← #27539307
Subgroups — #27539307
Order 13568.
544 conjugacy classes of subgroups.
The subgroup lattice has 314 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20285365 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20276906 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20276879 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20276755 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20276632 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #20276577 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C4×#13133160 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #20276522 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C53×#2950 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #16811228 | — | 2 | 1 | no | — |
| 11 | C2×#13133160 | C22 | 1 | 2 | yes | — |
| 12 | C2×#13133160 | C22 | 1 | 2 | yes | — |
| 13 | #16811025 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16811000 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16810988 | — | 2 | 1 | no | — |
| 16 | #16810876 | — | 2 | 1 | no | — |
| 17 | #16810859 | C22 | 1 | 2 | yes | — |
| 18 | #16810736 | — | 2 | 1 | no | — |
| 19 | C4×D424 | C22 | 1 | 2 | yes | — |
| 20 | #16810450 | C22 | 1 | 2 | yes | — |
| 21 | #16810448 | C22 | 1 | 2 | yes | — |
| 22 | C4⋊#11719908 | C22 | 1 | 2 | yes | — |
| 23 | #16810427 | C22 | 1 | 2 | yes | — |
| 24 | #16810427 | C22 | 1 | 2 | yes | — |
| 25 | #16810423 | C22 | 1 | 2 | yes | — |
| 26 | #16810423 | C22 | 1 | 2 | yes | — |
| 27 | C22⋊#11719908 | C22 | 1 | 2 | yes | — |
| 28 | #16810399 | C22 | 1 | 1 | yes | characteristic |
| 29 | C4×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 30 | C22×Q16×C53 | C22 | 1 | 1 | yes | characteristic |
| 31 | C53×#519 | — | 2 | 1 | no | — |
| 32 | C53×#441 | — | 2 | 1 | no | — |
| 33 | C53×#441 | — | 2 | 1 | no | — |
| 34 | C4×Q16×C53 | C22 | 1 | 2 | yes | — |
| 35 | C4×Q16×C53 | C22 | 1 | 2 | yes | — |
| 36 | C53×#427 | C22 | 1 | 1 | yes | characteristic |
| 37 | C53×#419 | — | 2 | 1 | no | — |
| 38 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×D424 | — | 2 | 1 | no | — |
| 40 | C2×D424 | C23 | 1 | 2 | yes | — |
| 41 | #13133214 | — | 2 | 1 | no | — |
| 42 | #13133209 | — | 2 | 1 | no | — |
| 43 | #13133202 | — | 4 | 2 | no | — |
| 44 | #13133191 | — | 2 | 2 | no | — |
| 45 | #13133190 | — | 4 | 2 | no | — |
| 46 | C4×D212 | — | 2 | 2 | no | — |
| 47 | #13133160 | — | 2 | 4 | no | — |
| 48 | #13133160 | — | 2 | 4 | no | — |
| 49 | C2×#11719908 | C23 | 1 | 2 | yes | — |
| 50 | C2×#11719908 | C23 | 1 | 2 | yes | — |
| 51 | #13133151 | C23 | 1 | 2 | yes | — |
| 52 | #13133149 | — | 2 | 1 | no | — |
| 53 | #13133139 | — | 2 | 2 | no | — |
| 54 | #13133139 | — | 2 | 2 | no | — |
| 55 | C22×Q8×C53 | — | 2 | 1 | no | — |
| 56 | C2×Q16×C53 | — | 2 | 4 | no | — |
| 57 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 58 | C2×Q16×C53 | C23 | 1 | 2 | yes | — |
| 59 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 60 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 61 | C4×Q8×C53 | — | 2 | 2 | no | — |
| 62 | C4×Q8×C53 | — | 2 | 2 | no | — |
| 63 | C4×Q8×C53 | — | 4 | 2 | no | — |
| 64 | #117×C53 | — | 2 | 1 | no | — |
| 65 | #117×C53 | — | 4 | 2 | no | — |
| 66 | #117×C53 | D4 | 1 | 1 | yes | characteristic |
| 67 | #107×C53 | C23 | 1 | 2 | yes | — |
| 68 | #106×C53 | — | 2 | 1 | no | — |
| 69 | #103×C53 | — | 2 | 2 | no | — |
| 70 | #103×C53 | — | 2 | 2 | no | — |
| 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 | C2×Q8×C53 | — | 2 | 4 | no | — |
| 76 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 77 | C2×Q8×C53 | — | 4 | 2 | no | — |
| 78 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 79 | C22×C4×C53 | — | 4 | 1 | no | — |
| 80 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 82 | C2×D212 | — | 2 | 2 | no | — |
| 83 | C2×D212 | — | 4 | 2 | no | — |
| 84 | C2×C4×D53 | — | 4 | 2 | no | — |
| 85 | Q16×C53 | — | 2 | 4 | no | — |
| 86 | Q16×C53 | — | 2 | 2 | no | — |
| 87 | Q16×C53 | #41 | 1 | 4 | yes | — |
| 88 | M16×C53 | — | 4 | 1 | no | — |
| 89 | C2×C8×C53 | — | 2 | 2 | no | — |
| 90 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C53 | C24 | 1 | 1 | yes | characteristic |
| 94 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 95 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 96 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 97 | C4⋊C4×C53 | D8 | 1 | 2 | yes | — |
| 98 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 99 | C22⋊C4×C53 | D8 | 1 | 2 | yes | — |
| 100 | C42×C53 | — | 4 | 1 | no | — |
| 101 | C42×C53 | — | 4 | 2 | no | — |
| 102 | C42×C53 | — | 4 | 2 | no | — |
| 103 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 105 | #11719921 | — | 4 | 2 | no | — |
| 106 | #11719921 | — | 4 | 2 | no | — |
| 107 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 108 | #11719919 | — | 4 | 2 | no | — |
| 109 | D424 | — | 2 | 2 | no | — |
| 110 | D424 | — | 2 | 2 | no | — |
| 111 | D424 | — | 4 | 2 | no | — |
| 112 | #11719908 | — | 2 | 2 | no | — |
| 113 | #11719908 | — | 2 | 2 | no | — |
| 114 | C23×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 115 | C22×D53 | — | 4 | 2 | no | — |
| 116 | Q8×C53 | — | 2 | 4 | no | — |
| 117 | Q8×C53 | — | 4 | 4 | no | — |
| 118 | Q8×C53 | — | 4 | 2 | no | — |
| 119 | C2×C4×C53 | — | 2 | 1 | no | — |
| 120 | C2×C4×C53 | — | 2 | 1 | no | — |
| 121 | C2×C4×C53 | — | 4 | 4 | no | — |
| 122 | C2×C4×C53 | — | 4 | 2 | no | — |
| 123 | C2×C4×C53 | — | 4 | 2 | no | — |
| 124 | C2×C4×C53 | C2×D8 | 1 | 2 | yes | — |
| 125 | C2×C4×C53 | C2×D8 | 1 | 2 | yes | — |
| 126 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 128 | C22⋊D53 | — | 16 | 2 | no | — |
| 129 | C2×Dic53 | — | 4 | 2 | no | — |
| 130 | D212 | — | 4 | 2 | no | — |
| 131 | D212 | — | 4 | 2 | no | — |
| 132 | D212 | — | 16 | 2 | no | — |
| 133 | C4×D53 | — | 16 | 2 | no | — |
| 134 | C8×C53 | — | 2 | 1 | no | — |
| 135 | C8×C53 | — | 2 | 1 | no | — |
| 136 | C8×C53 | — | 2 | 1 | no | — |
| 137 | C8×C53 | C2×#41 | 1 | 2 | yes | — |
| 138 | #33686 | — | 53 | 1 | no | maximal, Sylow |
| 139 | C22×C53 | — | 2 | 1 | no | — |
| 140 | C22×C53 | — | 2 | 2 | no | — |
| 141 | C22×C53 | C22×D8 | 1 | 1 | yes | characteristic |
| 142 | C2×D53 | — | 8 | 2 | no | — |
| 143 | C2×D53 | — | 8 | 2 | no | — |
| 144 | C2×D53 | — | 8 | 2 | no | — |
| 145 | C4×C53 | — | 2 | 1 | no | — |
| 146 | C4×C53 | — | 2 | 2 | no | — |
| 147 | C4×C53 | — | 2 | 2 | no | — |
| 148 | C4×C53 | — | 4 | 4 | no | — |
| 149 | C4×C53 | — | 8 | 2 | no | — |
| 150 | C4×C53 | #525 | 1 | 1 | yes | characteristic |
| 151 | C4×C53 | #575 | 1 | 1 | yes | characteristic |
| 152 | Dic53 | — | 8 | 2 | no | — |
| 153 | #3176 | — | 53 | 1 | no | — |
| 154 | #2950 | D53 | 1 | 1 | yes | characteristic, complemented |
| 155 | #2234 | — | 53 | 2 | no | — |
| 156 | #2229 | — | 53 | 2 | no | — |
| 157 | #2207 | — | 53 | 2 | no | — |
| 158 | #2185 | — | 53 | 2 | no | — |
| 159 | #2175 | — | 53 | 2 | no | — |
| 160 | C4×SD32 | — | 53 | 2 | no | — |
| 161 | #2164 | — | 53 | 1 | no | — |
| 162 | C2×C53 | — | 4 | 1 | no | — |
| 163 | C2×C53 | #3100 | 1 | 1 | yes | characteristic |
| 164 | C2×C53 | #3416 | 1 | 2 | yes | — |
| 165 | D53 | — | 16 | 2 | no | — |
| 166 | C22×Q16 | C2×D53 | 1 | 1 | yes | characteristic |
| 167 | #540 | — | 106 | 1 | no | — |
| 168 | #519 | — | 2 | 1 | no | — |
| 169 | C2×SD32 | — | 53 | 2 | no | — |
| 170 | C2×SD32 | — | 53 | 2 | no | — |
| 171 | #499 | — | 53 | 1 | no | — |
| 172 | #493 | — | 53 | 1 | no | — |
| 173 | #490 | — | 106 | 1 | no | — |
| 174 | #469 | — | 106 | 1 | no | — |
| 175 | #466 | — | 53 | 2 | no | — |
| 176 | #442 | — | 106 | 1 | no | — |
| 177 | #441 | — | 2 | 1 | no | — |
| 178 | #441 | — | 2 | 1 | no | — |
| 179 | C4×Q16 | C2×D53 | 1 | 2 | yes | — |
| 180 | C4×Q16 | C2×D53 | 1 | 2 | yes | — |
| 181 | C4×D8 | — | 53 | 2 | no | — |
| 182 | #427 | C2×D53 | 1 | 1 | yes | characteristic |
| 183 | #419 | — | 2 | 1 | no | — |
| 184 | #405 | C2×D53 | 1 | 1 | yes | characteristic |
| 185 | #367 | — | 53 | 2 | no | — |
| 186 | #366 | — | 53 | 2 | no | — |
| 187 | C4⋊C16 | — | 53 | 2 | no | — |
| 188 | #358 | — | 53 | 2 | no | — |
| 189 | #358 | — | 53 | 2 | no | — |
| 190 | #357 | — | 53 | 2 | no | — |
| 191 | #357 | — | 53 | 2 | no | — |
| 192 | C22⋊C16 | — | 53 | 2 | no | — |
| 193 | #346 | — | 53 | 1 | no | — |
| 194 | C4×C16 | — | 53 | 1 | no | — |
| 195 | C53 | #33686 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 196 | C22×Q8 | — | 2 | 1 | no | — |
| 197 | C2×Q16 | — | 2 | 4 | no | — |
| 198 | C2×Q16 | — | 2 | 1 | no | — |
| 199 | C2×Q16 | C22×D53 | 1 | 2 | yes | — |
| 200 | C2×D8 | — | 53 | 2 | no | — |
| 201 | C2×D8 | — | 106 | 1 | no | — |
| 202 | C2×M16 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 203 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 204 | #127 | — | 106 | 1 | no | — |
| 205 | #125 | — | 106 | 1 | no | — |
| 206 | #123 | — | 212 | 2 | no | — |
| 207 | #121 | — | 106 | 2 | no | — |
| 208 | #121 | — | 212 | 2 | no | — |
| 209 | C4×Q8 | — | 2 | 2 | no | — |
| 210 | C4×Q8 | — | 2 | 2 | no | — |
| 211 | C4×Q8 | — | 4 | 2 | no | — |
| 212 | C4×D4 | — | 106 | 2 | no | — |
| 213 | #117 | — | 2 | 1 | no | — |
| 214 | #117 | — | 4 | 2 | no | — |
| 215 | #117 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 216 | SD32 | — | 106 | 4 | no | — |
| 217 | SD32 | — | 106 | 4 | no | — |
| 218 | C2×C16 | — | 53 | 2 | no | — |
| 219 | C2×C16 | — | 53 | 2 | no | — |
| 220 | #107 | — | 53 | 2 | no | — |
| 221 | #107 | C22×D53 | 1 | 2 | yes | — |
| 222 | #106 | — | 2 | 1 | no | — |
| 223 | #106 | — | 106 | 1 | no | — |
| 224 | #103 | — | 2 | 2 | no | — |
| 225 | #103 | — | 2 | 2 | no | — |
| 226 | #102 | — | 106 | 2 | no | — |
| 227 | #102 | — | 106 | 2 | no | — |
| 228 | #97 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 229 | #97 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 230 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 231 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 232 | C2×Q8 | — | 2 | 4 | no | — |
| 233 | C2×Q8 | — | 2 | 2 | no | — |
| 234 | C2×Q8 | — | 4 | 2 | no | — |
| 235 | C2×Q8 | — | 4 | 1 | no | — |
| 236 | C2×D4 | — | 106 | 2 | no | — |
| 237 | C2×D4 | — | 212 | 2 | no | — |
| 238 | C2×D4 | — | 212 | 2 | no | — |
| 239 | C22×C4 | — | 4 | 1 | no | — |
| 240 | C22×C4 | — | 212 | 2 | no | — |
| 241 | C22×C4 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 242 | Q16 | — | 2 | 4 | no | — |
| 243 | Q16 | — | 2 | 2 | no | — |
| 244 | Q16 | #11719945 | 1 | 4 | yes | — |
| 245 | D8 | — | 106 | 2 | no | — |
| 246 | D8 | — | 106 | 2 | no | — |
| 247 | D8 | — | 212 | 2 | no | — |
| 248 | M16 | — | 4 | 1 | no | — |
| 249 | C2×C8 | — | 2 | 2 | no | — |
| 250 | C2×C8 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 251 | C2×C8 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 252 | C2×C8 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 253 | C2×C8 | C23×D53 | 1 | 1 | yes | characteristic |
| 254 | C4⋊C4 | — | 2 | 2 | no | — |
| 255 | C4⋊C4 | — | 4 | 2 | no | — |
| 256 | C4⋊C4 | — | 4 | 2 | no | — |
| 257 | C4⋊C4 | — | 106 | 2 | no | — |
| 258 | C4⋊C4 | — | 212 | 2 | no | — |
| 259 | C4⋊C4 | #11719922 | 1 | 2 | yes | — |
| 260 | C22⋊C4 | — | 4 | 1 | no | — |
| 261 | C22⋊C4 | — | 212 | 2 | no | — |
| 262 | C22⋊C4 | — | 212 | 2 | no | — |
| 263 | C22⋊C4 | #11719922 | 1 | 2 | yes | — |
| 264 | C42 | — | 4 | 1 | no | — |
| 265 | C42 | — | 4 | 2 | no | — |
| 266 | C42 | — | 4 | 2 | no | — |
| 267 | C42 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 268 | C42 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 269 | C16 | — | 106 | 2 | no | — |
| 270 | C16 | — | 106 | 2 | no | — |
| 271 | C23 | — | 212 | 2 | no | — |
| 272 | C23 | C2×#11719922 | 1 | 1 | yes | characteristic |
| 273 | Q8 | — | 2 | 4 | no | — |
| 274 | Q8 | — | 4 | 4 | no | — |
| 275 | Q8 | — | 4 | 2 | no | — |
| 276 | D4 | — | 848 | 2 | no | — |
| 277 | D4 | — | 848 | 2 | no | — |
| 278 | D4 | — | 212 | 2 | no | — |
| 279 | D4 | — | 212 | 2 | no | — |
| 280 | C2×C4 | — | 2 | 1 | no | — |
| 281 | C2×C4 | — | 2 | 1 | no | — |
| 282 | C2×C4 | — | 4 | 4 | no | — |
| 283 | C2×C4 | — | 4 | 2 | no | — |
| 284 | C2×C4 | — | 4 | 2 | no | — |
| 285 | C2×C4 | — | 848 | 2 | no | — |
| 286 | C2×C4 | — | 212 | 2 | no | — |
| 287 | C2×C4 | C2×#11719922 | 1 | 2 | yes | — |
| 288 | C2×C4 | C2×#11719922 | 1 | 2 | yes | — |
| 289 | C2×C4 | C2×#11719922 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | C22×C22⋊D53 | 1 | 1 | yes | characteristic |
| 291 | C8 | — | 2 | 1 | no | — |
| 292 | C8 | — | 2 | 1 | no | — |
| 293 | C8 | — | 2 | 1 | no | — |
| 294 | C8 | C2×#11719945 | 1 | 2 | yes | — |
| 295 | C22 | — | 2 | 1 | no | — |
| 296 | C22 | — | 2 | 2 | no | — |
| 297 | C22 | — | 424 | 2 | no | — |
| 298 | C22 | — | 424 | 2 | no | — |
| 299 | C22 | — | 424 | 2 | no | — |
| 300 | C22 | C22×#11719922 | 1 | 1 | yes | characteristic |
| 301 | C4 | — | 2 | 1 | no | — |
| 302 | C4 | — | 2 | 2 | no | — |
| 303 | C4 | — | 2 | 2 | no | — |
| 304 | C4 | — | 4 | 4 | no | — |
| 305 | C4 | — | 8 | 2 | no | — |
| 306 | C4 | — | 424 | 2 | no | — |
| 307 | C4 | #16811134 | 1 | 1 | yes | characteristic |
| 308 | C4 | #16811421 | 1 | 1 | yes | characteristic |
| 309 | C2 | — | 4 | 1 | no | — |
| 310 | C2 | — | 848 | 2 | no | — |
| 311 | C2 | #20284715 | 1 | 1 | yes | characteristic |
| 312 | C2 | #20287859 | 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