← #27538749
Subgroups — #27538749
Order 13568.
598 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20285229 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C4×D424 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20276873 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #20276863 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20276853 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #20276503 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #20276432 | C2 | 1 | 2 | yes | maximal |
| 8 | C53×#2150 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C22×D424 | C22 | 1 | 1 | yes | characteristic |
| 10 | #16811180 | — | 2 | 1 | no | — |
| 11 | C2×C4×D212 | — | 2 | 1 | no | — |
| 12 | #16811028 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | #16811028 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | #16811025 | — | 2 | 1 | no | — |
| 15 | #16811025 | C22 | 1 | 2 | yes | complemented |
| 16 | #16811021 | — | 2 | 1 | no | — |
| 17 | C4×D424 | — | 2 | 1 | no | — |
| 18 | C4×D424 | — | 2 | 1 | no | — |
| 19 | C4×D424 | C22 | 1 | 2 | yes | — |
| 20 | C4×D424 | C22 | 1 | 2 | yes | — |
| 21 | #16810661 | — | 2 | 1 | no | — |
| 22 | #16810661 | C22 | 1 | 2 | yes | — |
| 23 | C2×#13133151 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×#13133139 | — | 2 | 1 | no | — |
| 25 | #16810451 | C22 | 1 | 4 | yes | complemented |
| 26 | #16810449 | C22 | 1 | 4 | yes | complemented |
| 27 | #16810425 | C22 | 1 | 4 | yes | — |
| 28 | #16810425 | C22 | 1 | 4 | yes | — |
| 29 | C2×M32×C53 | C22 | 1 | 2 | yes | — |
| 30 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 31 | C53×C4⋊C16 | C22 | 1 | 2 | yes | complemented |
| 32 | C53×C4⋊C16 | C22 | 1 | 2 | yes | complemented |
| 33 | C22×D212 | — | 2 | 1 | no | — |
| 34 | C22×C4×D53 | — | 4 | 1 | no | — |
| 35 | C2×D424 | — | 2 | 1 | no | — |
| 36 | C2×D424 | — | 2 | 2 | no | — |
| 37 | C2×D424 | Q8 | 1 | 4 | yes | complemented |
| 38 | C2×D424 | C23 | 1 | 2 | yes | — |
| 39 | #13133217 | — | 2 | 1 | no | — |
| 40 | #13133217 | — | 2 | 1 | no | — |
| 41 | #13133209 | — | 2 | 2 | no | — |
| 42 | #13133196 | — | 4 | 1 | no | — |
| 43 | #13133196 | — | 4 | 1 | no | — |
| 44 | C4×Dic106 | — | 4 | 1 | no | — |
| 45 | C4×Dic106 | — | 4 | 1 | no | — |
| 46 | C4×D212 | — | 2 | 2 | no | — |
| 47 | C4×D212 | — | 2 | 2 | no | — |
| 48 | C4×D212 | — | 4 | 1 | no | — |
| 49 | C4×D212 | — | 4 | 1 | no | — |
| 50 | #13133176 | — | 2 | 2 | no | — |
| 51 | C2×C4⋊Dic53 | — | 2 | 1 | no | — |
| 52 | C2×#11719921 | — | 4 | 1 | no | — |
| 53 | #13133151 | — | 2 | 1 | no | — |
| 54 | #13133151 | — | 2 | 1 | no | — |
| 55 | #13133151 | — | 2 | 1 | no | — |
| 56 | #13133151 | C23 | 1 | 2 | yes | — |
| 57 | #13133151 | C23 | 1 | 2 | yes | — |
| 58 | #13133151 | C23 | 1 | 2 | yes | — |
| 59 | #13133149 | — | 2 | 2 | no | — |
| 60 | #13133149 | — | 2 | 2 | no | — |
| 61 | #13133139 | — | 2 | 2 | no | — |
| 62 | #13133139 | — | 2 | 2 | no | — |
| 63 | C22×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 64 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 65 | C2×C42×C53 | D4 | 1 | 1 | yes | characteristic |
| 66 | M32×C53 | — | 2 | 4 | no | — |
| 67 | C2×C16×C53 | C23 | 1 | 4 | yes | — |
| 68 | C4×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 69 | C4×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 70 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 71 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 72 | C23×D53 | — | 4 | 1 | no | — |
| 73 | C22×C4×C53 | D8 | 1 | 1 | yes | characteristic |
| 74 | C22×C4×C53 | D8 | 1 | 1 | yes | characteristic |
| 75 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C22×Dic53 | — | 4 | 1 | no | — |
| 77 | C2×D212 | — | 2 | 4 | no | — |
| 78 | C2×D212 | — | 2 | 2 | no | — |
| 79 | C2×D212 | — | 4 | 1 | no | — |
| 80 | C2×D212 | — | 4 | 2 | no | — |
| 81 | C2×C4×D53 | — | 4 | 2 | no | — |
| 82 | C2×C4×D53 | — | 4 | 2 | no | — |
| 83 | C2×C4×D53 | — | 8 | 4 | no | — |
| 84 | C2×Dic106 | — | 4 | 1 | no | — |
| 85 | C2×Dic106 | — | 4 | 1 | no | — |
| 86 | C2×C8×C53 | — | 2 | 2 | no | — |
| 87 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C53 | C2×Q8 | 1 | 2 | yes | — |
| 91 | C2×C8×C53 | #41 | 1 | 2 | yes | — |
| 92 | C2×C8×C53 | C24 | 1 | 1 | yes | characteristic |
| 93 | C42×C53 | D8 | 1 | 1 | yes | characteristic |
| 94 | C42×C53 | D8 | 1 | 1 | yes | characteristic |
| 95 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C22⋊Dic53 | — | 4 | 2 | no | — |
| 98 | #11719921 | — | 4 | 2 | no | — |
| 99 | #11719921 | — | 4 | 2 | no | — |
| 100 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 101 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 102 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 103 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 104 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 105 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 106 | #11719919 | — | 4 | 2 | no | — |
| 107 | #11719919 | — | 4 | 2 | no | — |
| 108 | C4×Dic53 | — | 4 | 2 | no | — |
| 109 | D424 | — | 2 | 4 | no | — |
| 110 | D424 | — | 2 | 2 | no | — |
| 111 | D424 | C2×Q8 | 1 | 4 | yes | — |
| 112 | C16×C53 | — | 2 | 4 | no | — |
| 113 | C23×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 114 | C22×D53 | — | 4 | 2 | no | — |
| 115 | C22×D53 | — | 4 | 4 | no | — |
| 116 | C22×D53 | — | 8 | 2 | no | — |
| 117 | C22×D53 | — | 8 | 2 | no | — |
| 118 | C2×C4×C53 | — | 2 | 2 | no | — |
| 119 | C2×C4×C53 | — | 4 | 1 | no | — |
| 120 | C2×C4×C53 | C22⋊Q8 | 1 | 2 | yes | — |
| 121 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 125 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×C53 | #137 | 1 | 2 | yes | — |
| 127 | C2×C4×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 128 | C2×Dic53 | — | 4 | 2 | no | — |
| 129 | C2×Dic53 | — | 4 | 2 | no | — |
| 130 | C2×Dic53 | — | 4 | 2 | no | — |
| 131 | C2×Dic53 | — | 8 | 2 | no | — |
| 132 | D212 | — | 2 | 4 | no | — |
| 133 | D212 | — | 4 | 2 | no | — |
| 134 | D212 | — | 4 | 4 | no | — |
| 135 | C4×D53 | — | 8 | 4 | no | — |
| 136 | C4×D53 | — | 8 | 4 | no | — |
| 137 | Dic106 | — | 8 | 2 | no | — |
| 138 | Dic106 | — | 8 | 2 | no | — |
| 139 | C8×C53 | — | 2 | 1 | no | — |
| 140 | C8×C53 | — | 2 | 1 | no | — |
| 141 | C8×C53 | C22⋊Q8 | 1 | 2 | yes | — |
| 142 | C8×C53 | C22×Q8 | 1 | 1 | yes | characteristic |
| 143 | C8×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 144 | #33627 | — | 53 | 1 | no | maximal, Sylow |
| 145 | C22×C53 | — | 2 | 1 | no | — |
| 146 | C22×C53 | — | 2 | 1 | no | — |
| 147 | C22×C53 | #474 | 1 | 2 | yes | — |
| 148 | C22×C53 | C22×D8 | 1 | 1 | yes | characteristic |
| 149 | C2×D53 | — | 4 | 4 | no | — |
| 150 | C2×D53 | — | 8 | 8 | no | — |
| 151 | C2×D53 | — | 8 | 2 | no | — |
| 152 | C2×D53 | — | 8 | 2 | no | — |
| 153 | C4×C53 | — | 2 | 1 | no | — |
| 154 | C4×C53 | — | 2 | 1 | no | — |
| 155 | C4×C53 | — | 2 | 1 | no | — |
| 156 | C4×C53 | #474 | 1 | 2 | yes | — |
| 157 | C4×C53 | #509 | 1 | 2 | yes | — |
| 158 | C4×C53 | C2×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 159 | C4×C53 | C2×#137 | 1 | 1 | yes | characteristic |
| 160 | Dic53 | — | 8 | 2 | no | — |
| 161 | Dic53 | — | 8 | 2 | no | — |
| 162 | Dic53 | — | 8 | 2 | no | — |
| 163 | #3160 | — | 53 | 1 | no | — |
| 164 | C2×C4×D8 | — | 53 | 1 | no | — |
| 165 | #2228 | — | 53 | 4 | no | — |
| 166 | #2226 | — | 53 | 2 | no | — |
| 167 | #2224 | — | 53 | 2 | no | — |
| 168 | #2159 | — | 53 | 2 | no | — |
| 169 | #2150 | D53 | 1 | 1 | yes | characteristic, complemented |
| 170 | #2140 | — | 53 | 2 | no | — |
| 171 | C2×C53 | — | 2 | 2 | no | — |
| 172 | C2×C53 | C2×#474 | 1 | 1 | yes | characteristic |
| 173 | C2×C53 | C2×#509 | 1 | 1 | yes | characteristic |
| 174 | C2×C53 | #3414 | 1 | 1 | yes | characteristic |
| 175 | D53 | — | 8 | 4 | no | — |
| 176 | C22×D8 | — | 53 | 1 | no | — |
| 177 | #533 | — | 106 | 1 | no | — |
| 178 | C2×C4×D4 | — | 106 | 1 | no | — |
| 179 | C2×M32 | C2×D53 | 1 | 2 | yes | — |
| 180 | #500 | — | 53 | 1 | no | — |
| 181 | #500 | — | 53 | 1 | no | — |
| 182 | #499 | — | 53 | 2 | no | — |
| 183 | #499 | — | 106 | 1 | no | — |
| 184 | #498 | — | 106 | 1 | no | — |
| 185 | C4×D8 | — | 53 | 2 | no | — |
| 186 | C4×D8 | — | 53 | 2 | no | — |
| 187 | C4×D8 | — | 106 | 1 | no | — |
| 188 | C4×D8 | — | 106 | 1 | no | — |
| 189 | #427 | — | 53 | 2 | no | — |
| 190 | #427 | — | 106 | 1 | no | — |
| 191 | C2×#107 | — | 53 | 1 | no | — |
| 192 | C2×#102 | — | 106 | 1 | no | — |
| 193 | C2×C4×C8 | C2×D53 | 1 | 1 | yes | characteristic |
| 194 | #367 | — | 53 | 4 | no | — |
| 195 | #366 | — | 53 | 4 | no | — |
| 196 | C4⋊C16 | C2×D53 | 1 | 2 | yes | complemented |
| 197 | C4⋊C16 | C2×D53 | 1 | 2 | yes | complemented |
| 198 | #357 | — | 53 | 4 | no | — |
| 199 | #357 | — | 53 | 4 | no | — |
| 200 | C53 | #33627 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 201 | C22×D4 | — | 106 | 1 | no | — |
| 202 | C23×C4 | — | 212 | 1 | no | — |
| 203 | C2×D8 | — | 53 | 4 | no | — |
| 204 | C2×D8 | — | 53 | 2 | no | — |
| 205 | C2×D8 | — | 106 | 1 | no | — |
| 206 | C2×D8 | — | 106 | 2 | no | — |
| 207 | C22×C8 | D212 | 1 | 1 | yes | characteristic |
| 208 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 209 | C4⋊Q8 | — | 106 | 1 | no | — |
| 210 | C4⋊Q8 | — | 106 | 1 | no | — |
| 211 | #125 | — | 106 | 2 | no | — |
| 212 | C22⋊Q8 | — | 212 | 1 | no | — |
| 213 | C22⋊Q8 | — | 212 | 1 | no | — |
| 214 | C4×Q8 | — | 212 | 1 | no | — |
| 215 | C4×Q8 | — | 212 | 1 | no | — |
| 216 | C4×D4 | — | 106 | 2 | no | — |
| 217 | C4×D4 | — | 106 | 2 | no | — |
| 218 | C4×D4 | — | 212 | 1 | no | — |
| 219 | C4×D4 | — | 212 | 1 | no | — |
| 220 | #117 | — | 106 | 2 | no | — |
| 221 | C2×C4⋊C4 | — | 106 | 1 | no | — |
| 222 | C2×C22⋊C4 | — | 212 | 1 | no | — |
| 223 | C2×C42 | D212 | 1 | 1 | yes | characteristic |
| 224 | M32 | — | 2 | 4 | no | — |
| 225 | C2×C16 | C22×D53 | 1 | 4 | yes | — |
| 226 | #107 | — | 53 | 2 | no | — |
| 227 | #107 | — | 53 | 2 | no | — |
| 228 | #107 | — | 53 | 2 | no | — |
| 229 | #107 | — | 106 | 1 | no | — |
| 230 | #107 | — | 106 | 1 | no | — |
| 231 | #107 | — | 106 | 1 | no | — |
| 232 | #106 | — | 106 | 2 | no | — |
| 233 | #106 | — | 106 | 2 | no | — |
| 234 | #102 | — | 106 | 2 | no | — |
| 235 | #102 | — | 106 | 2 | no | — |
| 236 | C4×C8 | D212 | 1 | 1 | yes | characteristic |
| 237 | C4×C8 | D212 | 1 | 1 | yes | characteristic |
| 238 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 239 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 240 | C24 | — | 212 | 1 | no | — |
| 241 | C2×Q8 | — | 212 | 1 | no | — |
| 242 | C2×Q8 | — | 212 | 1 | no | — |
| 243 | C2×D4 | — | 106 | 4 | no | — |
| 244 | C2×D4 | — | 106 | 2 | no | — |
| 245 | C2×D4 | — | 212 | 1 | no | — |
| 246 | C2×D4 | — | 212 | 2 | no | — |
| 247 | C22×C4 | — | 424 | 4 | no | — |
| 248 | C22×C4 | — | 212 | 1 | no | — |
| 249 | C22×C4 | — | 212 | 2 | no | — |
| 250 | C22×C4 | — | 212 | 2 | no | — |
| 251 | C22×C4 | D424 | 1 | 1 | yes | characteristic |
| 252 | C22×C4 | D424 | 1 | 1 | yes | characteristic |
| 253 | C22×C4 | C2×D212 | 1 | 1 | yes | characteristic |
| 254 | D8 | — | 53 | 4 | no | — |
| 255 | D8 | — | 106 | 4 | no | — |
| 256 | D8 | — | 106 | 2 | no | — |
| 257 | C2×C8 | — | 2 | 2 | no | — |
| 258 | C2×C8 | C2×D212 | 1 | 1 | yes | characteristic |
| 259 | C2×C8 | C2×D212 | 1 | 1 | yes | characteristic |
| 260 | C2×C8 | C2×D212 | 1 | 1 | yes | characteristic |
| 261 | C2×C8 | #11719947 | 1 | 2 | yes | — |
| 262 | C2×C8 | Q8×D53 | 1 | 2 | yes | — |
| 263 | C2×C8 | C23×D53 | 1 | 1 | yes | characteristic |
| 264 | C4⋊C4 | — | 106 | 2 | no | — |
| 265 | C4⋊C4 | — | 106 | 2 | no | — |
| 266 | C4⋊C4 | — | 106 | 2 | no | — |
| 267 | C4⋊C4 | — | 212 | 1 | no | — |
| 268 | C4⋊C4 | — | 212 | 2 | no | — |
| 269 | C4⋊C4 | — | 212 | 2 | no | — |
| 270 | C4⋊C4 | — | 212 | 1 | no | — |
| 271 | C4⋊C4 | — | 212 | 1 | no | — |
| 272 | C22⋊C4 | — | 212 | 2 | no | — |
| 273 | C22⋊C4 | — | 212 | 2 | no | — |
| 274 | C22⋊C4 | — | 212 | 2 | no | — |
| 275 | C42 | — | 212 | 2 | no | — |
| 276 | C42 | D424 | 1 | 1 | yes | characteristic |
| 277 | C42 | D424 | 1 | 1 | yes | characteristic |
| 278 | C42 | C2×D212 | 1 | 1 | yes | characteristic |
| 279 | C42 | C2×D212 | 1 | 1 | yes | characteristic |
| 280 | C16 | — | 2 | 4 | no | — |
| 281 | C23 | — | 424 | 2 | no | — |
| 282 | C23 | — | 424 | 2 | no | — |
| 283 | C23 | — | 212 | 2 | no | — |
| 284 | C23 | — | 212 | 4 | no | — |
| 285 | C23 | C2×D424 | 1 | 1 | yes | characteristic |
| 286 | Q8 | — | 424 | 2 | no | — |
| 287 | Q8 | — | 424 | 2 | no | — |
| 288 | D4 | — | 106 | 4 | no | — |
| 289 | D4 | — | 212 | 4 | no | — |
| 290 | D4 | — | 212 | 2 | no | — |
| 291 | C2×C4 | — | 2 | 2 | no | — |
| 292 | C2×C4 | — | 4 | 1 | no | — |
| 293 | C2×C4 | — | 424 | 4 | no | — |
| 294 | C2×C4 | — | 424 | 2 | no | — |
| 295 | C2×C4 | — | 424 | 4 | no | — |
| 296 | C2×C4 | — | 212 | 2 | no | — |
| 297 | C2×C4 | — | 212 | 2 | no | — |
| 298 | C2×C4 | — | 212 | 2 | no | — |
| 299 | C2×C4 | #13133197 | 1 | 2 | yes | — |
| 300 | C2×C4 | C2×D424 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | C2×D424 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | C2×D424 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | C2×D424 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | C2×D424 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | #13133250 | 1 | 2 | yes | — |
| 306 | C2×C4 | C22×D212 | 1 | 1 | yes | characteristic |
| 307 | C8 | — | 2 | 1 | no | — |
| 308 | C8 | — | 2 | 1 | no | — |
| 309 | C8 | #13133197 | 1 | 2 | yes | — |
| 310 | C8 | C2×Q8×D53 | 1 | 1 | yes | characteristic |
| 311 | C8 | C2×#11719947 | 1 | 1 | yes | characteristic |
| 312 | C22 | — | 2 | 1 | no | — |
| 313 | C22 | — | 2 | 1 | no | — |
| 314 | C22 | — | 424 | 8 | no | — |
| 315 | C22 | — | 424 | 2 | no | — |
| 316 | C22 | — | 424 | 2 | no | — |
| 317 | C22 | — | 212 | 4 | no | — |
| 318 | C22 | #16810904 | 1 | 2 | yes | — |
| 319 | C22 | C22×D424 | 1 | 1 | yes | characteristic |
| 320 | C4 | — | 2 | 1 | no | — |
| 321 | C4 | — | 2 | 1 | no | — |
| 322 | C4 | — | 2 | 1 | no | — |
| 323 | C4 | — | 424 | 2 | no | — |
| 324 | C4 | — | 424 | 2 | no | — |
| 325 | C4 | — | 424 | 2 | no | — |
| 326 | C4 | #16810904 | 1 | 2 | yes | — |
| 327 | C4 | #16811061 | 1 | 2 | yes | — |
| 328 | C4 | C2×#13133197 | 1 | 1 | yes | characteristic |
| 329 | C4 | C2×#13133250 | 1 | 1 | yes | characteristic |
| 330 | C2 | — | 2 | 2 | no | — |
| 331 | C2 | — | 424 | 4 | no | — |
| 332 | C2 | C2×#16810904 | 1 | 1 | yes | characteristic |
| 333 | C2 | C2×#16811061 | 1 | 1 | yes | characteristic |
| 334 | C2 | #20287851 | 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