← #27300719
Subgroups — #27300719
Order 13568.
424 conjugacy classes of subgroups.
The subgroup lattice has 320 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20276504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#16810436 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#16810424 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20275500 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20272810 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20272804 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×#2948 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16810666 | — | 2 | 1 | no | — |
| 12 | C2×#13133151 | C22 | 1 | 1 | yes | characteristic |
| 13 | #16810450 | — | 2 | 1 | no | — |
| 14 | #16810448 | — | 2 | 1 | no | — |
| 15 | #16810436 | C4 | 1 | 4 | yes | complemented |
| 16 | #16810424 | — | 2 | 2 | no | — |
| 17 | #16810391 | — | 2 | 1 | no | — |
| 18 | C22×D8×C53 | C22 | 1 | 1 | yes | characteristic |
| 19 | C53×#518 | — | 2 | 1 | no | — |
| 20 | C53×#442 | — | 2 | 2 | no | — |
| 21 | C4×D8×C53 | — | 2 | 1 | no | — |
| 22 | C4×D8×C53 | — | 2 | 1 | no | — |
| 23 | C53×#427 | C22 | 1 | 1 | yes | characteristic |
| 24 | C53×#418 | — | 2 | 1 | no | — |
| 25 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#11719917 | — | 2 | 1 | no | — |
| 27 | C2×C4⋊Dic53 | — | 2 | 1 | no | — |
| 28 | #13133157 | — | 2 | 1 | no | — |
| 29 | #13133157 | C2×C4 | 1 | 2 | yes | complemented |
| 30 | C2×#11719908 | — | 2 | 1 | no | — |
| 31 | C2×#11719908 | — | 2 | 1 | no | — |
| 32 | C2×#11719908 | — | 2 | 1 | no | — |
| 33 | C2×#11719908 | — | 2 | 1 | no | — |
| 34 | #13133153 | — | 2 | 1 | no | — |
| 35 | #13133153 | C2×C4 | 1 | 2 | yes | complemented |
| 36 | #13133151 | — | 2 | 1 | no | — |
| 37 | #13133151 | — | 2 | 1 | no | — |
| 38 | #13133150 | — | 4 | 1 | no | — |
| 39 | #13133148 | — | 4 | 1 | no | — |
| 40 | C22×D4×C53 | — | 2 | 1 | no | — |
| 41 | C2×D8×C53 | — | 2 | 1 | no | — |
| 42 | C2×D8×C53 | — | 2 | 1 | no | — |
| 43 | C2×D8×C53 | — | 2 | 1 | no | — |
| 44 | C2×D8×C53 | C2×C4 | 1 | 2 | yes | — |
| 45 | C2×D8×C53 | D4 | 1 | 2 | yes | — |
| 46 | C2×D8×C53 | Q8 | 1 | 2 | yes | — |
| 47 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 48 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 49 | C4×D4×C53 | — | 2 | 2 | no | — |
| 50 | C4×D4×C53 | — | 2 | 2 | no | — |
| 51 | C4×D4×C53 | — | 4 | 1 | no | — |
| 52 | C4×D4×C53 | — | 4 | 1 | no | — |
| 53 | #117×C53 | — | 2 | 1 | no | — |
| 54 | #117×C53 | D4 | 1 | 1 | yes | characteristic |
| 55 | C2×C22⋊C4×C53 | — | 4 | 2 | no | — |
| 56 | #107×C53 | D4 | 1 | 1 | yes | characteristic |
| 57 | #107×C53 | D4 | 1 | 1 | yes | characteristic |
| 58 | #106×C53 | — | 2 | 1 | no | — |
| 59 | #102×C53 | — | 2 | 2 | no | — |
| 60 | #102×C53 | — | 2 | 2 | no | — |
| 61 | #97×C53 | — | 2 | 1 | no | — |
| 62 | C4×C8×C53 | — | 2 | 1 | no | — |
| 63 | C24×C53 | — | 4 | 1 | no | — |
| 64 | C2×D4×C53 | — | 2 | 2 | no | — |
| 65 | C2×D4×C53 | — | 2 | 2 | no | — |
| 66 | C2×D4×C53 | — | 2 | 2 | no | — |
| 67 | C2×D4×C53 | — | 4 | 1 | no | — |
| 68 | C2×D4×C53 | — | 4 | 1 | no | — |
| 69 | C2×D4×C53 | — | 4 | 1 | no | — |
| 70 | C22×C4×C53 | — | 4 | 2 | no | — |
| 71 | C22×C4×C53 | — | 4 | 2 | no | — |
| 72 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | C22×Dic53 | — | 4 | 1 | no | — |
| 74 | D8×C53 | — | 2 | 2 | no | — |
| 75 | D8×C53 | — | 2 | 2 | no | — |
| 76 | D8×C53 | — | 2 | 2 | no | — |
| 77 | D8×C53 | C4⋊C4 | 1 | 4 | yes | — |
| 78 | M16×C53 | — | 2 | 1 | no | — |
| 79 | M16×C53 | — | 2 | 1 | no | — |
| 80 | C2×C8×C53 | — | 2 | 1 | no | — |
| 81 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 85 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 87 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 88 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 89 | C4⋊C4×C53 | D8 | 1 | 1 | yes | characteristic |
| 90 | C4⋊C4×C53 | D8 | 1 | 1 | yes | characteristic |
| 91 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 92 | C22⋊C4×C53 | — | 4 | 2 | no | — |
| 93 | C22⋊C4×C53 | — | 4 | 2 | no | — |
| 94 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 95 | C42×C53 | — | 2 | 1 | no | — |
| 96 | C42×C53 | — | 4 | 1 | no | — |
| 97 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 98 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 99 | #11719917 | — | 2 | 2 | no | — |
| 100 | #11719917 | — | 4 | 1 | no | — |
| 101 | C2×#88322 | — | 4 | 1 | no | — |
| 102 | #11719908 | — | 2 | 2 | no | — |
| 103 | #11719908 | — | 4 | 1 | no | — |
| 104 | C23×C53 | — | 4 | 2 | no | — |
| 105 | C23×C53 | — | 4 | 2 | no | — |
| 106 | C23×C53 | — | 4 | 2 | no | — |
| 107 | C23×C53 | — | 16 | 1 | no | — |
| 108 | C23×C53 | — | 16 | 1 | no | — |
| 109 | C23×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 110 | D4×C53 | — | 2 | 4 | no | — |
| 111 | D4×C53 | — | 4 | 2 | no | — |
| 112 | D4×C53 | — | 4 | 2 | no | — |
| 113 | D4×C53 | — | 4 | 2 | no | — |
| 114 | C2×C4×C53 | — | 2 | 1 | no | — |
| 115 | C2×C4×C53 | — | 2 | 1 | no | — |
| 116 | C2×C4×C53 | — | 4 | 1 | no | — |
| 117 | C2×C4×C53 | — | 4 | 1 | no | — |
| 118 | C2×C4×C53 | — | 16 | 2 | no | — |
| 119 | C2×C4×C53 | — | 16 | 2 | no | — |
| 120 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 125 | C2×C4×C53 | #137 | 1 | 1 | yes | characteristic |
| 126 | C2×Dic53 | — | 8 | 1 | no | — |
| 127 | C2×Dic53 | — | 16 | 1 | no | — |
| 128 | C8×C53 | — | 4 | 1 | no | — |
| 129 | C8×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 130 | C8×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 131 | C8×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 132 | C8×C53 | #123 | 1 | 1 | yes | characteristic |
| 133 | #88322 | — | 4 | 2 | no | — |
| 134 | #12940 | — | 53 | 1 | no | maximal, Sylow |
| 135 | C22×C53 | — | 2 | 1 | no | — |
| 136 | C22×C53 | — | 2 | 1 | no | — |
| 137 | C22×C53 | — | 4 | 4 | no | — |
| 138 | C22×C53 | — | 8 | 2 | no | — |
| 139 | C22×C53 | — | 8 | 2 | no | — |
| 140 | C22×C53 | — | 16 | 2 | no | — |
| 141 | C22×C53 | — | 16 | 2 | no | — |
| 142 | C22×C53 | #440 | 1 | 1 | yes | characteristic |
| 143 | C22×C53 | #447 | 1 | 1 | yes | characteristic |
| 144 | C22×C53 | #478 | 1 | 1 | yes | characteristic |
| 145 | C4×C53 | — | 4 | 1 | no | — |
| 146 | C4×C53 | — | 4 | 1 | no | — |
| 147 | C4×C53 | — | 8 | 1 | no | — |
| 148 | C4×C53 | — | 8 | 1 | no | — |
| 149 | C4×C53 | #385 | 1 | 1 | yes | characteristic |
| 150 | C4×C53 | C4×D8 | 1 | 1 | yes | characteristic |
| 151 | C4×C53 | #452 | 1 | 1 | yes | characteristic |
| 152 | C4×C53 | #474 | 1 | 1 | yes | characteristic |
| 153 | Dic53 | — | 16 | 1 | no | — |
| 154 | #2948 | D53 | 1 | 1 | yes | characteristic |
| 155 | #2159 | — | 53 | 1 | no | — |
| 156 | C2×#361 | — | 53 | 1 | no | — |
| 157 | C2×#357 | — | 53 | 1 | no | — |
| 158 | #1952 | — | 53 | 1 | no | — |
| 159 | #1382 | — | 53 | 1 | no | — |
| 160 | #1379 | — | 53 | 1 | no | — |
| 161 | C2×C53 | — | 2 | 1 | no | — |
| 162 | C2×C53 | — | 2 | 1 | no | — |
| 163 | C2×C53 | — | 8 | 4 | no | — |
| 164 | C2×C53 | #1862 | 1 | 1 | yes | characteristic |
| 165 | C2×C53 | #2192 | 1 | 1 | yes | characteristic |
| 166 | C2×C53 | #2194 | 1 | 1 | yes | characteristic |
| 167 | C22×D8 | C2×D53 | 1 | 1 | yes | characteristic |
| 168 | #518 | — | 2 | 1 | no | — |
| 169 | C2×M32 | — | 53 | 1 | no | — |
| 170 | C22×C16 | — | 53 | 1 | no | — |
| 171 | #442 | — | 2 | 2 | no | — |
| 172 | C4×D8 | — | 2 | 1 | no | — |
| 173 | C4×D8 | — | 2 | 1 | no | — |
| 174 | C2×#108 | — | 53 | 1 | no | — |
| 175 | #428 | — | 106 | 1 | no | — |
| 176 | #427 | C2×D53 | 1 | 1 | yes | characteristic |
| 177 | C2×#107 | — | 53 | 1 | no | — |
| 178 | #418 | — | 2 | 1 | no | — |
| 179 | #405 | C2×D53 | 1 | 1 | yes | characteristic |
| 180 | #367 | — | 106 | 1 | no | — |
| 181 | #366 | — | 106 | 1 | no | — |
| 182 | #361 | — | 53 | 4 | no | — |
| 183 | #357 | — | 106 | 2 | no | — |
| 184 | #343 | — | 106 | 1 | no | — |
| 185 | C53 | #12940 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 186 | C22×D4 | — | 2 | 1 | no | — |
| 187 | C2×D8 | — | 2 | 1 | no | — |
| 188 | C2×D8 | — | 2 | 1 | no | — |
| 189 | C2×D8 | — | 2 | 1 | no | — |
| 190 | C2×D8 | Dic106 | 1 | 2 | yes | — |
| 191 | C2×D8 | C4×D53 | 1 | 2 | yes | — |
| 192 | C2×D8 | C22⋊D53 | 1 | 2 | yes | — |
| 193 | C2×M16 | — | 106 | 1 | no | — |
| 194 | C2×M16 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 195 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 196 | C4×D4 | — | 2 | 2 | no | — |
| 197 | C4×D4 | — | 2 | 2 | no | — |
| 198 | C4×D4 | — | 4 | 1 | no | — |
| 199 | C4×D4 | — | 4 | 1 | no | — |
| 200 | #117 | — | 2 | 1 | no | — |
| 201 | #117 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 202 | C2×C4⋊C4 | — | 106 | 1 | no | — |
| 203 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 204 | M32 | — | 53 | 2 | no | — |
| 205 | M32 | — | 106 | 1 | no | — |
| 206 | C2×C16 | — | 106 | 1 | no | — |
| 207 | C2×C16 | — | 106 | 1 | no | — |
| 208 | C2×C16 | — | 106 | 1 | no | — |
| 209 | C2×C16 | — | 106 | 1 | no | — |
| 210 | #108 | — | 53 | 2 | no | — |
| 211 | #108 | — | 106 | 1 | no | — |
| 212 | #107 | — | 106 | 1 | no | — |
| 213 | #107 | — | 106 | 1 | no | — |
| 214 | #107 | — | 212 | 1 | no | — |
| 215 | #107 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 216 | #107 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 217 | #106 | — | 2 | 1 | no | — |
| 218 | #106 | — | 212 | 1 | no | — |
| 219 | #102 | — | 2 | 2 | no | — |
| 220 | #102 | — | 2 | 2 | no | — |
| 221 | #97 | — | 2 | 1 | no | — |
| 222 | C4×C8 | — | 2 | 1 | no | — |
| 223 | C24 | — | 4 | 1 | no | — |
| 224 | C2×D4 | — | 2 | 2 | no | — |
| 225 | C2×D4 | — | 2 | 2 | no | — |
| 226 | C2×D4 | — | 2 | 2 | no | — |
| 227 | C2×D4 | — | 4 | 1 | no | — |
| 228 | C2×D4 | — | 4 | 1 | no | — |
| 229 | C2×D4 | — | 4 | 1 | no | — |
| 230 | C22×C4 | — | 4 | 2 | no | — |
| 231 | C22×C4 | — | 4 | 2 | no | — |
| 232 | C22×C4 | — | 212 | 1 | no | — |
| 233 | C22×C4 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 234 | D8 | — | 2 | 2 | no | — |
| 235 | D8 | — | 2 | 2 | no | — |
| 236 | D8 | — | 2 | 2 | no | — |
| 237 | D8 | #11719919 | 1 | 4 | yes | — |
| 238 | M16 | — | 2 | 1 | no | — |
| 239 | M16 | — | 2 | 1 | no | — |
| 240 | M16 | — | 106 | 2 | no | — |
| 241 | M16 | — | 212 | 1 | no | — |
| 242 | C2×C8 | — | 2 | 1 | no | — |
| 243 | C2×C8 | — | 212 | 1 | no | — |
| 244 | C2×C8 | C2×Dic106 | 1 | 1 | yes | characteristic |
| 245 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 246 | C2×C8 | #11719945 | 1 | 1 | yes | characteristic |
| 247 | C2×C8 | #11719945 | 1 | 1 | yes | characteristic |
| 248 | C2×C8 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 250 | C4⋊C4 | — | 2 | 1 | no | — |
| 251 | C4⋊C4 | — | 2 | 1 | no | — |
| 252 | C4⋊C4 | — | 212 | 1 | no | — |
| 253 | C4⋊C4 | — | 212 | 1 | no | — |
| 254 | C4⋊C4 | #11719922 | 1 | 1 | yes | characteristic |
| 255 | C4⋊C4 | #11719922 | 1 | 1 | yes | characteristic |
| 256 | C22⋊C4 | — | 2 | 1 | no | — |
| 257 | C22⋊C4 | — | 4 | 2 | no | — |
| 258 | C22⋊C4 | — | 4 | 2 | no | — |
| 259 | C22⋊C4 | — | 4 | 1 | no | — |
| 260 | C42 | — | 2 | 1 | no | — |
| 261 | C42 | — | 4 | 1 | no | — |
| 262 | C16 | — | 106 | 2 | no | — |
| 263 | C16 | — | 212 | 1 | no | — |
| 264 | C23 | — | 4 | 2 | no | — |
| 265 | C23 | — | 4 | 2 | no | — |
| 266 | C23 | — | 4 | 2 | no | — |
| 267 | C23 | — | 16 | 1 | no | — |
| 268 | C23 | — | 16 | 1 | no | — |
| 269 | C23 | #13133243 | 1 | 1 | yes | characteristic |
| 270 | D4 | — | 2 | 4 | no | — |
| 271 | D4 | — | 4 | 2 | no | — |
| 272 | D4 | — | 4 | 2 | no | — |
| 273 | D4 | — | 4 | 2 | no | — |
| 274 | C2×C4 | — | 2 | 1 | no | — |
| 275 | C2×C4 | — | 2 | 1 | no | — |
| 276 | C2×C4 | — | 4 | 1 | no | — |
| 277 | C2×C4 | — | 4 | 1 | no | — |
| 278 | C2×C4 | — | 16 | 2 | no | — |
| 279 | C2×C4 | — | 16 | 2 | no | — |
| 280 | C2×C4 | — | 848 | 1 | no | — |
| 281 | C2×C4 | — | 424 | 1 | no | — |
| 282 | C2×C4 | C4×C22⋊D53 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | #13133186 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #13133196 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | C2×#11719922 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | #13133237 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | #13133247 | 1 | 1 | yes | characteristic |
| 288 | C8 | — | 4 | 1 | no | — |
| 289 | C8 | — | 212 | 2 | no | — |
| 290 | C8 | C2×#11719919 | 1 | 1 | yes | characteristic |
| 291 | C8 | C4×C22⋊D53 | 1 | 1 | yes | characteristic |
| 292 | C8 | #13133196 | 1 | 1 | yes | characteristic |
| 293 | C8 | #13133200 | 1 | 1 | yes | characteristic |
| 294 | C22 | — | 2 | 1 | no | — |
| 295 | C22 | — | 2 | 1 | no | — |
| 296 | C22 | — | 4 | 4 | no | — |
| 297 | C22 | — | 8 | 2 | no | — |
| 298 | C22 | — | 8 | 2 | no | — |
| 299 | C22 | — | 16 | 2 | no | — |
| 300 | C22 | — | 16 | 2 | no | — |
| 301 | C22 | #16810723 | 1 | 1 | yes | characteristic |
| 302 | C22 | #16810757 | 1 | 1 | yes | characteristic |
| 303 | C22 | #16810923 | 1 | 1 | yes | characteristic |
| 304 | C4 | — | 4 | 1 | no | — |
| 305 | C4 | — | 4 | 1 | no | — |
| 306 | C4 | — | 8 | 1 | no | — |
| 307 | C4 | — | 8 | 1 | no | — |
| 308 | C4 | — | 848 | 1 | no | — |
| 309 | C4 | #16810496 | 1 | 1 | yes | characteristic |
| 310 | C4 | C4×#11719922 | 1 | 1 | yes | characteristic |
| 311 | C4 | #16810785 | 1 | 1 | yes | characteristic |
| 312 | C4 | #16810903 | 1 | 1 | yes | characteristic |
| 313 | C2 | — | 2 | 1 | no | — |
| 314 | C2 | — | 2 | 1 | no | — |
| 315 | C2 | — | 8 | 4 | no | — |
| 316 | C2 | #20274962 | 1 | 1 | yes | characteristic |
| 317 | C2 | #20276672 | 1 | 1 | yes | characteristic |
| 318 | C2 | #20276686 | 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