #13133176 | C23 | 1 | 2 | yes | — |
| 40 | C2×C4×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 41 | #13133153 | — | 2 | 1 | no | — |
| 42 | #13133153 | — | 2 | 1 | no | — |
| 43 | #13133152 | — | 2 | 1 | no | — |
| 44 | #13133152 | — | 2 | 1 | no | — |
| 45 | #13133152 | — | 2 | 2 | no | — |
| 46 | #13133152 | — | 2 | 2 | no | — |
| 47 | #13133152 | — | 4 | 2 | no | — |
| 48 | #13133151 | — | 2 | 1 | no | — |
| 49 | #13133150 | — | 2 | 1 | no | — |
| 50 | #13133149 | — | 2 | 1 | no | — |
| 51 | #13133148 | — | 2 | 1 | no | — |
| 52 | #13133146 | — | 4 | 2 | no | — |
| 53 | #13133127 | — | 2 | 1 | no | — |
| 54 | #13133127 | — | 2 | 2 | no | — |
| 55 | C8×Dic53 | — | 2 | 1 | no | — |
| 56 | C8×Dic53 | — | 2 | 2 | no | — |
| 57 | C8×Dic53 | Q8 | 1 | 1 | yes | characteristic |
| 58 | C8×Dic53 | Q8 | 1 | 1 | yes | characteristic |
| 59 | C8×Dic53 | Q8 | 1 | 1 | yes | characteristic |
| 60 | C8×Dic53 | Q8 | 1 | 1 | yes | characteristic |
| 61 | C2×M16×C53 | C23 | 1 | 2 | yes | — |
| 62 | C2×M16×C53 | C23 | 1 | 2 | yes | — |
| 63 | C2×M16×C53 | C23 | 1 | 1 | yes | characteristic |
| 64 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 65 | #117×C53 | C23 | 1 | 1 | yes | characteristic |
| 66 | #108×C53 | — | 2 | 1 | no | — |
| 67 | #108×C53 | — | 2 | 1 | no | — |
| 68 | #108×C53 | — | 2 | 1 | no | — |
| 69 | #108×C53 | — | 2 | 1 | no | — |
| 70 | #108×C53 | — | 4 | 2 | no | — |
| 71 | #97×C53 | — | 2 | 1 | no | — |
| 72 | C4×C8×C53 | — | 2 | 1 | no | — |
| 73 | C22×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 74 | C22×Dic53 | — | 2 | 1 | no | — |
| 75 | M16×C53 | — | 2 | 2 | no | — |
| 76 | M16×C53 | — | 2 | 2 | no | — |
| 77 | M16×C53 | — | 2 | 1 | no | — |
| 78 | M16×C53 | — | 2 | 2 | no | — |
| 79 | M16×C53 | — | 2 | 1 | no | — |
| 80 | M16×C53 | — | 2 | 2 | no | — |
| 81 | C2×C8×C53 | — | 2 | 1 | no | — |
| 82 | C2×C8×C53 | — | 2 | 2 | no | — |
| 83 | C2×C8×C53 | — | 2 | 2 | no | — |
| 84 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 85 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 87 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 88 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 89 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 90 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 91 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 92 | C42×C53 | — | 2 | 1 | no | — |
| 93 | C22⋊Dic53 | — | 2 | 1 | no | — |
| 94 | C22⋊Dic53 | — | 2 | 2 | no | — |
| 95 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 96 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 97 | C4×Dic53 | — | 2 | 1 | no | — |
| 98 | C4×Dic53 | — | 2 | 2 | no | — |
| 99 | C4×Dic53 | — | 2 | 1 | no | — |
| 100 | C4×Dic53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 101 | C4×Dic53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 102 | #11719917 | — | 2 | 1 | no | — |
| 103 | #11719917 | — | 2 | 2 | no | — |
| 104 | #11719917 | — | 2 | 2 | no | — |
| 105 | #11719917 | — | 2 | 1 | no | — |
| 106 | C2×#88322 | — | 2 | 1 | no | — |
| 107 | C2×#88322 | — | 2 | 2 | no | — |
| 108 | C2×#88322 | C2×Q8 | 1 | 1 | yes | characteristic |
| 109 | C2×#88322 | C2×Q8 | 1 | 1 | yes | characteristic |
| 110 | C2×#88322 | #41 | 1 | 1 | yes | characteristic |
| 111 | C2×#88322 | #41 | 1 | 1 | yes | characteristic |
| 112 | C2×#88322 | #41 | 1 | 1 | yes | characteristic |
| 113 | C2×#88322 | #41 | 1 | 1 | yes | characteristic |
| 114 | C23×C53 | 2−1+4 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C53 | — | 4 | 1 | no | — |
| 116 | C2×C4×C53 | C22×Q8 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 122 | C2×Dic53 | — | 2 | 1 | no | — |
| 123 | C2×Dic53 | — | 2 | 1 | no | — |
| 124 | C2×Dic53 | — | 4 | 2 | no | — |
| 125 | C2×Dic53 | — | 4 | 1 | no | — |
| 126 | C2×Dic53 | — | 4 | 2 | no | — |
| 127 | C8×C53 | — | 4 | 1 | no | — |
| 128 | C8×C53 | — | 4 | 2 | no | — |
| 129 | C8×C53 | — | 4 | 2 | no | — |
| 130 | C8×C53 | #125 | 1 | 1 | yes | characteristic |
| 131 | C8×C53 | #125 | 1 | 1 | yes | characteristic |
| 132 | C8×C53 | #125 | 1 | 1 | yes | characteristic |
| 133 | C8×C53 | #125 | 1 | 1 | yes | characteristic |
| 134 | #88322 | — | 4 | 1 | no | — |
| 135 | #88322 | — | 4 | 2 | no | — |
| 136 | #88322 | #125 | 1 | 1 | yes | characteristic |
| 137 | #88322 | #125 | 1 | 1 | yes | characteristic |
| 138 | #88322 | #125 | 1 | 1 | yes | characteristic |
| 139 | #88322 | #125 | 1 | 1 | yes | characteristic |
| 140 | #29230 | — | 53 | 1 | no | maximal, Sylow |
| 141 | C22×C53 | — | 4 | 1 | no | — |
| 142 | C22×C53 | #544 | 1 | 1 | yes | characteristic |
| 143 | C22×C53 | #556 | 1 | 1 | yes | characteristic |
| 144 | C22×C53 | #556 | 1 | 1 | yes | characteristic |
| 145 | C4×C53 | — | 8 | 1 | no | — |
| 146 | C4×C53 | C2×#125 | 1 | 1 | yes | characteristic |
| 147 | C4×C53 | C2×#125 | 1 | 1 | yes | characteristic |
| 148 | C4×C53 | #555 | 1 | 1 | yes | characteristic |
| 149 | C4×C53 | #559 | 1 | 1 | yes | characteristic |
| 150 | Dic53 | — | 2 | 2 | no | — |
| 151 | Dic53 | — | 4 | 1 | no | — |
| 152 | Dic53 | — | 8 | 2 | no | — |
| 153 | Dic53 | — | 8 | 1 | no | — |
| 154 | #2102 | — | 53 | 2 | no | — |
| 155 | #2102 | — | 53 | 2 | no | — |
| 156 | #1953 | — | 53 | 2 | no | — |
| 157 | #1953 | — | 53 | 1 | no | — |
| 158 | #1953 | D53 | 1 | 1 | yes | characteristic |
| 159 | #1848 | — | 53 | 1 | no | — |
| 160 | #1833 | — | 53 | 1 | no | — |
| 161 | #1833 | — | 53 | 1 | no | — |
| 162 | #1833 | — | 53 | 2 | no | — |
| 163 | #1772 | — | 53 | 1 | no | — |
| 164 | #1772 | — | 53 | 1 | no | — |
| 165 | C2×C53 | — | 2 | 1 | no | — |
| 166 | C2×C53 | — | 2 | 1 | no | — |
| 167 | C2×C53 | — | 2 | 1 | no | — |
| 168 | C2×C53 | #2571 | 1 | 1 | yes | characteristic |
| 169 | #430 | — | 53 | 2 | no | — |
| 170 | #430 | C2×D53 | 1 | 2 | yes | — |
| 171 | C2×#108 | — | 53 | 2 | no | — |
| 172 | C2×#108 | — | 53 | 1 | no | — |
| 173 | C2×#108 | — | 53 | 1 | no | — |
| 174 | C2×#108 | C2×D53 | 1 | 1 | yes | characteristic |
| 175 | C2×#108 | C2×D53 | 1 | 1 | yes | characteristic |
| 176 | #427 | — | 53 | 1 | no | — |
| 177 | #427 | — | 53 | 1 | no | — |
| 178 | #423 | — | 53 | 2 | no | — |
| 179 | #405 | — | 53 | 2 | no | — |
| 180 | #405 | — | 53 | 1 | no | — |
| 181 | #405 | C2×D53 | 1 | 1 | yes | characteristic |
| 182 | C2×C4×C8 | — | 53 | 1 | no | — |
| 183 | #344 | — | 53 | 2 | no | — |
| 184 | #344 | — | 53 | 2 | no | — |
| 185 | #344 | — | 53 | 2 | no | — |
| 186 | #344 | C2×D53 | 1 | 2 | yes | — |
| 187 | #339 | — | 53 | 1 | no | — |
| 188 | #339 | — | 53 | 2 | no | — |
| 189 | #339 | — | 53 | 1 | no | — |
| 190 | H(3,ℤ/4) | — | 53 | 2 | no | — |
| 191 | H(3,ℤ/4) | — | 53 | 2 | no | — |
| 192 | C53 | #29230 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 193 | C2×M16 | — | 53 | 2 | no | — |
| 194 | C2×M16 | — | 53 | 1 | no | — |
| 195 | C2×M16 | C22×D53 | 1 | 2 | yes | — |
| 196 | C2×M16 | C22×D53 | 1 | 1 | yes | characteristic |
| 197 | C2×M16 | C22×D53 | 1 | 2 | yes | — |
| 198 | C22×C8 | — | 53 | 1 | no | — |
| 199 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 200 | #117 | — | 53 | 2 | no | — |
| 201 | #117 | — | 53 | 1 | no | — |
| 202 | #117 | C22×D53 | 1 | 1 | yes | characteristic |
| 203 | C2×C42 | — | 53 | 1 | no | — |
| 204 | #108 | — | 2 | 1 | no | — |
| 205 | #108 | — | 2 | 1 | no | — |
| 206 | #108 | — | 2 | 1 | no | — |
| 207 | #108 | — | 2 | 1 | no | — |
| 208 | #108 | — | 4 | 2 | no | — |
| 209 | #108 | — | 106 | 1 | no | — |
| 210 | #108 | — | 106 | 1 | no | — |
| 211 | #108 | — | 106 | 2 | no | — |
| 212 | #108 | — | 106 | 2 | no | — |
| 213 | #108 | — | 106 | 1 | no | — |
| 214 | #108 | — | 106 | 1 | no | — |
| 215 | #108 | — | 212 | 2 | no | — |
| 216 | #107 | — | 106 | 1 | no | — |
| 217 | #107 | — | 106 | 1 | no | — |
| 218 | #106 | — | 106 | 1 | no | — |
| 219 | #106 | — | 106 | 1 | no | — |
| 220 | C4⋊C8 | — | 212 | 2 | no | — |
| 221 | #97 | — | 2 | 1 | no | — |
| 222 | #97 | — | 106 | 2 | no | — |
| 223 | #97 | — | 106 | 1 | no | — |
| 224 | C4×C8 | — | 2 | 1 | no | — |
| 225 | C4×C8 | — | 53 | 1 | no | — |
| 226 | C4×C8 | — | 53 | 1 | no | — |
| 227 | C4×C8 | — | 53 | 1 | no | — |
| 228 | C4×C8 | — | 53 | 1 | no | — |
| 229 | C4×C8 | — | 106 | 2 | no | — |
| 230 | C4×C8 | — | 106 | 1 | no | — |
| 231 | C22×C4 | — | 106 | 1 | no | — |
| 232 | C22×C4 | C23×D53 | 1 | 1 | yes | characteristic |
| 233 | M16 | — | 2 | 1 | no | — |
| 234 | M16 | — | 2 | 2 | no | — |
| 235 | M16 | — | 2 | 2 | no | — |
| 236 | M16 | — | 2 | 2 | no | — |
| 237 | M16 | — | 2 | 1 | no | — |
| 238 | M16 | — | 2 | 2 | no | — |
| 239 | M16 | — | 106 | 1 | no | — |
| 240 | M16 | — | 106 | 2 | no | — |
| 241 | M16 | — | 106 | 2 | no | — |
| 242 | M16 | — | 106 | 1 | no | — |
| 243 | C2×C8 | — | 2 | 2 | no | — |
| 244 | C2×C8 | — | 2 | 1 | no | — |
| 245 | C2×C8 | — | 2 | 2 | no | — |
| 246 | C2×C8 | — | 53 | 1 | no | — |
| 247 | C2×C8 | — | 53 | 1 | no | — |
| 248 | C2×C8 | — | 53 | 1 | no | — |
| 249 | C2×C8 | — | 53 | 1 | no | — |
| 250 | C2×C8 | — | 53 | 1 | no | — |
| 251 | C2×C8 | — | 53 | 1 | no | — |
| 252 | C2×C8 | — | 106 | 1 | no | — |
| 253 | C2×C8 | — | 106 | 2 | no | — |
| 254 | C2×C8 | #11719945 | 1 | 1 | yes | characteristic |
| 255 | C2×C8 | #11719945 | 1 | 1 | yes | characteristic |
| 256 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 258 | C2×C8 | Q8×D53 | 1 | 1 | yes | characteristic |
| 259 | C2×C8 | Q8×D53 | 1 | 1 | yes | characteristic |
| 260 | C4⋊C4 | — | 2 | 1 | no | — |
| 261 | C4⋊C4 | — | 106 | 1 | no | — |
| 262 | C4⋊C4 | — | 106 | 2 | no | — |
| 263 | C22⋊C4 | — | 2 | 1 | no | — |
| 264 | C22⋊C4 | — | 106 | 1 | no | — |
| 265 | C22⋊C4 | — | 106 | 2 | no | — |
| 266 | C42 | — | 2 | 1 | no | — |
| 267 | C42 | — | 53 | 1 | no | — |
| 268 | C42 | — | 53 | 1 | no | — |
| 269 | C42 | — | 106 | 1 | no | — |
| 270 | C42 | — | 106 | 2 | no | — |
| 271 | C42 | — | 106 | 1 | no | — |
| 272 | C23 | #13133265 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | — | 4 | 1 | no | — |
| 274 | C2×C4 | — | 106 | 1 | no | — |
| 275 | C2×C4 | — | 106 | 1 | no | — |
| 276 | C2×C4 | — | 212 | 2 | no | — |
| 277 | C2×C4 | — | 212 | 1 | no | — |
| 278 | C2×C4 | — | 212 | 2 | no | — |
| 279 | C2×C4 | C2×Q8×D53 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | #41×D53 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | #41×D53 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | C2×#11719947 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | C2×#11719945 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #13133262 | 1 | 1 | yes | characteristic |
| 285 | C8 | — | 4 | 2 | no | — |
| 286 | C8 | — | 4 | 1 | no | — |
| 287 | C8 | — | 4 | 2 | no | — |
| 288 | C8 | — | 53 | 1 | no | — |
| 289 | C8 | — | 53 | 1 | no | — |
| 290 | C8 | — | 53 | 1 | no | — |
| 291 | C8 | — | 53 | 1 | no | — |
| 292 | C8 | — | 212 | 1 | no | — |
| 293 | C8 | — | 212 | 2 | no | — |
| 294 | C8 | #13133207 | 1 | 1 | yes | characteristic |
| 295 | C8 | #13133207 | 1 | 1 | yes | characteristic |
| 296 | C8 | #13133207 | 1 | 1 | yes | characteristic |
| 297 | C8 | #13133207 | 1 | 1 | yes | characteristic |
| 298 | C22 | — | 4 | 1 | no | — |
| 299 | C22 | #16811250 | 1 | 1 | yes | characteristic |
| 300 | C22 | #16811344 | 1 | 1 | yes | characteristic |
| 301 | C22 | #16811345 | 1 | 1 | yes | characteristic |
| 302 | C4 | — | 8 | 1 | no | — |
| 303 | C4 | — | 106 | 2 | no | — |
| 304 | C4 | — | 424 | 1 | no | — |
| 305 | C4 | — | 424 | 2 | no | — |
| 306 | C4 | — | 212 | 1 | no | — |
| 307 | C4 | #125×D53 | 1 | 1 | yes | characteristic |
| 308 | C4 | C2×#13133207 | 1 | 1 | yes | characteristic |
| 309 | C4 | #16811337 | 1 | 1 | yes | characteristic |
| 310 | C4 | #16811356 | 1 | 1 | yes | characteristic |
| 311 | C2 | — | 2 | 1 | no | — |
| 312 | C2 | — | 2 | 1 | no | — |
| 313 | C2 | — | 2 | 1 | no | — |
| 314 | C2 | #20279956 | 1 | 1 | yes | characteristic |
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