← #34444697
Subgroups — #34444697
Order 20224.
464 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 | #23774059 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23765564 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #23765561 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23765506 | C2 | 1 | 2 | yes | maximal |
| 5 | #23765489 | C2 | 1 | 2 | yes | maximal |
| 6 | #23765450 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #23765447 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #23765159 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C79×#3177 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #19099175 | — | 2 | 1 | no | — |
| 11 | C2×#14579107 | C22 | 1 | 2 | yes | — |
| 12 | C2×#14579107 | C22 | 1 | 2 | yes | — |
| 13 | C2×#14579101 | C22 | 1 | 1 | yes | characteristic |
| 14 | C22×C79⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 15 | #19098965 | C22 | 1 | 1 | yes | characteristic |
| 16 | #19098943 | C22 | 1 | 1 | yes | characteristic |
| 17 | #19098932 | — | 2 | 1 | no | — |
| 18 | #19098821 | — | 2 | 1 | no | — |
| 19 | #19098804 | C22 | 1 | 2 | yes | — |
| 20 | #19098671 | — | 2 | 1 | no | — |
| 21 | C4×Dic316 | C22 | 1 | 2 | yes | — |
| 22 | #19098390 | C22 | 1 | 2 | yes | — |
| 23 | #19098388 | C22 | 1 | 2 | yes | — |
| 24 | C4⋊(C79⋊C16) | C22 | 1 | 2 | yes | — |
| 25 | C4⋊(C79⋊C16) | C22 | 1 | 2 | yes | — |
| 26 | #19098367 | C22 | 1 | 2 | yes | — |
| 27 | #19098367 | C22 | 1 | 2 | yes | — |
| 28 | #19098366 | C22 | 1 | 2 | yes | — |
| 29 | #19098366 | C22 | 1 | 2 | yes | — |
| 30 | #541×C79 | — | 2 | 1 | no | — |
| 31 | #499×C79 | C22 | 1 | 1 | yes | characteristic |
| 32 | #494×C79 | C22 | 1 | 1 | yes | characteristic |
| 33 | #491×C79 | — | 2 | 1 | no | — |
| 34 | #470×C79 | — | 2 | 1 | no | — |
| 35 | #467×C79 | C22 | 1 | 2 | yes | — |
| 36 | #441×C79 | — | 2 | 1 | no | — |
| 37 | C4×Q16×C79 | C22 | 1 | 2 | yes | — |
| 38 | #405×C79 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×Dic316 | — | 2 | 1 | no | — |
| 40 | C2×Dic316 | C23 | 1 | 2 | yes | — |
| 41 | #14579161 | — | 2 | 1 | no | — |
| 42 | #14579154 | — | 4 | 2 | no | — |
| 43 | #14579153 | — | 2 | 1 | no | — |
| 44 | #14579151 | — | 4 | 2 | no | — |
| 45 | #14579140 | — | 2 | 2 | no | — |
| 46 | C4×Dic158 | — | 2 | 2 | no | — |
| 47 | #14579107 | — | 4 | 2 | no | — |
| 48 | #14579107 | — | 4 | 2 | no | — |
| 49 | #14579101 | — | 4 | 1 | no | — |
| 50 | C2×C79⋊C16 | — | 4 | 1 | no | — |
| 51 | C2×C79⋊C16 | C23 | 1 | 2 | yes | — |
| 52 | C2×C79⋊C16 | C23 | 1 | 2 | yes | — |
| 53 | #14579095 | C23 | 1 | 2 | yes | — |
| 54 | #14579093 | — | 2 | 1 | no | — |
| 55 | #14579086 | — | 2 | 2 | no | — |
| 56 | #14579086 | — | 2 | 2 | no | — |
| 57 | C2×Q16×C79 | — | 2 | 1 | no | — |
| 58 | C2×Q16×C79 | C23 | 1 | 2 | yes | — |
| 59 | C2×M16×C79 | D4 | 1 | 1 | yes | characteristic |
| 60 | C22×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 61 | C4⋊Q8×C79 | — | 2 | 1 | no | — |
| 62 | #126×C79 | — | 4 | 2 | no | — |
| 63 | #125×C79 | — | 2 | 1 | no | — |
| 64 | #125×C79 | — | 4 | 2 | no | — |
| 65 | C22⋊Q8×C79 | — | 2 | 2 | no | — |
| 66 | C4×Q8×C79 | — | 2 | 2 | no | — |
| 67 | #117×C79 | D4 | 1 | 1 | yes | characteristic |
| 68 | #107×C79 | C23 | 1 | 2 | yes | — |
| 69 | #106×C79 | — | 2 | 1 | no | — |
| 70 | #103×C79 | — | 2 | 2 | no | — |
| 71 | #103×C79 | — | 2 | 2 | no | — |
| 72 | #97×C79 | D4 | 1 | 1 | yes | characteristic |
| 73 | #97×C79 | D4 | 1 | 1 | yes | characteristic |
| 74 | C4×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 75 | C4×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 76 | C2×Dic158 | — | 2 | 2 | no | — |
| 77 | Dic316 | — | 2 | 2 | no | — |
| 78 | Dic316 | — | 2 | 2 | no | — |
| 79 | Dic316 | — | 4 | 2 | no | — |
| 80 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 81 | C4⋊Dic79 | — | 4 | 2 | no | — |
| 82 | Dic79⋊C4 | — | 4 | 2 | no | — |
| 83 | Dic79⋊C4 | — | 4 | 2 | no | — |
| 84 | Dic79⋊C4 | — | 4 | 2 | no | — |
| 85 | C22⋊Dic79 | — | 4 | 2 | no | — |
| 86 | C4×Dic79 | — | 4 | 2 | no | — |
| 87 | C79⋊C16 | — | 2 | 2 | no | — |
| 88 | C79⋊C16 | — | 2 | 2 | no | — |
| 89 | C2×Q8×C79 | — | 2 | 2 | no | — |
| 90 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | Q16×C79 | — | 2 | 2 | no | — |
| 92 | Q16×C79 | — | 2 | 2 | no | — |
| 93 | Q16×C79 | — | 4 | 2 | no | — |
| 94 | M16×C79 | — | 4 | 1 | no | — |
| 95 | C2×C8×C79 | — | 2 | 1 | no | — |
| 96 | C2×C8×C79 | — | 2 | 1 | no | — |
| 97 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C2×C8×C79 | C24 | 1 | 1 | yes | characteristic |
| 101 | C4⋊C4×C79 | — | 2 | 2 | no | — |
| 102 | C4⋊C4×C79 | — | 4 | 2 | no | — |
| 103 | C4⋊C4×C79 | — | 4 | 2 | no | — |
| 104 | C4⋊C4×C79 | — | 4 | 2 | no | — |
| 105 | C4⋊C4×C79 | — | 4 | 2 | no | — |
| 106 | C4⋊C4×C79 | D8 | 1 | 2 | yes | — |
| 107 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 108 | C22⋊C4×C79 | D8 | 1 | 2 | yes | — |
| 109 | C42×C79 | — | 4 | 2 | no | — |
| 110 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C23×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 113 | Q8×C79 | — | 4 | 2 | no | — |
| 114 | Q8×C79 | — | 4 | 2 | no | — |
| 115 | C2×C4×C79 | — | 2 | 1 | no | — |
| 116 | C2×C4×C79 | — | 2 | 1 | no | — |
| 117 | C2×C4×C79 | — | 4 | 2 | no | — |
| 118 | C2×C4×C79 | — | 4 | 2 | no | — |
| 119 | C2×C4×C79 | C2×D8 | 1 | 2 | yes | — |
| 120 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C79 | C2×D8 | 1 | 2 | yes | — |
| 122 | C2×C4×C79 | C22×D4 | 1 | 1 | yes | characteristic |
| 123 | C2×Dic79 | — | 4 | 2 | no | — |
| 124 | C2×Dic79 | — | 4 | 2 | no | — |
| 125 | Dic158 | — | 4 | 2 | no | — |
| 126 | Dic158 | — | 4 | 2 | no | — |
| 127 | C8×C79 | — | 2 | 1 | no | — |
| 128 | C8×C79 | — | 2 | 1 | no | — |
| 129 | C8×C79 | — | 2 | 1 | no | — |
| 130 | C8×C79 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 131 | C8×C79 | 2−1+4 | 1 | 1 | yes | characteristic |
| 132 | C22×C79 | — | 2 | 2 | no | — |
| 133 | C22×C79 | — | 2 | 1 | no | — |
| 134 | C22×C79 | C22×D8 | 1 | 1 | yes | characteristic |
| 135 | C4×C79 | — | 2 | 2 | no | — |
| 136 | C4×C79 | — | 2 | 2 | no | — |
| 137 | C4×C79 | — | 2 | 1 | no | — |
| 138 | C4×C79 | — | 8 | 2 | no | — |
| 139 | C4×C79 | — | 8 | 2 | no | — |
| 140 | C4×C79 | #537 | 1 | 1 | yes | characteristic |
| 141 | C4×C79 | #575 | 1 | 1 | yes | characteristic |
| 142 | Dic79 | — | 8 | 2 | no | — |
| 143 | Dic79 | — | 8 | 2 | no | — |
| 144 | #33827 | — | 79 | 1 | no | maximal, Sylow |
| 145 | C2×C79 | — | 4 | 1 | no | — |
| 146 | C2×C79 | #3245 | 1 | 1 | yes | characteristic |
| 147 | C2×C79 | #3417 | 1 | 2 | yes | — |
| 148 | #3177 | — | 79 | 1 | no | — |
| 149 | #3177 | D79 | 1 | 1 | yes | characteristic |
| 150 | #2229 | — | 79 | 2 | no | — |
| 151 | #2229 | — | 79 | 2 | no | — |
| 152 | #2219 | — | 79 | 2 | no | — |
| 153 | #2216 | — | 79 | 2 | no | — |
| 154 | #2209 | — | 79 | 2 | no | — |
| 155 | #2209 | — | 79 | 2 | no | — |
| 156 | #2151 | — | 79 | 1 | no | — |
| 157 | C79 | #33827 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 158 | #541 | — | 2 | 1 | no | — |
| 159 | #541 | — | 158 | 1 | no | — |
| 160 | C2×Q32 | — | 79 | 2 | no | — |
| 161 | C2×Q32 | — | 79 | 2 | no | — |
| 162 | C2×M32 | — | 79 | 1 | no | — |
| 163 | C22×C16 | — | 79 | 1 | no | — |
| 164 | #499 | — | 79 | 1 | no | — |
| 165 | #499 | C2×D79 | 1 | 1 | yes | characteristic |
| 166 | #494 | — | 79 | 1 | no | — |
| 167 | #494 | C2×D79 | 1 | 1 | yes | characteristic |
| 168 | #491 | — | 2 | 1 | no | — |
| 169 | #491 | — | 158 | 1 | no | — |
| 170 | #470 | — | 2 | 1 | no | — |
| 171 | #470 | — | 158 | 1 | no | — |
| 172 | #467 | — | 79 | 2 | no | — |
| 173 | #467 | C2×D79 | 1 | 2 | yes | — |
| 174 | #441 | — | 2 | 1 | no | — |
| 175 | #441 | — | 158 | 1 | no | — |
| 176 | C4×Q16 | — | 79 | 2 | no | — |
| 177 | C4×Q16 | C2×D79 | 1 | 2 | yes | — |
| 178 | #405 | C2×D79 | 1 | 1 | yes | characteristic |
| 179 | #367 | — | 79 | 2 | no | — |
| 180 | #366 | — | 79 | 2 | no | — |
| 181 | C4⋊C16 | — | 79 | 2 | no | — |
| 182 | C4⋊C16 | — | 79 | 2 | no | — |
| 183 | #358 | — | 79 | 2 | no | — |
| 184 | #358 | — | 79 | 2 | no | — |
| 185 | #358 | — | 79 | 2 | no | — |
| 186 | #358 | — | 79 | 2 | no | — |
| 187 | C2×Q16 | — | 2 | 1 | no | — |
| 188 | C2×Q16 | — | 79 | 2 | no | — |
| 189 | C2×Q16 | — | 158 | 1 | no | — |
| 190 | C2×Q16 | C22×D79 | 1 | 2 | yes | — |
| 191 | C2×M16 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 192 | C22×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 193 | C4⋊Q8 | — | 2 | 1 | no | — |
| 194 | C4⋊Q8 | — | 158 | 1 | no | — |
| 195 | #126 | — | 4 | 2 | no | — |
| 196 | #126 | — | 316 | 2 | no | — |
| 197 | #125 | — | 2 | 1 | no | — |
| 198 | #125 | — | 4 | 2 | no | — |
| 199 | #125 | — | 316 | 2 | no | — |
| 200 | #125 | — | 158 | 1 | no | — |
| 201 | C22⋊Q8 | — | 2 | 2 | no | — |
| 202 | C22⋊Q8 | — | 158 | 2 | no | — |
| 203 | C4×Q8 | — | 2 | 2 | no | — |
| 204 | C4×Q8 | — | 158 | 2 | no | — |
| 205 | #117 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 206 | Q32 | — | 316 | 2 | no | — |
| 207 | Q32 | — | 316 | 2 | no | — |
| 208 | M32 | — | 316 | 1 | no | — |
| 209 | C2×C16 | — | 316 | 1 | no | — |
| 210 | C2×C16 | — | 79 | 2 | no | — |
| 211 | C2×C16 | — | 79 | 2 | no | — |
| 212 | #107 | — | 79 | 2 | no | — |
| 213 | #107 | C22×D79 | 1 | 2 | yes | — |
| 214 | #106 | — | 2 | 1 | no | — |
| 215 | #106 | — | 158 | 1 | no | — |
| 216 | #103 | — | 2 | 2 | no | — |
| 217 | #103 | — | 2 | 2 | no | — |
| 218 | #103 | — | 158 | 2 | no | — |
| 219 | #103 | — | 158 | 2 | no | — |
| 220 | #97 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 221 | #97 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 222 | C4×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 223 | C4×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 224 | C2×Q8 | — | 2 | 2 | no | — |
| 225 | C2×Q8 | — | 158 | 2 | no | — |
| 226 | C22×C4 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 227 | Q16 | — | 2 | 2 | no | — |
| 228 | Q16 | — | 2 | 2 | no | — |
| 229 | Q16 | — | 4 | 2 | no | — |
| 230 | Q16 | — | 316 | 2 | no | — |
| 231 | Q16 | — | 158 | 2 | no | — |
| 232 | Q16 | — | 158 | 2 | no | — |
| 233 | M16 | — | 4 | 1 | no | — |
| 234 | C2×C8 | — | 2 | 1 | no | — |
| 235 | C2×C8 | — | 2 | 1 | no | — |
| 236 | C2×C8 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | C23×D79 | 1 | 1 | yes | characteristic |
| 240 | C4⋊C4 | — | 2 | 2 | no | — |
| 241 | C4⋊C4 | — | 4 | 2 | no | — |
| 242 | C4⋊C4 | — | 4 | 2 | no | — |
| 243 | C4⋊C4 | — | 4 | 2 | no | — |
| 244 | C4⋊C4 | — | 4 | 2 | no | — |
| 245 | C4⋊C4 | — | 316 | 2 | no | — |
| 246 | C4⋊C4 | — | 316 | 2 | no | — |
| 247 | C4⋊C4 | — | 316 | 2 | no | — |
| 248 | C4⋊C4 | — | 316 | 2 | no | — |
| 249 | C4⋊C4 | — | 158 | 2 | no | — |
| 250 | C4⋊C4 | #11929609 | 1 | 2 | yes | — |
| 251 | C22⋊C4 | — | 4 | 2 | no | — |
| 252 | C22⋊C4 | — | 316 | 2 | no | — |
| 253 | C22⋊C4 | #11929609 | 1 | 2 | yes | — |
| 254 | C42 | — | 4 | 2 | no | — |
| 255 | C42 | — | 316 | 2 | no | — |
| 256 | C42 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 257 | C42 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 258 | C16 | — | 158 | 2 | no | — |
| 259 | C16 | — | 158 | 2 | no | — |
| 260 | C23 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 261 | Q8 | — | 4 | 2 | no | — |
| 262 | Q8 | — | 4 | 2 | no | — |
| 263 | Q8 | — | 316 | 2 | no | — |
| 264 | Q8 | — | 316 | 2 | no | — |
| 265 | C2×C4 | — | 2 | 1 | no | — |
| 266 | C2×C4 | — | 2 | 1 | no | — |
| 267 | C2×C4 | — | 4 | 2 | no | — |
| 268 | C2×C4 | — | 4 | 2 | no | — |
| 269 | C2×C4 | — | 316 | 2 | no | — |
| 270 | C2×C4 | — | 316 | 2 | no | — |
| 271 | C2×C4 | C2×#11929609 | 1 | 2 | yes | — |
| 272 | C2×C4 | C2×#11929609 | 1 | 2 | yes | — |
| 273 | C2×C4 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | C22×C22⋊D79 | 1 | 1 | yes | characteristic |
| 275 | C8 | — | 2 | 1 | no | — |
| 276 | C8 | — | 2 | 1 | no | — |
| 277 | C8 | — | 2 | 1 | no | — |
| 278 | C8 | #14579207 | 1 | 1 | yes | characteristic |
| 279 | C8 | #14579208 | 1 | 1 | yes | characteristic |
| 280 | C22 | — | 2 | 2 | no | — |
| 281 | C22 | — | 2 | 1 | no | — |
| 282 | C22 | C22×#11929609 | 1 | 1 | yes | characteristic |
| 283 | C4 | — | 2 | 2 | no | — |
| 284 | C4 | — | 2 | 2 | no | — |
| 285 | C4 | — | 2 | 1 | no | — |
| 286 | C4 | — | 8 | 2 | no | — |
| 287 | C4 | — | 8 | 2 | no | — |
| 288 | C4 | — | 632 | 2 | no | — |
| 289 | C4 | — | 632 | 2 | no | — |
| 290 | C4 | #19099141 | 1 | 1 | yes | characteristic |
| 291 | C4 | #19099361 | 1 | 1 | yes | characteristic |
| 292 | C2 | — | 4 | 1 | no | — |
| 293 | C2 | #23774707 | 1 | 1 | yes | characteristic |
| 294 | C2 | #23776551 | 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