← #27542131
Subgroups — #27542131
Order 13568.
498 conjugacy classes of subgroups.
The subgroup lattice has 344 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20287720 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20287701 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20276985 | C2 | 1 | 2 | yes | maximal |
| 4 | #20276968 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20276776 | C2 | 1 | 2 | yes | maximal |
| 6 | #20276765 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #20276764 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #20276760 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #20276623 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #20276585 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | C4×SD32×C53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #16811351 | — | 2 | 1 | no | — |
| 13 | #16811343 | — | 2 | 1 | no | — |
| 14 | C2×Dic424 | C22 | 1 | 2 | yes | — |
| 15 | C2×Dic424 | C22 | 1 | 2 | yes | — |
| 16 | C2×#13133163 | C22 | 1 | 2 | yes | — |
| 17 | C2×#13133161 | C22 | 1 | 2 | yes | — |
| 18 | #16811006 | C22 | 1 | 2 | yes | — |
| 19 | #16811003 | C22 | 1 | 1 | yes | characteristic |
| 20 | #16811003 | C22 | 1 | 1 | yes | characteristic |
| 21 | #16811001 | C22 | 1 | 2 | yes | — |
| 22 | #16810844 | — | 2 | 2 | no | — |
| 23 | #16810833 | — | 2 | 1 | no | — |
| 24 | #16810832 | — | 2 | 2 | no | — |
| 25 | #16810828 | — | 2 | 1 | no | — |
| 26 | C8×Dic106 | C22 | 1 | 1 | yes | characteristic |
| 27 | #16810717 | C22 | 1 | 2 | yes | — |
| 28 | #16810717 | C22 | 1 | 2 | yes | — |
| 29 | C4⋊#11719908 | C22 | 1 | 1 | yes | characteristic |
| 30 | #16810441 | C22 | 1 | 2 | yes | — |
| 31 | #16810426 | C22 | 1 | 2 | yes | — |
| 32 | #16810426 | C22 | 1 | 2 | yes | — |
| 33 | #16810398 | C22 | 1 | 2 | yes | — |
| 34 | C2×SD32×C53 | C22 | 1 | 1 | yes | characteristic |
| 35 | C4×Q16×C53 | C22 | 1 | 1 | yes | characteristic |
| 36 | C4×D8×C53 | C22 | 1 | 1 | yes | characteristic |
| 37 | C53×#367 | C22 | 1 | 1 | yes | characteristic |
| 38 | C53×#358 | C22 | 1 | 1 | yes | characteristic |
| 39 | C53×#357 | C22 | 1 | 1 | yes | characteristic |
| 40 | C4×C16×C53 | C22 | 1 | 1 | yes | characteristic |
| 41 | C2×Dic212 | — | 2 | 1 | no | — |
| 42 | C2×Dic212 | — | 2 | 1 | no | — |
| 43 | C2×Dic212 | C23 | 1 | 2 | yes | — |
| 44 | C2×Dic212 | C23 | 1 | 2 | yes | — |
| 45 | C2×#11719925 | — | 2 | 2 | no | — |
| 46 | C2×#11719923 | — | 2 | 2 | no | — |
| 47 | #13133217 | — | 2 | 1 | no | — |
| 48 | #13133217 | — | 2 | 1 | no | — |
| 49 | #13133215 | — | 2 | 2 | no | — |
| 50 | #13133210 | — | 4 | 2 | no | — |
| 51 | #13133209 | — | 4 | 1 | no | — |
| 52 | #13133207 | — | 4 | 2 | no | — |
| 53 | #13133204 | — | 2 | 2 | no | — |
| 54 | #13133196 | — | 4 | 1 | no | — |
| 55 | C4×Dic106 | — | 4 | 1 | no | — |
| 56 | C4×Dic106 | D4 | 1 | 1 | yes | characteristic |
| 57 | #13133182 | — | 2 | 2 | no | — |
| 58 | #13133182 | — | 2 | 2 | no | — |
| 59 | #13133176 | — | 4 | 1 | no | — |
| 60 | Dic424 | — | 2 | 4 | no | — |
| 61 | Dic424 | — | 4 | 2 | no | — |
| 62 | #13133163 | — | 4 | 2 | no | — |
| 63 | #13133161 | — | 4 | 2 | no | — |
| 64 | C2×#11719908 | C23 | 1 | 2 | yes | — |
| 65 | C4⋊#88322 | D4 | 1 | 1 | yes | characteristic |
| 66 | #13133146 | D4 | 1 | 2 | yes | — |
| 67 | #13133140 | — | 2 | 2 | no | — |
| 68 | #13133140 | — | 2 | 2 | no | — |
| 69 | C8×Dic53 | C23 | 1 | 2 | yes | — |
| 70 | C2×Q16×C53 | C23 | 1 | 1 | yes | characteristic |
| 71 | C2×D8×C53 | C23 | 1 | 1 | yes | characteristic |
| 72 | C4×Q8×C53 | — | 2 | 1 | no | — |
| 73 | C4×D4×C53 | — | 2 | 1 | no | — |
| 74 | SD32×C53 | — | 2 | 2 | no | — |
| 75 | C2×C16×C53 | C23 | 1 | 1 | yes | characteristic |
| 76 | C2×C16×C53 | C23 | 1 | 1 | yes | characteristic |
| 77 | #107×C53 | C23 | 1 | 1 | yes | characteristic |
| 78 | #107×C53 | C23 | 1 | 1 | yes | characteristic |
| 79 | #103×C53 | — | 2 | 1 | no | — |
| 80 | #102×C53 | — | 2 | 1 | no | — |
| 81 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 82 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 83 | C2×D4×C53 | — | 2 | 1 | no | — |
| 84 | C22×C4×C53 | — | 4 | 1 | no | — |
| 85 | C2×Dic106 | — | 2 | 2 | no | — |
| 86 | C2×Dic106 | — | 2 | 2 | no | — |
| 87 | C2×Dic106 | D8 | 1 | 1 | yes | characteristic |
| 88 | Q16×C53 | — | 2 | 1 | no | — |
| 89 | Q16×C53 | — | 2 | 1 | no | — |
| 90 | D8×C53 | — | 2 | 1 | no | — |
| 91 | D8×C53 | — | 2 | 1 | no | — |
| 92 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C53 | C24 | 1 | 1 | yes | characteristic |
| 94 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 95 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 96 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 97 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 98 | C42×C53 | — | 4 | 1 | no | — |
| 99 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C22⋊Dic53 | — | 4 | 2 | no | — |
| 101 | C22⋊Dic53 | — | 4 | 2 | no | — |
| 102 | #11719925 | — | 8 | 2 | no | — |
| 103 | #11719923 | — | 8 | 2 | no | — |
| 104 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 105 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 106 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 107 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 108 | C4⋊Dic53 | D8 | 1 | 1 | yes | characteristic |
| 109 | #11719919 | — | 4 | 2 | no | — |
| 110 | #11719919 | — | 4 | 2 | no | — |
| 111 | #11719919 | — | 4 | 2 | no | — |
| 112 | #11719919 | — | 4 | 2 | no | — |
| 113 | #11719919 | D8 | 1 | 2 | yes | — |
| 114 | C4×Dic53 | — | 4 | 2 | no | — |
| 115 | C4×Dic53 | — | 4 | 2 | no | — |
| 116 | C4×Dic53 | C2×D4 | 1 | 2 | yes | — |
| 117 | C2×#88322 | C2×D4 | 1 | 2 | yes | — |
| 118 | Dic212 | — | 2 | 2 | no | — |
| 119 | Dic212 | — | 2 | 2 | no | — |
| 120 | Dic212 | — | 2 | 2 | no | — |
| 121 | Dic212 | — | 2 | 2 | no | — |
| 122 | Dic212 | — | 8 | 1 | no | — |
| 123 | Dic212 | — | 8 | 1 | no | — |
| 124 | C16×C53 | — | 2 | 1 | no | — |
| 125 | C16×C53 | #41 | 1 | 2 | yes | — |
| 126 | #11719908 | — | 2 | 2 | no | — |
| 127 | C23×C53 | — | 4 | 1 | no | — |
| 128 | Q8×C53 | — | 4 | 1 | no | — |
| 129 | Q8×C53 | — | 4 | 1 | no | — |
| 130 | D4×C53 | — | 4 | 1 | no | — |
| 131 | D4×C53 | — | 4 | 1 | no | — |
| 132 | C2×C4×C53 | — | 4 | 1 | no | — |
| 133 | C2×C4×C53 | — | 4 | 1 | no | — |
| 134 | C2×C4×C53 | — | 4 | 1 | no | — |
| 135 | C2×C4×C53 | — | 8 | 2 | no | — |
| 136 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 137 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 138 | C2×C4×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 139 | C2×Dic53 | — | 4 | 2 | no | — |
| 140 | C2×Dic53 | — | 4 | 2 | no | — |
| 141 | C2×Dic53 | C2×D8 | 1 | 2 | yes | — |
| 142 | C2×Dic53 | C2×D8 | 1 | 2 | yes | — |
| 143 | Dic106 | — | 2 | 2 | no | — |
| 144 | Dic106 | — | 4 | 2 | no | — |
| 145 | Dic106 | — | 4 | 2 | no | — |
| 146 | Dic106 | — | 4 | 2 | no | — |
| 147 | Dic106 | — | 4 | 2 | no | — |
| 148 | C8×C53 | — | 2 | 1 | no | — |
| 149 | C8×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 150 | C8×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 151 | #88322 | — | 2 | 2 | no | — |
| 152 | #33965 | — | 53 | 1 | no | maximal, Sylow |
| 153 | C22×C53 | — | 8 | 1 | no | — |
| 154 | C22×C53 | — | 8 | 1 | no | — |
| 155 | C22×C53 | — | 8 | 1 | no | — |
| 156 | C22×C53 | C22×D8 | 1 | 1 | yes | characteristic |
| 157 | C4×C53 | — | 2 | 1 | no | — |
| 158 | C4×C53 | — | 8 | 1 | no | — |
| 159 | C4×C53 | — | 8 | 1 | no | — |
| 160 | C4×C53 | — | 8 | 1 | no | — |
| 161 | C4×C53 | #510 | 1 | 2 | yes | — |
| 162 | C4×C53 | #550 | 1 | 1 | yes | characteristic |
| 163 | C4×C53 | #576 | 1 | 1 | yes | characteristic |
| 164 | Dic53 | — | 2 | 2 | no | — |
| 165 | Dic53 | — | 2 | 2 | no | — |
| 166 | Dic53 | — | 8 | 2 | no | — |
| 167 | Dic53 | — | 8 | 2 | no | — |
| 168 | #3395 | — | 53 | 1 | no | — |
| 169 | #3393 | — | 53 | 1 | no | — |
| 170 | #2251 | — | 53 | 2 | no | — |
| 171 | #2247 | — | 53 | 1 | no | — |
| 172 | #2211 | — | 53 | 2 | no | — |
| 173 | #2209 | — | 53 | 1 | no | — |
| 174 | #2209 | — | 53 | 2 | no | — |
| 175 | #2208 | — | 53 | 1 | no | — |
| 176 | #2183 | — | 53 | 1 | no | — |
| 177 | #2176 | — | 53 | 2 | no | — |
| 178 | C4×SD32 | D53 | 1 | 1 | yes | characteristic |
| 179 | C2×C53 | — | 16 | 1 | no | — |
| 180 | C2×C53 | #3391 | 1 | 1 | yes | characteristic |
| 181 | C2×C53 | C2×#510 | 1 | 1 | yes | characteristic |
| 182 | C2×C53 | #3417 | 1 | 1 | yes | characteristic |
| 183 | #557 | — | 106 | 1 | no | — |
| 184 | #556 | — | 106 | 1 | no | — |
| 185 | C2×Q32 | — | 53 | 2 | no | — |
| 186 | C2×Q32 | — | 53 | 2 | no | — |
| 187 | C2×Q32 | — | 53 | 2 | no | — |
| 188 | C2×SD32 | — | 53 | 2 | no | — |
| 189 | C2×SD32 | C2×D53 | 1 | 1 | yes | characteristic |
| 190 | #495 | — | 53 | 2 | no | — |
| 191 | #494 | — | 53 | 2 | no | — |
| 192 | #494 | — | 53 | 1 | no | — |
| 193 | #494 | — | 53 | 1 | no | — |
| 194 | #464 | — | 106 | 2 | no | — |
| 195 | #462 | — | 106 | 2 | no | — |
| 196 | #462 | — | 106 | 1 | no | — |
| 197 | #461 | — | 106 | 1 | no | — |
| 198 | C8×Q8 | — | 53 | 1 | no | — |
| 199 | C4×Q16 | — | 53 | 2 | no | — |
| 200 | C4×Q16 | — | 53 | 2 | no | — |
| 201 | C4×Q16 | C2×D53 | 1 | 1 | yes | characteristic |
| 202 | C4×D8 | C2×D53 | 1 | 1 | yes | characteristic |
| 203 | #367 | C2×D53 | 1 | 1 | yes | characteristic |
| 204 | C4⋊C16 | — | 53 | 2 | no | — |
| 205 | C4⋊C16 | — | 53 | 1 | no | — |
| 206 | #358 | — | 53 | 2 | no | — |
| 207 | #358 | — | 53 | 2 | no | — |
| 208 | #358 | C2×D53 | 1 | 1 | yes | characteristic |
| 209 | #357 | C2×D53 | 1 | 1 | yes | characteristic |
| 210 | #346 | — | 53 | 2 | no | — |
| 211 | C4×C16 | C2×D53 | 1 | 1 | yes | characteristic |
| 212 | C53 | #33965 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 213 | C2×Q16 | — | 53 | 2 | no | — |
| 214 | C2×Q16 | — | 53 | 2 | no | — |
| 215 | C2×Q16 | — | 106 | 2 | no | — |
| 216 | C2×Q16 | — | 106 | 1 | no | — |
| 217 | C2×Q16 | — | 106 | 1 | no | — |
| 218 | C2×Q16 | C22×D53 | 1 | 1 | yes | characteristic |
| 219 | C2×SD16 | — | 106 | 2 | no | — |
| 220 | C2×D8 | C22×D53 | 1 | 1 | yes | characteristic |
| 221 | C4⋊Q8 | — | 106 | 2 | no | — |
| 222 | C4⋊Q8 | — | 106 | 1 | no | — |
| 223 | C4⋊Q8 | — | 106 | 1 | no | — |
| 224 | #126 | — | 212 | 2 | no | — |
| 225 | #125 | — | 212 | 1 | no | — |
| 226 | #125 | — | 212 | 2 | no | — |
| 227 | #124 | — | 106 | 2 | no | — |
| 228 | C22⋊Q8 | — | 212 | 1 | no | — |
| 229 | C4×Q8 | — | 2 | 1 | no | — |
| 230 | C4×Q8 | — | 53 | 1 | no | — |
| 231 | C4×Q8 | — | 106 | 2 | no | — |
| 232 | C4×Q8 | — | 106 | 2 | no | — |
| 233 | C4×Q8 | — | 212 | 1 | no | — |
| 234 | C4×D4 | — | 2 | 1 | no | — |
| 235 | #117 | — | 212 | 1 | no | — |
| 236 | Q32 | — | 106 | 4 | no | — |
| 237 | Q32 | — | 212 | 2 | no | — |
| 238 | Q32 | — | 212 | 2 | no | — |
| 239 | SD32 | — | 2 | 2 | no | — |
| 240 | SD32 | — | 212 | 2 | no | — |
| 241 | C2×C16 | — | 53 | 2 | no | — |
| 242 | C2×C16 | C22×D53 | 1 | 1 | yes | characteristic |
| 243 | C2×C16 | C22×D53 | 1 | 1 | yes | characteristic |
| 244 | #107 | C22×D53 | 1 | 1 | yes | characteristic |
| 245 | #107 | C22×D53 | 1 | 1 | yes | characteristic |
| 246 | C4⋊C8 | — | 53 | 2 | no | — |
| 247 | C4⋊C8 | — | 53 | 1 | no | — |
| 248 | #103 | — | 2 | 1 | no | — |
| 249 | #103 | — | 106 | 2 | no | — |
| 250 | #103 | — | 106 | 2 | no | — |
| 251 | #102 | — | 2 | 1 | no | — |
| 252 | C4×C8 | — | 53 | 2 | no | — |
| 253 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 254 | C2×Q8 | — | 2 | 1 | no | — |
| 255 | C2×Q8 | — | 53 | 1 | no | — |
| 256 | C2×Q8 | — | 106 | 2 | no | — |
| 257 | C2×Q8 | — | 106 | 2 | no | — |
| 258 | C2×D4 | — | 2 | 1 | no | — |
| 259 | C22×C4 | — | 4 | 1 | no | — |
| 260 | Q16 | — | 2 | 1 | no | — |
| 261 | Q16 | — | 2 | 1 | no | — |
| 262 | Q16 | — | 106 | 2 | no | — |
| 263 | Q16 | — | 106 | 2 | no | — |
| 264 | Q16 | — | 106 | 2 | no | — |
| 265 | Q16 | — | 106 | 2 | no | — |
| 266 | Q16 | — | 424 | 1 | no | — |
| 267 | Q16 | — | 424 | 1 | no | — |
| 268 | Q16 | — | 424 | 2 | no | — |
| 269 | SD16 | — | 424 | 2 | no | — |
| 270 | D8 | — | 2 | 1 | no | — |
| 271 | D8 | — | 2 | 1 | no | — |
| 272 | C2×C8 | — | 53 | 2 | no | — |
| 273 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 274 | C2×C8 | C23×D53 | 1 | 1 | yes | characteristic |
| 275 | C4⋊C4 | — | 2 | 1 | no | — |
| 276 | C4⋊C4 | — | 2 | 1 | no | — |
| 277 | C4⋊C4 | — | 4 | 1 | no | — |
| 278 | C4⋊C4 | — | 53 | 1 | no | — |
| 279 | C4⋊C4 | — | 53 | 2 | no | — |
| 280 | C4⋊C4 | — | 212 | 2 | no | — |
| 281 | C4⋊C4 | — | 212 | 2 | no | — |
| 282 | C4⋊C4 | — | 212 | 2 | no | — |
| 283 | C4⋊C4 | — | 212 | 2 | no | — |
| 284 | C4⋊C4 | — | 212 | 1 | no | — |
| 285 | C4⋊C4 | — | 212 | 1 | no | — |
| 286 | C4⋊C4 | — | 212 | 1 | no | — |
| 287 | C4⋊C4 | — | 212 | 1 | no | — |
| 288 | C22⋊C4 | — | 4 | 1 | no | — |
| 289 | C22⋊C4 | — | 212 | 2 | no | — |
| 290 | C22⋊C4 | — | 212 | 2 | no | — |
| 291 | C42 | — | 4 | 1 | no | — |
| 292 | C42 | — | 53 | 2 | no | — |
| 293 | C42 | — | 212 | 2 | no | — |
| 294 | C42 | — | 212 | 2 | no | — |
| 295 | C42 | D4×D53 | 1 | 1 | yes | characteristic |
| 296 | C16 | — | 2 | 1 | no | — |
| 297 | C16 | — | 106 | 2 | no | — |
| 298 | C16 | #11719945 | 1 | 2 | yes | — |
| 299 | C23 | — | 4 | 1 | no | — |
| 300 | Q8 | — | 4 | 1 | no | — |
| 301 | Q8 | — | 4 | 1 | no | — |
| 302 | Q8 | — | 106 | 2 | no | — |
| 303 | Q8 | — | 212 | 2 | no | — |
| 304 | Q8 | — | 212 | 2 | no | — |
| 305 | Q8 | — | 212 | 2 | no | — |
| 306 | Q8 | — | 212 | 2 | no | — |
| 307 | D4 | — | 4 | 1 | no | — |
| 308 | D4 | — | 4 | 1 | no | — |
| 309 | C2×C4 | — | 4 | 1 | no | — |
| 310 | C2×C4 | — | 4 | 1 | no | — |
| 311 | C2×C4 | — | 4 | 1 | no | — |
| 312 | C2×C4 | — | 8 | 2 | no | — |
| 313 | C2×C4 | — | 53 | 2 | no | — |
| 314 | C2×C4 | — | 53 | 2 | no | — |
| 315 | C2×C4 | — | 212 | 2 | no | — |
| 316 | C2×C4 | — | 212 | 2 | no | — |
| 317 | C2×C4 | D8×D53 | 1 | 1 | yes | characteristic |
| 318 | C2×C4 | D8×D53 | 1 | 1 | yes | characteristic |
| 319 | C2×C4 | C2×D4×D53 | 1 | 1 | yes | characteristic |
| 320 | C8 | — | 2 | 1 | no | — |
| 321 | C8 | — | 106 | 2 | no | — |
| 322 | C8 | C2×#11719945 | 1 | 1 | yes | characteristic |
| 323 | C8 | #13133263 | 1 | 1 | yes | characteristic |
| 324 | C22 | — | 8 | 1 | no | — |
| 325 | C22 | — | 8 | 1 | no | — |
| 326 | C22 | — | 8 | 1 | no | — |
| 327 | C22 | C2×D8×D53 | 1 | 1 | yes | characteristic |
| 328 | C4 | — | 2 | 1 | no | — |
| 329 | C4 | — | 8 | 1 | no | — |
| 330 | C4 | — | 8 | 1 | no | — |
| 331 | C4 | — | 8 | 1 | no | — |
| 332 | C4 | — | 106 | 2 | no | — |
| 333 | C4 | — | 106 | 2 | no | — |
| 334 | C4 | — | 424 | 2 | no | — |
| 335 | C4 | — | 424 | 2 | no | — |
| 336 | C4 | #16811062 | 1 | 2 | yes | — |
| 337 | C4 | #16811294 | 1 | 1 | yes | characteristic |
| 338 | C4 | #16811425 | 1 | 1 | yes | characteristic |
| 339 | C2 | — | 16 | 1 | no | — |
| 340 | C2 | #20287689 | 1 | 1 | yes | characteristic |
| 341 | C2 | C2×#16811062 | 1 | 1 | yes | characteristic |
| 342 | C2 | #20287865 | 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