← #27468203
Subgroups — #27468203
Order 13568.
438 conjugacy classes of subgroups.
The subgroup lattice has 342 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20276297 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20276169 | C2 | 1 | 2 | yes | maximal |
| 3 | #20275505 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #20275468 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20275381 | C2 | 1 | 2 | yes | maximal |
| 6 | #20274861 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #20274839 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #20274769 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #20274765 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #20274621 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | C53×#1765 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #16810671 | C22 | 1 | 2 | yes | — |
| 13 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16810661 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16810649 | C22 | 1 | 2 | yes | — |
| 18 | #16810640 | C22 | 1 | 2 | yes | — |
| 19 | #16810607 | C22 | 1 | 2 | yes | — |
| 20 | C4×#11719917 | C22 | 1 | 2 | yes | — |
| 21 | #16810395 | C22 | 1 | 2 | yes | — |
| 22 | #16810394 | C22 | 1 | 2 | yes | — |
| 23 | #16810393 | C22 | 1 | 2 | yes | — |
| 24 | #16810388 | C22 | 1 | 2 | yes | — |
| 25 | #16810383 | C22 | 1 | 2 | yes | — |
| 26 | #16810381 | C22 | 1 | 2 | yes | — |
| 27 | #16810379 | C22 | 1 | 2 | yes | — |
| 28 | C2×#131×C53 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×#104×C53 | C22 | 1 | 1 | yes | characteristic |
| 30 | C53×#416 | C22 | 1 | 1 | yes | characteristic |
| 31 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 32 | C2×#97×C53 | C22 | 1 | 1 | yes | characteristic |
| 33 | C53×#341 | C22 | 1 | 1 | yes | characteristic |
| 34 | C53×#339 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×#11719917 | C23 | 1 | 2 | yes | — |
| 36 | C2×#11719917 | C23 | 1 | 2 | yes | — |
| 37 | C2×#11719917 | C23 | 1 | 2 | yes | — |
| 38 | #13133176 | C23 | 1 | 2 | yes | — |
| 39 | #13133153 | — | 2 | 1 | no | — |
| 40 | #13133153 | — | 2 | 1 | no | — |
| 41 | #13133153 | — | 2 | 1 | no | — |
| 42 | #13133153 | — | 2 | 1 | no | — |
| 43 | #13133153 | — | 2 | 1 | no | — |
| 44 | #13133153 | — | 2 | 1 | no | — |
| 45 | #13133152 | — | 4 | 2 | no | — |
| 46 | #13133151 | — | 2 | 1 | no | — |
| 47 | #13133149 | — | 2 | 1 | no | — |
| 48 | #13133146 | — | 4 | 2 | no | — |
| 49 | #13133144 | — | 4 | 2 | no | — |
| 50 | #13133136 | — | 2 | 2 | no | — |
| 51 | #13133134 | — | 2 | 2 | no | — |
| 52 | #13133128 | — | 2 | 2 | no | — |
| 53 | C4×#88322 | — | 2 | 2 | no | — |
| 54 | C2×#41×C53 | C23 | 1 | 1 | yes | characteristic |
| 55 | #131×C53 | — | 2 | 1 | no | — |
| 56 | #131×C53 | — | 2 | 1 | no | — |
| 57 | #131×C53 | — | 2 | 1 | no | — |
| 58 | #131×C53 | — | 2 | 1 | no | — |
| 59 | C2×M16×C53 | — | 2 | 1 | no | — |
| 60 | C2×M16×C53 | C23 | 1 | 1 | yes | characteristic |
| 61 | C2×M16×C53 | C23 | 1 | 1 | yes | characteristic |
| 62 | C22×C8×C53 | — | 2 | 1 | no | — |
| 63 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 64 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 65 | #117×C53 | C23 | 1 | 1 | yes | characteristic |
| 66 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 67 | #104×C53 | — | 2 | 2 | no | — |
| 68 | #103×C53 | — | 2 | 1 | no | — |
| 69 | #102×C53 | — | 2 | 1 | no | — |
| 70 | #97×C53 | — | 2 | 1 | no | — |
| 71 | #97×C53 | — | 2 | 1 | no | — |
| 72 | #97×C53 | — | 2 | 1 | no | — |
| 73 | C4×C8×C53 | — | 2 | 1 | no | — |
| 74 | #41×C53 | — | 2 | 1 | no | — |
| 75 | #41×C53 | — | 2 | 1 | no | — |
| 76 | #41×C53 | — | 2 | 1 | no | — |
| 77 | #41×C53 | — | 2 | 1 | no | — |
| 78 | C2×Q8×C53 | #41 | 1 | 1 | yes | characteristic |
| 79 | C2×D4×C53 | — | 2 | 1 | no | — |
| 80 | C2×D4×C53 | #41 | 1 | 1 | yes | characteristic |
| 81 | C22×C4×C53 | — | 2 | 1 | no | — |
| 82 | C22×C4×C53 | — | 2 | 1 | no | — |
| 83 | C22×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 84 | M16×C53 | — | 2 | 1 | no | — |
| 85 | M16×C53 | — | 2 | 1 | no | — |
| 86 | M16×C53 | — | 2 | 1 | no | — |
| 87 | M16×C53 | — | 2 | 1 | no | — |
| 88 | M16×C53 | — | 4 | 1 | no | — |
| 89 | M16×C53 | — | 4 | 1 | no | — |
| 90 | C2×C8×C53 | — | 2 | 1 | no | — |
| 91 | C2×C8×C53 | — | 2 | 1 | no | — |
| 92 | C2×C8×C53 | — | 2 | 1 | no | — |
| 93 | C2×C8×C53 | — | 2 | 1 | no | — |
| 94 | C2×C8×C53 | — | 2 | 1 | no | — |
| 95 | C2×C8×C53 | — | 4 | 1 | no | — |
| 96 | C2×C8×C53 | — | 4 | 1 | no | — |
| 97 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 98 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 99 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 100 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 101 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 102 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 103 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 104 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 105 | C42×C53 | — | 2 | 1 | no | — |
| 106 | C42×C53 | — | 2 | 1 | no | — |
| 107 | C42×C53 | #41 | 1 | 2 | yes | — |
| 108 | C22⋊Dic53 | — | 2 | 2 | no | — |
| 109 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 110 | C4×Dic53 | — | 2 | 2 | no | — |
| 111 | #11719917 | — | 2 | 2 | no | — |
| 112 | #11719917 | — | 2 | 2 | no | — |
| 113 | #11719917 | — | 2 | 2 | no | — |
| 114 | #11719917 | — | 2 | 2 | no | — |
| 115 | #11719917 | — | 2 | 2 | no | — |
| 116 | #11719917 | — | 2 | 2 | no | — |
| 117 | C2×#88322 | — | 2 | 2 | no | — |
| 118 | C2×#88322 | — | 2 | 2 | no | — |
| 119 | C2×#88322 | — | 2 | 2 | no | — |
| 120 | C23×C53 | — | 2 | 1 | no | — |
| 121 | C23×C53 | 2−1+4 | 1 | 1 | yes | characteristic |
| 122 | Q8×C53 | — | 2 | 1 | no | — |
| 123 | Q8×C53 | — | 2 | 1 | no | — |
| 124 | D4×C53 | — | 2 | 1 | no | — |
| 125 | D4×C53 | — | 2 | 1 | no | — |
| 126 | D4×C53 | — | 4 | 1 | no | — |
| 127 | D4×C53 | — | 4 | 1 | no | — |
| 128 | C2×C4×C53 | — | 2 | 1 | no | — |
| 129 | C2×C4×C53 | — | 2 | 2 | no | — |
| 130 | C2×C4×C53 | — | 4 | 1 | no | — |
| 131 | C2×C4×C53 | — | 4 | 1 | no | — |
| 132 | C2×C4×C53 | — | 4 | 1 | no | — |
| 133 | C2×C4×C53 | — | 8 | 1 | no | — |
| 134 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 135 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 136 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 137 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 138 | C2×C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 139 | C2×C4×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 140 | C2×Dic53 | — | 4 | 2 | no | — |
| 141 | C8×C53 | — | 4 | 1 | no | — |
| 142 | C8×C53 | — | 4 | 1 | no | — |
| 143 | C8×C53 | — | 4 | 1 | no | — |
| 144 | C8×C53 | #126 | 1 | 1 | yes | characteristic |
| 145 | C8×C53 | #126 | 1 | 1 | yes | characteristic |
| 146 | C8×C53 | #126 | 1 | 1 | yes | characteristic |
| 147 | C8×C53 | #126 | 1 | 1 | yes | characteristic |
| 148 | #88322 | — | 4 | 2 | no | — |
| 149 | #88322 | — | 4 | 2 | no | — |
| 150 | #88322 | — | 4 | 2 | no | — |
| 151 | #27834 | — | 53 | 1 | no | maximal, Sylow |
| 152 | C22×C53 | — | 4 | 1 | no | — |
| 153 | C22×C53 | — | 4 | 1 | no | — |
| 154 | C22×C53 | — | 8 | 1 | no | — |
| 155 | C22×C53 | #542 | 1 | 1 | yes | characteristic |
| 156 | C22×C53 | #552 | 1 | 1 | yes | characteristic |
| 157 | C22×C53 | #552 | 1 | 1 | yes | characteristic |
| 158 | C4×C53 | — | 4 | 2 | no | — |
| 159 | C4×C53 | — | 4 | 1 | no | — |
| 160 | C4×C53 | — | 8 | 1 | no | — |
| 161 | C4×C53 | C2×#126 | 1 | 1 | yes | characteristic |
| 162 | C4×C53 | #529 | 1 | 1 | yes | characteristic |
| 163 | C4×C53 | #553 | 1 | 1 | yes | characteristic |
| 164 | C4×C53 | #555 | 1 | 1 | yes | characteristic |
| 165 | Dic53 | — | 8 | 2 | no | — |
| 166 | #2103 | — | 53 | 2 | no | — |
| 167 | #2080 | — | 53 | 2 | no | — |
| 168 | #1953 | — | 53 | 1 | no | — |
| 169 | #1946 | — | 53 | 1 | no | — |
| 170 | #1932 | — | 53 | 2 | no | — |
| 171 | #1841 | — | 53 | 2 | no | — |
| 172 | #1833 | — | 53 | 1 | no | — |
| 173 | #1814 | — | 53 | 1 | no | — |
| 174 | #1813 | — | 53 | 1 | no | — |
| 175 | #1778 | — | 53 | 1 | no | — |
| 176 | #1765 | D53 | 1 | 1 | yes | characteristic |
| 177 | C2×C53 | — | 2 | 1 | no | — |
| 178 | C2×C53 | — | 2 | 1 | no | — |
| 179 | C2×C53 | — | 2 | 1 | no | — |
| 180 | C2×C53 | — | 8 | 1 | no | — |
| 181 | C2×C53 | #2495 | 1 | 1 | yes | characteristic |
| 182 | C2×#131 | C2×D53 | 1 | 1 | yes | characteristic |
| 183 | #430 | — | 53 | 2 | no | — |
| 184 | C2×#108 | — | 53 | 1 | no | — |
| 185 | C2×#108 | — | 53 | 1 | no | — |
| 186 | C2×#108 | — | 53 | 1 | no | — |
| 187 | #427 | — | 53 | 1 | no | — |
| 188 | #424 | — | 53 | 2 | no | — |
| 189 | #421 | — | 53 | 2 | no | — |
| 190 | C2×#104 | C2×D53 | 1 | 1 | yes | characteristic |
| 191 | #416 | C2×D53 | 1 | 1 | yes | characteristic |
| 192 | #413 | — | 53 | 2 | no | — |
| 193 | #405 | C2×D53 | 1 | 1 | yes | characteristic |
| 194 | C4×M16 | — | 53 | 2 | no | — |
| 195 | C2×#97 | C2×D53 | 1 | 1 | yes | characteristic |
| 196 | #344 | — | 53 | 2 | no | — |
| 197 | #344 | — | 53 | 2 | no | — |
| 198 | #344 | — | 53 | 2 | no | — |
| 199 | #341 | — | 53 | 2 | no | — |
| 200 | #341 | C2×D53 | 1 | 1 | yes | characteristic |
| 201 | #339 | — | 53 | 2 | no | — |
| 202 | #339 | C2×D53 | 1 | 1 | yes | characteristic |
| 203 | #338 | — | 53 | 2 | no | — |
| 204 | H(3,ℤ/4) | — | 53 | 2 | no | — |
| 205 | C53 | #27834 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 206 | C2×#41 | C22×D53 | 1 | 1 | yes | characteristic |
| 207 | #131 | — | 2 | 1 | no | — |
| 208 | #131 | — | 2 | 1 | no | — |
| 209 | #131 | — | 2 | 1 | no | — |
| 210 | #131 | — | 2 | 1 | no | — |
| 211 | C2×M16 | — | 2 | 1 | no | — |
| 212 | C2×M16 | — | 53 | 2 | no | — |
| 213 | C2×M16 | — | 53 | 2 | no | — |
| 214 | C2×M16 | — | 53 | 2 | no | — |
| 215 | C2×M16 | C22×D53 | 1 | 1 | yes | characteristic |
| 216 | C2×M16 | C22×D53 | 1 | 1 | yes | characteristic |
| 217 | C22×C8 | — | 2 | 1 | no | — |
| 218 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 219 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 220 | #117 | — | 53 | 2 | no | — |
| 221 | #117 | C22×D53 | 1 | 1 | yes | characteristic |
| 222 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 223 | #108 | — | 106 | 1 | no | — |
| 224 | #108 | — | 106 | 1 | no | — |
| 225 | #108 | — | 106 | 1 | no | — |
| 226 | #108 | — | 106 | 1 | no | — |
| 227 | #108 | — | 106 | 1 | no | — |
| 228 | #108 | — | 106 | 1 | no | — |
| 229 | #108 | — | 212 | 2 | no | — |
| 230 | #107 | — | 106 | 1 | no | — |
| 231 | #106 | — | 106 | 1 | no | — |
| 232 | C4⋊C8 | — | 212 | 2 | no | — |
| 233 | #104 | — | 2 | 2 | no | — |
| 234 | #104 | — | 212 | 2 | no | — |
| 235 | #103 | — | 2 | 1 | no | — |
| 236 | #102 | — | 2 | 1 | no | — |
| 237 | #101 | — | 106 | 2 | no | — |
| 238 | #100 | — | 106 | 2 | no | — |
| 239 | #97 | — | 2 | 1 | no | — |
| 240 | #97 | — | 2 | 1 | no | — |
| 241 | #97 | — | 2 | 1 | no | — |
| 242 | #97 | — | 106 | 2 | no | — |
| 243 | C4×C8 | — | 2 | 1 | no | — |
| 244 | C4×C8 | — | 106 | 2 | no | — |
| 245 | #41 | — | 2 | 1 | no | — |
| 246 | #41 | — | 2 | 1 | no | — |
| 247 | #41 | — | 2 | 1 | no | — |
| 248 | #41 | — | 2 | 1 | no | — |
| 249 | C2×Q8 | #11719945 | 1 | 1 | yes | characteristic |
| 250 | C2×D4 | — | 2 | 1 | no | — |
| 251 | C2×D4 | #11719945 | 1 | 1 | yes | characteristic |
| 252 | C22×C4 | — | 2 | 1 | no | — |
| 253 | C22×C4 | — | 2 | 1 | no | — |
| 254 | C22×C4 | C23×D53 | 1 | 1 | yes | characteristic |
| 255 | M16 | — | 2 | 1 | no | — |
| 256 | M16 | — | 2 | 1 | no | — |
| 257 | M16 | — | 2 | 1 | no | — |
| 258 | M16 | — | 2 | 1 | no | — |
| 259 | M16 | — | 4 | 1 | no | — |
| 260 | M16 | — | 4 | 1 | no | — |
| 261 | M16 | — | 106 | 2 | no | — |
| 262 | M16 | — | 106 | 2 | no | — |
| 263 | M16 | — | 106 | 2 | no | — |
| 264 | M16 | — | 106 | 2 | no | — |
| 265 | M16 | — | 106 | 2 | no | — |
| 266 | M16 | — | 106 | 2 | no | — |
| 267 | C2×C8 | — | 2 | 1 | no | — |
| 268 | C2×C8 | — | 2 | 1 | no | — |
| 269 | C2×C8 | — | 2 | 1 | no | — |
| 270 | C2×C8 | — | 2 | 1 | no | — |
| 271 | C2×C8 | — | 2 | 1 | no | — |
| 272 | C2×C8 | — | 4 | 1 | no | — |
| 273 | C2×C8 | — | 4 | 1 | no | — |
| 274 | C2×C8 | — | 106 | 2 | no | — |
| 275 | C2×C8 | — | 106 | 2 | no | — |
| 276 | C2×C8 | — | 106 | 2 | no | — |
| 277 | C2×C8 | #11719945 | 1 | 1 | yes | characteristic |
| 278 | C2×C8 | #11719945 | 1 | 1 | yes | characteristic |
| 279 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 280 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 281 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 282 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 283 | C4⋊C4 | — | 2 | 1 | no | — |
| 284 | C4⋊C4 | — | 106 | 2 | no | — |
| 285 | C22⋊C4 | — | 2 | 1 | no | — |
| 286 | C22⋊C4 | — | 106 | 2 | no | — |
| 287 | C42 | — | 2 | 1 | no | — |
| 288 | C42 | — | 2 | 1 | no | — |
| 289 | C42 | — | 106 | 2 | no | — |
| 290 | C42 | #11719945 | 1 | 2 | yes | — |
| 291 | C23 | — | 2 | 1 | no | — |
| 292 | C23 | #13133264 | 1 | 1 | yes | characteristic |
| 293 | Q8 | — | 2 | 1 | no | — |
| 294 | Q8 | — | 2 | 1 | no | — |
| 295 | D4 | — | 2 | 1 | no | — |
| 296 | D4 | — | 2 | 1 | no | — |
| 297 | D4 | — | 4 | 1 | no | — |
| 298 | D4 | — | 4 | 1 | no | — |
| 299 | C2×C4 | — | 2 | 2 | no | — |
| 300 | C2×C4 | — | 2 | 1 | no | — |
| 301 | C2×C4 | — | 4 | 1 | no | — |
| 302 | C2×C4 | — | 4 | 1 | no | — |
| 303 | C2×C4 | — | 4 | 1 | no | — |
| 304 | C2×C4 | — | 8 | 1 | no | — |
| 305 | C2×C4 | — | 212 | 2 | no | — |
| 306 | C2×C4 | C2×#11719947 | 1 | 1 | yes | characteristic |
| 307 | C2×C4 | C2×#11719947 | 1 | 1 | yes | characteristic |
| 308 | C2×C4 | C2×#11719945 | 1 | 1 | yes | characteristic |
| 309 | C2×C4 | C2×#11719945 | 1 | 1 | yes | characteristic |
| 310 | C2×C4 | C2×#11719945 | 1 | 1 | yes | characteristic |
| 311 | C2×C4 | #13133263 | 1 | 1 | yes | characteristic |
| 312 | C8 | — | 4 | 1 | no | — |
| 313 | C8 | — | 4 | 1 | no | — |
| 314 | C8 | — | 4 | 1 | no | — |
| 315 | C8 | — | 212 | 2 | no | — |
| 316 | C8 | — | 212 | 2 | no | — |
| 317 | C8 | — | 212 | 2 | no | — |
| 318 | C8 | #13133210 | 1 | 1 | yes | characteristic |
| 319 | C8 | #13133210 | 1 | 1 | yes | characteristic |
| 320 | C8 | #13133210 | 1 | 1 | yes | characteristic |
| 321 | C8 | #13133210 | 1 | 1 | yes | characteristic |
| 322 | C22 | — | 4 | 1 | no | — |
| 323 | C22 | — | 4 | 1 | no | — |
| 324 | C22 | — | 8 | 1 | no | — |
| 325 | C22 | #16811244 | 1 | 1 | yes | characteristic |
| 326 | C22 | #16811313 | 1 | 1 | yes | characteristic |
| 327 | C22 | #16811313 | 1 | 1 | yes | characteristic |
| 328 | C4 | — | 4 | 2 | no | — |
| 329 | C4 | — | 4 | 1 | no | — |
| 330 | C4 | — | 8 | 1 | no | — |
| 331 | C4 | — | 424 | 2 | no | — |
| 332 | C4 | C2×#13133210 | 1 | 1 | yes | characteristic |
| 333 | C4 | #16811157 | 1 | 1 | yes | characteristic |
| 334 | C4 | #16811319 | 1 | 1 | yes | characteristic |
| 335 | C4 | #16811338 | 1 | 1 | yes | characteristic |
| 336 | C2 | — | 2 | 1 | no | — |
| 337 | C2 | — | 2 | 1 | no | — |
| 338 | C2 | — | 2 | 1 | no | — |
| 339 | C2 | — | 8 | 1 | no | — |
| 340 | C2 | #20279126 | 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