← #27299329
Subgroups — #27299329
Order 13568.
438 conjugacy classes of subgroups.
The subgroup lattice has 296 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16810740 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16810430 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C16×Dic53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #20274596 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20272810 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20272809 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×#2138 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#13133224 | — | 2 | 1 | no | — |
| 9 | C22×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 10 | #16810740 | — | 2 | 2 | no | — |
| 11 | #16810740 | C4 | 1 | 4 | yes | complemented |
| 12 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#13133144 | — | 2 | 1 | no | — |
| 15 | #16810579 | — | 2 | 1 | no | — |
| 16 | C2×C8×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16810430 | C4 | 1 | 4 | yes | complemented |
| 18 | C16×Dic53 | — | 2 | 2 | no | — |
| 19 | #16810382 | — | 2 | 1 | no | — |
| 20 | C2×#135×C53 | C22 | 1 | 1 | yes | characteristic |
| 21 | C22×C16×C53 | C22 | 1 | 1 | yes | characteristic |
| 22 | C53×#427 | C22 | 1 | 1 | yes | characteristic |
| 23 | C53×#358 | C4 | 1 | 2 | yes | complemented |
| 24 | C53×#357 | C4 | 1 | 2 | yes | complemented |
| 25 | #13133224 | — | 2 | 4 | no | — |
| 26 | #13133224 | — | 4 | 2 | no | — |
| 27 | C2×#11719917 | — | 2 | 1 | no | — |
| 28 | C2×#11719917 | — | 2 | 1 | no | — |
| 29 | C2×#11719917 | — | 4 | 1 | no | — |
| 30 | C22×#88322 | — | 4 | 1 | no | — |
| 31 | C22×#88322 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×C4×Dic53 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×#11719908 | — | 2 | 1 | no | — |
| 34 | C2×#11719908 | — | 2 | 1 | no | — |
| 35 | C2×#11719908 | C2×C4 | 1 | 2 | yes | — |
| 36 | #13133153 | — | 2 | 1 | no | — |
| 37 | #13133153 | — | 2 | 1 | no | — |
| 38 | #13133153 | C2×C4 | 1 | 2 | yes | — |
| 39 | #13133153 | C2×C4 | 1 | 2 | yes | — |
| 40 | #13133144 | — | 2 | 4 | no | — |
| 41 | #13133128 | — | 4 | 1 | no | — |
| 42 | C4×#88322 | — | 4 | 1 | no | — |
| 43 | C8×Dic53 | C2×C4 | 1 | 2 | yes | — |
| 44 | C8×Dic53 | D4 | 1 | 2 | yes | — |
| 45 | C2×#41×C53 | — | 2 | 1 | no | — |
| 46 | #135×C53 | — | 2 | 1 | no | — |
| 47 | #135×C53 | — | 2 | 1 | no | — |
| 48 | #135×C53 | C2×C4 | 1 | 2 | yes | — |
| 49 | #135×C53 | C2×C4 | 1 | 2 | yes | — |
| 50 | C2×Q16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C2×SD16×C53 | — | 2 | 1 | no | — |
| 52 | C2×D8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 54 | #117×C53 | — | 2 | 1 | no | — |
| 55 | C2×C16×C53 | — | 2 | 1 | no | — |
| 56 | C2×C16×C53 | — | 2 | 1 | no | — |
| 57 | C2×C16×C53 | C2×C4 | 1 | 2 | yes | — |
| 58 | #107×C53 | C2×C4 | 1 | 2 | yes | — |
| 59 | #106×C53 | — | 2 | 1 | no | — |
| 60 | #41×C53 | — | 2 | 2 | no | — |
| 61 | #41×C53 | — | 2 | 2 | no | — |
| 62 | #41×C53 | — | 4 | 1 | no | — |
| 63 | #41×C53 | — | 4 | 1 | no | — |
| 64 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 65 | C2×D4×C53 | — | 2 | 1 | no | — |
| 66 | C2×D4×C53 | — | 4 | 1 | no | — |
| 67 | C22×C4×C53 | — | 4 | 1 | no | — |
| 68 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C22×Dic53 | — | 2 | 1 | no | — |
| 70 | Q16×C53 | — | 2 | 1 | no | — |
| 71 | Q16×C53 | C42 | 1 | 2 | yes | — |
| 72 | SD16×C53 | — | 2 | 2 | no | — |
| 73 | SD16×C53 | — | 2 | 2 | no | — |
| 74 | D8×C53 | — | 2 | 1 | no | — |
| 75 | D8×C53 | C42 | 1 | 2 | yes | — |
| 76 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 81 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 82 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 83 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 84 | C42×C53 | — | 4 | 1 | no | — |
| 85 | C4×Dic53 | — | 2 | 1 | no | — |
| 86 | C4×Dic53 | C22⋊C4 | 1 | 2 | yes | — |
| 87 | #11719917 | — | 2 | 2 | no | — |
| 88 | #11719917 | — | 2 | 2 | no | — |
| 89 | #11719917 | — | 4 | 1 | no | — |
| 90 | #11719917 | — | 4 | 1 | no | — |
| 91 | #11719917 | — | 4 | 4 | no | — |
| 92 | C2×#88322 | — | 4 | 1 | no | — |
| 93 | C2×#88322 | — | 4 | 1 | no | — |
| 94 | C2×#88322 | — | 4 | 4 | no | — |
| 95 | C2×#88322 | C22⋊C4 | 1 | 2 | yes | — |
| 96 | C2×#88322 | D8 | 1 | 1 | yes | characteristic |
| 97 | C2×#88322 | D8 | 1 | 1 | yes | characteristic |
| 98 | C2×#88322 | SD16 | 1 | 2 | yes | — |
| 99 | C16×C53 | — | 2 | 2 | no | — |
| 100 | #11719908 | — | 2 | 2 | no | — |
| 101 | C23×C53 | — | 4 | 1 | no | — |
| 102 | C23×C53 | #135 | 1 | 1 | yes | characteristic |
| 103 | Q8×C53 | — | 2 | 2 | no | — |
| 104 | Q8×C53 | — | 4 | 1 | no | — |
| 105 | D4×C53 | — | 2 | 2 | no | — |
| 106 | D4×C53 | — | 4 | 2 | no | — |
| 107 | D4×C53 | — | 4 | 2 | no | — |
| 108 | D4×C53 | — | 4 | 1 | no | — |
| 109 | C2×C4×C53 | — | 4 | 1 | no | — |
| 110 | C2×C4×C53 | — | 4 | 2 | no | — |
| 111 | C2×C4×C53 | — | 4 | 2 | no | — |
| 112 | C2×C4×C53 | — | 4 | 2 | no | — |
| 113 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C53 | C2×SD16 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 119 | C2×Dic53 | — | 2 | 4 | no | — |
| 120 | C2×Dic53 | — | 2 | 2 | no | — |
| 121 | C8×C53 | C2×C42 | 1 | 1 | yes | characteristic |
| 122 | C8×C53 | #117 | 1 | 1 | yes | characteristic |
| 123 | C8×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 124 | C8×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 125 | #88322 | — | 4 | 2 | no | — |
| 126 | #88322 | — | 4 | 2 | no | — |
| 127 | #88322 | #102 | 1 | 2 | yes | — |
| 128 | #88322 | #102 | 1 | 2 | yes | — |
| 129 | #12641 | — | 53 | 1 | no | maximal, Sylow |
| 130 | C22×C53 | — | 2 | 2 | no | — |
| 131 | C22×C53 | — | 4 | 2 | no | — |
| 132 | C22×C53 | — | 8 | 2 | no | — |
| 133 | C22×C53 | #416 | 1 | 1 | yes | characteristic |
| 134 | C22×C53 | C4×D8 | 1 | 1 | yes | characteristic |
| 135 | C22×C53 | C4×SD16 | 1 | 1 | yes | characteristic |
| 136 | C4×C53 | — | 4 | 2 | no | — |
| 137 | C4×C53 | — | 8 | 2 | no | — |
| 138 | C4×C53 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 139 | C4×C53 | C2×#102 | 1 | 1 | yes | characteristic |
| 140 | C4×C53 | C4×D8 | 1 | 1 | yes | characteristic |
| 141 | C4×C53 | C4×SD16 | 1 | 1 | yes | characteristic |
| 142 | Dic53 | — | 2 | 4 | no | — |
| 143 | C2×#443 | — | 53 | 1 | no | — |
| 144 | C2×#359 | — | 53 | 1 | no | — |
| 145 | #2138 | D53 | 1 | 1 | yes | characteristic |
| 146 | C2×C4×C16 | — | 53 | 1 | no | — |
| 147 | #1772 | — | 53 | 1 | no | — |
| 148 | #1382 | — | 53 | 1 | no | — |
| 149 | #1381 | — | 53 | 1 | no | — |
| 150 | C2×C53 | — | 2 | 1 | no | — |
| 151 | C2×C53 | — | 2 | 1 | no | — |
| 152 | C2×C53 | — | 8 | 2 | no | — |
| 153 | C2×C53 | C4×#102 | 1 | 1 | yes | characteristic |
| 154 | C2×C53 | #2178 | 1 | 2 | yes | — |
| 155 | C2×#135 | C2×D53 | 1 | 1 | yes | characteristic |
| 156 | C2×#131 | — | 106 | 1 | no | — |
| 157 | C22×C16 | — | 53 | 1 | no | — |
| 158 | C22×C16 | C2×D53 | 1 | 1 | yes | characteristic |
| 159 | #443 | — | 53 | 4 | no | — |
| 160 | #443 | — | 106 | 2 | no | — |
| 161 | C2×#108 | — | 53 | 1 | no | — |
| 162 | C2×#108 | — | 53 | 1 | no | — |
| 163 | #427 | C2×D53 | 1 | 1 | yes | characteristic |
| 164 | C2×#104 | — | 106 | 1 | no | — |
| 165 | #405 | — | 106 | 1 | no | — |
| 166 | C2×C4×C8 | — | 53 | 1 | no | — |
| 167 | #359 | — | 53 | 4 | no | — |
| 168 | #358 | Dic53 | 1 | 2 | yes | complemented |
| 169 | #357 | Dic53 | 1 | 2 | yes | complemented |
| 170 | C4×C16 | — | 106 | 2 | no | — |
| 171 | #339 | — | 106 | 1 | no | — |
| 172 | C53 | #12641 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 173 | C2×#41 | — | 2 | 1 | no | — |
| 174 | #135 | — | 2 | 1 | no | — |
| 175 | #135 | — | 2 | 1 | no | — |
| 176 | #135 | C4×D53 | 1 | 2 | yes | — |
| 177 | #135 | C4×D53 | 1 | 2 | yes | — |
| 178 | C2×Q16 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 179 | C2×SD16 | — | 2 | 1 | no | — |
| 180 | C2×D8 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 181 | #131 | — | 106 | 4 | no | — |
| 182 | #131 | — | 212 | 2 | no | — |
| 183 | C2×M16 | — | 106 | 1 | no | — |
| 184 | C2×M16 | — | 106 | 1 | no | — |
| 185 | C2×M16 | — | 212 | 1 | no | — |
| 186 | C22×C8 | — | 53 | 1 | no | — |
| 187 | C22×C8 | — | 212 | 1 | no | — |
| 188 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 189 | #117 | — | 2 | 1 | no | — |
| 190 | C2×C42 | — | 53 | 1 | no | — |
| 191 | C2×C16 | — | 2 | 1 | no | — |
| 192 | C2×C16 | — | 2 | 1 | no | — |
| 193 | C2×C16 | — | 53 | 2 | no | — |
| 194 | C2×C16 | — | 106 | 1 | no | — |
| 195 | C2×C16 | — | 106 | 1 | no | — |
| 196 | C2×C16 | C2×Dic53 | 1 | 2 | yes | — |
| 197 | #108 | — | 53 | 2 | no | — |
| 198 | #108 | — | 53 | 2 | no | — |
| 199 | #108 | — | 106 | 1 | no | — |
| 200 | #108 | — | 106 | 1 | no | — |
| 201 | #107 | C2×Dic53 | 1 | 2 | yes | — |
| 202 | #106 | — | 2 | 1 | no | — |
| 203 | #104 | — | 106 | 4 | no | — |
| 204 | #97 | — | 212 | 1 | no | — |
| 205 | C4×C8 | — | 53 | 2 | no | — |
| 206 | C4×C8 | — | 53 | 2 | no | — |
| 207 | C4×C8 | — | 212 | 1 | no | — |
| 208 | #41 | — | 2 | 2 | no | — |
| 209 | #41 | — | 2 | 2 | no | — |
| 210 | #41 | — | 4 | 1 | no | — |
| 211 | #41 | — | 4 | 1 | no | — |
| 212 | C2×Q8 | — | 2 | 1 | no | — |
| 213 | C2×D4 | — | 2 | 1 | no | — |
| 214 | C2×D4 | — | 4 | 1 | no | — |
| 215 | C22×C4 | — | 4 | 1 | no | — |
| 216 | C22×C4 | — | 106 | 1 | no | — |
| 217 | C22×C4 | D4×D53 | 1 | 1 | yes | characteristic |
| 218 | Q16 | — | 2 | 1 | no | — |
| 219 | Q16 | C4×Dic53 | 1 | 2 | yes | — |
| 220 | SD16 | — | 2 | 2 | no | — |
| 221 | SD16 | — | 2 | 2 | no | — |
| 222 | D8 | — | 2 | 1 | no | — |
| 223 | D8 | C4×Dic53 | 1 | 2 | yes | — |
| 224 | M16 | — | 106 | 2 | no | — |
| 225 | M16 | — | 106 | 2 | no | — |
| 226 | M16 | — | 212 | 4 | no | — |
| 227 | M16 | — | 212 | 1 | no | — |
| 228 | M16 | — | 212 | 1 | no | — |
| 229 | C2×C8 | — | 53 | 2 | no | — |
| 230 | C2×C8 | — | 53 | 1 | no | — |
| 231 | C2×C8 | — | 53 | 1 | no | — |
| 232 | C2×C8 | — | 53 | 2 | no | — |
| 233 | C2×C8 | — | 212 | 4 | no | — |
| 234 | C2×C8 | — | 212 | 1 | no | — |
| 235 | C2×C8 | — | 212 | 1 | no | — |
| 236 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 240 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 241 | C2×C8 | C22×Dic53 | 1 | 1 | yes | characteristic |
| 242 | C4⋊C4 | — | 2 | 2 | no | — |
| 243 | C22⋊C4 | — | 4 | 1 | no | — |
| 244 | C42 | — | 4 | 1 | no | — |
| 245 | C42 | — | 53 | 2 | no | — |
| 246 | C42 | — | 106 | 1 | no | — |
| 247 | C16 | — | 2 | 2 | no | — |
| 248 | C16 | — | 106 | 2 | no | — |
| 249 | C23 | — | 4 | 1 | no | — |
| 250 | C23 | #13133236 | 1 | 1 | yes | characteristic |
| 251 | Q8 | — | 2 | 2 | no | — |
| 252 | Q8 | — | 4 | 1 | no | — |
| 253 | D4 | — | 2 | 2 | no | — |
| 254 | D4 | — | 4 | 2 | no | — |
| 255 | D4 | — | 4 | 2 | no | — |
| 256 | D4 | — | 4 | 1 | no | — |
| 257 | C2×C4 | — | 4 | 1 | no | — |
| 258 | C2×C4 | — | 4 | 2 | no | — |
| 259 | C2×C4 | — | 4 | 2 | no | — |
| 260 | C2×C4 | — | 4 | 2 | no | — |
| 261 | C2×C4 | — | 106 | 4 | no | — |
| 262 | C2×C4 | — | 106 | 2 | no | — |
| 263 | C2×C4 | C22⋊C4×D53 | 1 | 1 | yes | characteristic |
| 264 | C2×C4 | #13133177 | 1 | 1 | yes | characteristic |
| 265 | C2×C4 | D4×Dic53 | 1 | 1 | yes | characteristic |
| 266 | C2×C4 | D8×D53 | 1 | 1 | yes | characteristic |
| 267 | C2×C4 | SD16×D53 | 1 | 1 | yes | characteristic |
| 268 | C2×C4 | #13133238 | 1 | 1 | yes | characteristic |
| 269 | C8 | — | 53 | 2 | no | — |
| 270 | C8 | — | 53 | 2 | no | — |
| 271 | C8 | — | 212 | 2 | no | — |
| 272 | C8 | — | 212 | 2 | no | — |
| 273 | C8 | C2×C4×Dic53 | 1 | 1 | yes | characteristic |
| 274 | C8 | #13133173 | 1 | 1 | yes | characteristic |
| 275 | C8 | #13133177 | 1 | 1 | yes | characteristic |
| 276 | C8 | D4×Dic53 | 1 | 1 | yes | characteristic |
| 277 | C22 | — | 2 | 2 | no | — |
| 278 | C22 | — | 4 | 2 | no | — |
| 279 | C22 | — | 8 | 2 | no | — |
| 280 | C22 | #16810616 | 1 | 1 | yes | characteristic |
| 281 | C22 | D8×Dic53 | 1 | 1 | yes | characteristic |
| 282 | C22 | #16810714 | 1 | 1 | yes | characteristic |
| 283 | C4 | — | 4 | 2 | no | — |
| 284 | C4 | — | 8 | 2 | no | — |
| 285 | C4 | — | 106 | 4 | no | — |
| 286 | C4 | C22⋊C4×Dic53 | 1 | 1 | yes | characteristic |
| 287 | C4 | #102×D53 | 1 | 1 | yes | characteristic |
| 288 | C4 | #16810703 | 1 | 1 | yes | characteristic |
| 289 | C4 | SD16×Dic53 | 1 | 1 | yes | characteristic |
| 290 | C2 | — | 2 | 1 | no | — |
| 291 | C2 | — | 2 | 1 | no | — |
| 292 | C2 | — | 8 | 2 | no | — |
| 293 | C2 | #102×Dic53 | 1 | 1 | yes | characteristic |
| 294 | C2 | #20276596 | 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