← #27539317
Subgroups — #27539317
Order 13568.
520 conjugacy classes of subgroups.
The subgroup lattice has 302 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20285374 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20283584 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20276924 | C2 | 1 | 2 | yes | maximal |
| 4 | #20276880 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20276766 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #20276651 | C2 | 1 | 2 | yes | maximal |
| 7 | #20276587 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C4×Dic424 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C53×#2164 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C22×Dic212 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16811235 | — | 2 | 1 | no | — |
| 12 | #16811100 | — | 2 | 1 | no | — |
| 13 | C2×Dic424 | C22 | 1 | 2 | yes | — |
| 14 | C2×Dic424 | C22 | 1 | 2 | yes | — |
| 15 | #16811025 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16811003 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16810992 | — | 2 | 1 | no | — |
| 18 | #16810881 | — | 2 | 1 | no | — |
| 19 | #16810864 | C22 | 1 | 2 | yes | — |
| 20 | #16810731 | — | 2 | 1 | no | — |
| 21 | #16810731 | — | 2 | 1 | no | — |
| 22 | #16810731 | — | 2 | 1 | no | — |
| 23 | C4×Dic212 | C22 | 1 | 2 | yes | — |
| 24 | C4×Dic212 | C22 | 1 | 2 | yes | — |
| 25 | C4×Dic212 | C22 | 1 | 2 | yes | — |
| 26 | #16810661 | C22 | 1 | 1 | yes | characteristic |
| 27 | #16810633 | — | 2 | 1 | no | — |
| 28 | #16810451 | C22 | 1 | 2 | yes | — |
| 29 | #16810449 | C22 | 1 | 2 | yes | — |
| 30 | #16810428 | C22 | 1 | 2 | yes | — |
| 31 | #16810428 | C22 | 1 | 2 | yes | — |
| 32 | #16810428 | C22 | 1 | 2 | yes | — |
| 33 | #16810428 | C22 | 1 | 2 | yes | — |
| 34 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 35 | C53×C4⋊C16 | C22 | 1 | 2 | yes | — |
| 36 | C53×C22⋊C16 | C22 | 1 | 2 | yes | — |
| 37 | C53×#346 | C22 | 1 | 1 | yes | characteristic |
| 38 | C4×C16×C53 | C22 | 1 | 1 | yes | characteristic |
| 39 | C22×Dic106 | — | 2 | 1 | no | — |
| 40 | C2×Dic212 | — | 2 | 1 | no | — |
| 41 | C2×Dic212 | — | 2 | 4 | no | — |
| 42 | C2×Dic212 | — | 2 | 1 | no | — |
| 43 | C2×Dic212 | C23 | 1 | 2 | yes | — |
| 44 | C2×Dic212 | C23 | 1 | 2 | yes | — |
| 45 | #13133217 | — | 2 | 1 | no | — |
| 46 | #13133210 | — | 4 | 2 | no | — |
| 47 | #13133209 | — | 2 | 1 | no | — |
| 48 | #13133207 | — | 4 | 2 | no | — |
| 49 | #13133196 | — | 2 | 2 | no | — |
| 50 | C4×Dic106 | — | 2 | 2 | no | — |
| 51 | C4×Dic106 | — | 2 | 2 | no | — |
| 52 | C4×Dic106 | — | 2 | 2 | no | — |
| 53 | #13133182 | — | 4 | 2 | no | — |
| 54 | #13133176 | — | 2 | 1 | no | — |
| 55 | #13133173 | — | 4 | 2 | no | — |
| 56 | Dic424 | — | 2 | 4 | no | — |
| 57 | Dic424 | — | 2 | 4 | no | — |
| 58 | #13133151 | C23 | 1 | 2 | yes | — |
| 59 | #13133151 | C23 | 1 | 2 | yes | — |
| 60 | #13133149 | — | 2 | 1 | no | — |
| 61 | #13133149 | — | 2 | 1 | no | — |
| 62 | #13133142 | — | 2 | 2 | no | — |
| 63 | #13133142 | — | 2 | 2 | no | — |
| 64 | #13133142 | — | 2 | 2 | no | — |
| 65 | #13133142 | — | 2 | 2 | no | — |
| 66 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 67 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 68 | #117×C53 | D4 | 1 | 1 | yes | characteristic |
| 69 | C2×C16×C53 | C23 | 1 | 2 | yes | — |
| 70 | C2×C16×C53 | C23 | 1 | 2 | yes | — |
| 71 | #97×C53 | D4 | 1 | 1 | yes | characteristic |
| 72 | #97×C53 | D4 | 1 | 1 | yes | characteristic |
| 73 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 74 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 75 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C22×Dic53 | — | 4 | 1 | no | — |
| 77 | C2×Dic106 | — | 2 | 4 | no | — |
| 78 | C2×Dic106 | — | 2 | 2 | no | — |
| 79 | C2×Dic106 | — | 2 | 2 | no | — |
| 80 | C2×Dic106 | — | 4 | 2 | no | — |
| 81 | C2×Dic106 | — | 4 | 1 | no | — |
| 82 | M16×C53 | — | 4 | 1 | no | — |
| 83 | C2×C8×C53 | — | 2 | 2 | no | — |
| 84 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C2×C8×C53 | C24 | 1 | 1 | yes | characteristic |
| 88 | C4⋊C4×C53 | D8 | 1 | 2 | yes | — |
| 89 | C22⋊C4×C53 | D8 | 1 | 2 | yes | — |
| 90 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 93 | C22⋊Dic53 | — | 4 | 2 | no | — |
| 94 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 95 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 96 | C4⋊Dic53 | — | 4 | 2 | no | — |
| 97 | #11719919 | — | 4 | 2 | no | — |
| 98 | #11719919 | — | 4 | 2 | no | — |
| 99 | #11719919 | — | 4 | 2 | no | — |
| 100 | #11719919 | — | 4 | 2 | no | — |
| 101 | #11719919 | — | 4 | 2 | no | — |
| 102 | C4×Dic53 | — | 4 | 2 | no | — |
| 103 | C4×Dic53 | — | 4 | 2 | no | — |
| 104 | C4×Dic53 | — | 4 | 1 | no | — |
| 105 | C4×Dic53 | — | 4 | 2 | no | — |
| 106 | Dic212 | — | 2 | 4 | no | — |
| 107 | Dic212 | — | 2 | 2 | no | — |
| 108 | Dic212 | — | 2 | 2 | no | — |
| 109 | Dic212 | — | 2 | 2 | no | — |
| 110 | Dic212 | — | 4 | 2 | no | — |
| 111 | Dic212 | #41 | 1 | 4 | yes | — |
| 112 | C16×C53 | — | 2 | 2 | no | — |
| 113 | C16×C53 | — | 2 | 2 | no | — |
| 114 | C23×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C53 | — | 2 | 1 | no | — |
| 116 | C2×C4×C53 | — | 2 | 1 | no | — |
| 117 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C53 | C2×D8 | 1 | 2 | yes | — |
| 119 | C2×C4×C53 | C2×D8 | 1 | 2 | yes | — |
| 120 | C2×C4×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 121 | C2×Dic53 | — | 4 | 4 | no | — |
| 122 | C2×Dic53 | — | 4 | 2 | no | — |
| 123 | C2×Dic53 | — | 4 | 2 | no | — |
| 124 | C2×Dic53 | — | 4 | 2 | no | — |
| 125 | C2×Dic53 | — | 4 | 2 | no | — |
| 126 | Dic106 | — | 2 | 4 | no | — |
| 127 | Dic106 | — | 4 | 4 | no | — |
| 128 | Dic106 | — | 4 | 2 | no | — |
| 129 | Dic106 | — | 4 | 2 | no | — |
| 130 | Dic106 | — | 4 | 2 | no | — |
| 131 | C8×C53 | — | 2 | 1 | no | — |
| 132 | C8×C53 | — | 2 | 1 | no | — |
| 133 | C8×C53 | — | 2 | 1 | no | — |
| 134 | C8×C53 | C2×#41 | 1 | 2 | yes | — |
| 135 | #33687 | — | 53 | 1 | no | maximal, Sylow |
| 136 | C22×C53 | — | 2 | 1 | no | — |
| 137 | C22×C53 | — | 2 | 2 | no | — |
| 138 | C22×C53 | C22×D8 | 1 | 1 | yes | characteristic |
| 139 | C4×C53 | — | 2 | 1 | no | — |
| 140 | C4×C53 | — | 2 | 2 | no | — |
| 141 | C4×C53 | — | 2 | 2 | no | — |
| 142 | C4×C53 | #525 | 1 | 1 | yes | characteristic |
| 143 | C4×C53 | #575 | 1 | 1 | yes | characteristic |
| 144 | Dic53 | — | 4 | 4 | no | — |
| 145 | Dic53 | — | 8 | 2 | no | — |
| 146 | Dic53 | — | 8 | 2 | no | — |
| 147 | Dic53 | — | 8 | 2 | no | — |
| 148 | #3177 | — | 53 | 1 | no | — |
| 149 | #2950 | — | 53 | 1 | no | — |
| 150 | #2237 | — | 53 | 2 | no | — |
| 151 | #2229 | — | 53 | 2 | no | — |
| 152 | #2209 | — | 53 | 2 | no | — |
| 153 | #2188 | — | 53 | 2 | no | — |
| 154 | #2176 | — | 53 | 2 | no | — |
| 155 | C4×Q32 | — | 53 | 2 | no | — |
| 156 | #2164 | D53 | 1 | 1 | yes | characteristic |
| 157 | C2×C53 | — | 4 | 1 | no | — |
| 158 | C2×C53 | #3100 | 1 | 1 | yes | characteristic |
| 159 | C2×C53 | #3417 | 1 | 2 | yes | — |
| 160 | C22×Q16 | — | 53 | 1 | no | — |
| 161 | #541 | — | 106 | 1 | no | — |
| 162 | #519 | — | 106 | 1 | no | — |
| 163 | C2×Q32 | — | 53 | 2 | no | — |
| 164 | C2×Q32 | — | 53 | 2 | no | — |
| 165 | #499 | — | 53 | 1 | no | — |
| 166 | #494 | — | 53 | 1 | no | — |
| 167 | #491 | — | 106 | 1 | no | — |
| 168 | #470 | — | 106 | 1 | no | — |
| 169 | #467 | — | 53 | 2 | no | — |
| 170 | #441 | — | 106 | 1 | no | — |
| 171 | #441 | — | 106 | 1 | no | — |
| 172 | #441 | — | 106 | 1 | no | — |
| 173 | C4×Q16 | — | 53 | 2 | no | — |
| 174 | C4×Q16 | — | 53 | 2 | no | — |
| 175 | C4×Q16 | — | 53 | 2 | no | — |
| 176 | #427 | — | 53 | 1 | no | — |
| 177 | #419 | — | 106 | 1 | no | — |
| 178 | #405 | C2×D53 | 1 | 1 | yes | characteristic |
| 179 | #367 | — | 53 | 2 | no | — |
| 180 | #366 | — | 53 | 2 | no | — |
| 181 | C4⋊C16 | C2×D53 | 1 | 2 | yes | — |
| 182 | #358 | — | 53 | 2 | no | — |
| 183 | #358 | — | 53 | 2 | no | — |
| 184 | #358 | — | 53 | 2 | no | — |
| 185 | #358 | — | 53 | 2 | no | — |
| 186 | C22⋊C16 | C2×D53 | 1 | 2 | yes | — |
| 187 | #346 | C2×D53 | 1 | 1 | yes | characteristic |
| 188 | C4×C16 | C2×D53 | 1 | 1 | yes | characteristic |
| 189 | C53 | #33687 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 190 | C22×Q8 | — | 106 | 1 | no | — |
| 191 | C2×Q16 | — | 53 | 2 | no | — |
| 192 | C2×Q16 | — | 53 | 2 | no | — |
| 193 | C2×Q16 | — | 106 | 4 | no | — |
| 194 | C2×Q16 | — | 106 | 1 | no | — |
| 195 | C2×Q16 | — | 106 | 1 | no | — |
| 196 | C2×M16 | D212 | 1 | 1 | yes | characteristic |
| 197 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 198 | C4⋊Q8 | — | 106 | 1 | no | — |
| 199 | #126 | — | 212 | 2 | no | — |
| 200 | #125 | — | 106 | 1 | no | — |
| 201 | #125 | — | 212 | 2 | no | — |
| 202 | C22⋊Q8 | — | 106 | 2 | no | — |
| 203 | C4×Q8 | — | 106 | 2 | no | — |
| 204 | C4×Q8 | — | 106 | 2 | no | — |
| 205 | C4×Q8 | — | 106 | 2 | no | — |
| 206 | C4×Q8 | — | 212 | 2 | no | — |
| 207 | #117 | — | 106 | 1 | no | — |
| 208 | #117 | — | 212 | 2 | no | — |
| 209 | #117 | D212 | 1 | 1 | yes | characteristic |
| 210 | Q32 | — | 106 | 4 | no | — |
| 211 | Q32 | — | 106 | 4 | no | — |
| 212 | C2×C16 | C22×D53 | 1 | 2 | yes | — |
| 213 | C2×C16 | C22×D53 | 1 | 2 | yes | — |
| 214 | #107 | — | 53 | 2 | no | — |
| 215 | #107 | — | 53 | 2 | no | — |
| 216 | #106 | — | 106 | 1 | no | — |
| 217 | #106 | — | 106 | 1 | no | — |
| 218 | #103 | — | 106 | 2 | no | — |
| 219 | #103 | — | 106 | 2 | no | — |
| 220 | #103 | — | 106 | 2 | no | — |
| 221 | #103 | — | 106 | 2 | no | — |
| 222 | #97 | D212 | 1 | 1 | yes | characteristic |
| 223 | #97 | D212 | 1 | 1 | yes | characteristic |
| 224 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 225 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 226 | C2×Q8 | — | 106 | 4 | no | — |
| 227 | C2×Q8 | — | 106 | 2 | no | — |
| 228 | C2×Q8 | — | 106 | 2 | no | — |
| 229 | C2×Q8 | — | 212 | 2 | no | — |
| 230 | C2×Q8 | — | 212 | 1 | no | — |
| 231 | C22×C4 | — | 212 | 1 | no | — |
| 232 | C22×C4 | C2×D212 | 1 | 1 | yes | characteristic |
| 233 | Q16 | — | 53 | 4 | no | — |
| 234 | Q16 | — | 106 | 4 | no | — |
| 235 | Q16 | — | 106 | 2 | no | — |
| 236 | Q16 | — | 106 | 2 | no | — |
| 237 | Q16 | — | 106 | 2 | no | — |
| 238 | Q16 | — | 212 | 2 | no | — |
| 239 | M16 | — | 4 | 1 | no | — |
| 240 | C2×C8 | — | 2 | 2 | no | — |
| 241 | C2×C8 | C2×D212 | 1 | 1 | yes | characteristic |
| 242 | C2×C8 | C2×D212 | 1 | 1 | yes | characteristic |
| 243 | C2×C8 | C2×D212 | 1 | 1 | yes | characteristic |
| 244 | C2×C8 | C23×D53 | 1 | 1 | yes | characteristic |
| 245 | C4⋊C4 | — | 106 | 2 | no | — |
| 246 | C4⋊C4 | — | 106 | 2 | no | — |
| 247 | C4⋊C4 | — | 212 | 2 | no | — |
| 248 | C4⋊C4 | — | 212 | 2 | no | — |
| 249 | C4⋊C4 | — | 212 | 2 | no | — |
| 250 | C4⋊C4 | — | 212 | 2 | no | — |
| 251 | C4⋊C4 | — | 212 | 2 | no | — |
| 252 | C4⋊C4 | — | 212 | 2 | no | — |
| 253 | C4⋊C4 | D424 | 1 | 2 | yes | — |
| 254 | C22⋊C4 | — | 212 | 1 | no | — |
| 255 | C22⋊C4 | — | 212 | 2 | no | — |
| 256 | C22⋊C4 | D424 | 1 | 2 | yes | — |
| 257 | C42 | — | 212 | 1 | no | — |
| 258 | C42 | — | 212 | 2 | no | — |
| 259 | C42 | — | 212 | 2 | no | — |
| 260 | C42 | — | 212 | 2 | no | — |
| 261 | C42 | C2×D212 | 1 | 1 | yes | characteristic |
| 262 | C42 | C2×D212 | 1 | 1 | yes | characteristic |
| 263 | C16 | — | 2 | 2 | no | — |
| 264 | C16 | — | 2 | 2 | no | — |
| 265 | C23 | C2×D424 | 1 | 1 | yes | characteristic |
| 266 | Q8 | — | 106 | 4 | no | — |
| 267 | Q8 | — | 212 | 4 | no | — |
| 268 | Q8 | — | 212 | 2 | no | — |
| 269 | Q8 | — | 212 | 2 | no | — |
| 270 | Q8 | — | 212 | 2 | no | — |
| 271 | C2×C4 | — | 2 | 1 | no | — |
| 272 | C2×C4 | — | 2 | 1 | no | — |
| 273 | C2×C4 | — | 212 | 2 | no | — |
| 274 | C2×C4 | — | 212 | 2 | no | — |
| 275 | C2×C4 | — | 212 | 2 | no | — |
| 276 | C2×C4 | — | 212 | 4 | no | — |
| 277 | C2×C4 | — | 212 | 2 | no | — |
| 278 | C2×C4 | C2×D424 | 1 | 2 | yes | — |
| 279 | C2×C4 | C2×D424 | 1 | 2 | yes | — |
| 280 | C2×C4 | C2×D424 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | C22×D212 | 1 | 1 | yes | characteristic |
| 282 | C8 | — | 2 | 1 | no | — |
| 283 | C8 | — | 2 | 1 | no | — |
| 284 | C8 | — | 2 | 1 | no | — |
| 285 | C8 | #41×D53 | 1 | 2 | yes | — |
| 286 | C22 | — | 2 | 1 | no | — |
| 287 | C22 | — | 2 | 2 | no | — |
| 288 | C22 | C22×D424 | 1 | 1 | yes | characteristic |
| 289 | C4 | — | 2 | 1 | no | — |
| 290 | C4 | — | 2 | 2 | no | — |
| 291 | C4 | — | 2 | 2 | no | — |
| 292 | C4 | — | 424 | 2 | no | — |
| 293 | C4 | — | 424 | 2 | no | — |
| 294 | C4 | — | 424 | 2 | no | — |
| 295 | C4 | — | 212 | 4 | no | — |
| 296 | C4 | #16811135 | 1 | 1 | yes | characteristic |
| 297 | C4 | #16811423 | 1 | 1 | yes | characteristic |
| 298 | C2 | — | 4 | 1 | no | — |
| 299 | C2 | #20284716 | 1 | 1 | yes | characteristic |
| 300 | C2 | #20287868 | 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