Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
418 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #23761328 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C79⋊#1304 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C79×#2021 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #19098419 | — | 2 | 1 | no | — |
| 5 | #19098287 | — | 2 | 1 | no | — |
| 6 | #534×C79 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×#120×C79 | — | 2 | 1 | no | — |
| 8 | C2×#99×C79 | — | 2 | 1 | no | — |
| 9 | #379×C79 | C22 | 1 | 1 | yes | characteristic |
| 10 | #342×C79 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×C22⋊Dic79 | — | 2 | 2 | no | — |
| 12 | C2×C22⋊Dic79 | — | 4 | 1 | no | — |
| 13 | C2×C22⋊Dic79 | — | 4 | 1 | no | — |
| 14 | C22⋊(C79⋊C8) | — | 4 | 1 | no | — |
| 15 | C25×C79 | D4 | 1 | 1 | yes | characteristic |
| 16 | C22×D4×C79 | — | 2 | 1 | no | — |
| 17 | C22×D4×C79 | — | 4 | 1 | no | — |
| 18 | #123×C79 | — | 4 | 1 | no | — |
| 19 | #121×C79 | — | 4 | 1 | no | — |
| 20 | #120×C79 | — | 4 | 2 | no | — |
| 21 | #120×C79 | — | 4 | 1 | no | — |
| 22 | #120×C79 | — | 4 | 1 | no | — |
| 23 | #120×C79 | — | 4 | 1 | no | — |
| 24 | C2×C22⋊C4×C79 | — | 2 | 1 | no | — |
| 25 | C2×C22⋊C4×C79 | — | 4 | 1 | no | — |
| 26 | C2×C22⋊C4×C79 | — | 4 | 1 | no | — |
| 27 | C2×C22⋊C4×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C2×C22⋊C4×C79 | D4 | 1 | 1 | yes | characteristic |
| 29 | #99×C79 | — | 4 | 2 | no | — |
| 30 | C22×Dic79 | — | 4 | 2 | no | — |
| 31 | C2×C79⋊C8 | — | 8 | 1 | no | — |
| 32 | C22⋊Dic79 | — | 8 | 2 | no | — |
| 33 | C22⋊Dic79 | — | 8 | 2 | no | — |
| 34 | C22⋊Dic79 | — | 8 | 1 | no | — |
| 35 | C22⋊Dic79 | — | 8 | 1 | no | — |
| 36 | C24×C79 | — | 2 | 1 | no | — |
| 37 | C24×C79 | — | 2 | 1 | no | — |
| 38 | C24×C79 | — | 2 | 1 | no | — |
| 39 | C24×C79 | — | 4 | 1 | no | — |
| 40 | C24×C79 | — | 4 | 1 | no | — |
| 41 | C24×C79 | — | 8 | 2 | no | — |
| 42 | C24×C79 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 43 | C24×C79 | D8 | 1 | 1 | yes | characteristic |
| 44 | C24×C79 | SD16 | 1 | 1 | yes | characteristic |
| 45 | C2×D4×C79 | — | 4 | 1 | no | — |
| 46 | C2×D4×C79 | — | 4 | 1 | no | — |
| 47 | C2×D4×C79 | — | 4 | 1 | no | — |
| 48 | C2×D4×C79 | — | 4 | 2 | no | — |
| 49 | C2×D4×C79 | — | 4 | 2 | no | — |
| 50 | C2×D4×C79 | — | 4 | 1 | no | — |
| 51 | C2×D4×C79 | — | 4 | 1 | no | — |
| 52 | C2×D4×C79 | — | 4 | 1 | no | — |
| 53 | C2×D4×C79 | — | 4 | 1 | no | — |
| 54 | C2×D4×C79 | — | 8 | 4 | no | — |
| 55 | C22×C4×C79 | — | 2 | 1 | no | — |
| 56 | C22×C4×C79 | — | 2 | 1 | no | — |
| 57 | C22×C4×C79 | — | 2 | 1 | no | — |
| 58 | C22×C4×C79 | — | 4 | 1 | no | — |
| 59 | C4⋊C4×C79 | — | 4 | 1 | no | — |
| 60 | C22⋊C4×C79 | — | 2 | 2 | no | — |
| 61 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 62 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 63 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 64 | C22⋊C4×C79 | — | 4 | 1 | no | — |
| 65 | C22⋊C4×C79 | — | 4 | 1 | no | — |
| 66 | C22⋊C4×C79 | — | 4 | 1 | no | — |
| 67 | C22⋊C4×C79 | — | 8 | 2 | no | — |
| 68 | C23×C79 | — | 2 | 1 | no | — |
| 69 | C23×C79 | — | 2 | 1 | no | — |
| 70 | C23×C79 | — | 2 | 1 | no | — |
| 71 | C23×C79 | — | 2 | 1 | no | — |
| 72 | C23×C79 | — | 4 | 2 | no | — |
| 73 | C23×C79 | — | 4 | 1 | no | — |
| 74 | C23×C79 | — | 4 | 1 | no | — |
| 75 | C23×C79 | — | 4 | 2 | no | — |
| 76 | C23×C79 | — | 4 | 2 | no | — |
| 77 | C23×C79 | — | 4 | 1 | no | — |
| 78 | C23×C79 | — | 4 | 1 | no | — |
| 79 | C23×C79 | — | 4 | 4 | no | — |
| 80 | C23×C79 | — | 4 | 1 | no | — |
| 81 | C23×C79 | — | 4 | 1 | no | — |
| 82 | C23×C79 | — | 4 | 1 | no | — |
| 83 | C23×C79 | — | 8 | 1 | no | — |
| 84 | C23×C79 | — | 8 | 1 | no | — |
| 85 | C23×C79 | — | 8 | 1 | no | — |
| 86 | C23×C79 | — | 8 | 1 | no | — |
| 87 | C23×C79 | — | 8 | 1 | no | — |
| 88 | C23×C79 | — | 8 | 2 | no | — |
| 89 | C23×C79 | — | 8 | 2 | no | — |
| 90 | C23×C79 | — | 8 | 2 | no | — |
| 91 | C23×C79 | — | 8 | 2 | no | — |
| 92 | C23×C79 | #99 | 1 | 1 | yes | characteristic |
| 93 | C23×C79 | #102 | 1 | 1 | yes | characteristic |
| 94 | C23×C79 | #104 | 1 | 1 | yes | characteristic |
| 95 | D4×C79 | — | 8 | 1 | no | — |
| 96 | D4×C79 | — | 8 | 1 | no | — |
| 97 | D4×C79 | — | 8 | 4 | no | — |
| 98 | D4×C79 | — | 8 | 4 | no | — |
| 99 | D4×C79 | — | 8 | 1 | no | — |
| 100 | D4×C79 | — | 8 | 1 | no | — |
| 101 | D4×C79 | — | 16 | 1 | no | — |
| 102 | C2×C4×C79 | — | 4 | 2 | no | — |
| 103 | C2×C4×C79 | — | 4 | 1 | no | — |
| 104 | C2×C4×C79 | — | 4 | 1 | no | — |
| 105 | C2×C4×C79 | — | 4 | 1 | no | — |
| 106 | C2×C4×C79 | — | 4 | 1 | no | — |
| 107 | C2×C4×C79 | — | 4 | 1 | no | — |
| 108 | C2×C4×C79 | — | 8 | 2 | no | — |
| 109 | C2×C4×C79 | — | 8 | 1 | no | — |
| 110 | C2×C4×C79 | — | 8 | 1 | no | — |
| 111 | C2×Dic79 | — | 8 | 2 | no | — |
| 112 | C2×Dic79 | — | 8 | 2 | no | — |
| 113 | C2×Dic79 | — | 8 | 2 | no | — |
| 114 | C79⋊C8 | — | 16 | 1 | no | — |
| 115 | C22×C79 | — | 2 | 2 | no | — |
| 116 | C22×C79 | — | 2 | 1 | no | — |
| 117 | C22×C79 | — | 2 | 1 | no | — |
| 118 | C22×C79 | — | 4 | 1 | no | — |
| 119 | C22×C79 | — | 4 | 1 | no | — |
| 120 | C22×C79 | — | 4 | 1 | no | — |
| 121 | C22×C79 | — | 4 | 2 | no | — |
| 122 | C22×C79 | — | 4 | 1 | no | — |
| 123 | C22×C79 | — | 4 | 1 | no | — |
| 124 | C22×C79 | — | 4 | 1 | no | — |
| 125 | C22×C79 | — | 4 | 2 | no | — |
| 126 | C22×C79 | — | 4 | 1 | no | — |
| 127 | C22×C79 | — | 4 | 2 | no | — |
| 128 | C22×C79 | — | 4 | 1 | no | — |
| 129 | C22×C79 | — | 8 | 1 | no | — |
| 130 | C22×C79 | — | 8 | 2 | no | — |
| 131 | C22×C79 | — | 8 | 2 | no | — |
| 132 | C22×C79 | — | 8 | 2 | no | — |
| 133 | C22×C79 | — | 8 | 1 | no | — |
| 134 | C22×C79 | — | 8 | 4 | no | — |
| 135 | C22×C79 | — | 8 | 1 | no | — |
| 136 | C22×C79 | — | 8 | 2 | no | — |
| 137 | C22×C79 | — | 8 | 1 | no | — |
| 138 | C22×C79 | — | 16 | 1 | no | — |
| 139 | C22×C79 | #327 | 1 | 1 | yes | characteristic |
| 140 | C22×C79 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 141 | C22×C79 | #353 | 1 | 1 | yes | characteristic |
| 142 | C4×C79 | — | 8 | 1 | no | — |
| 143 | C4×C79 | — | 8 | 1 | no | — |
| 144 | C4×C79 | — | 8 | 2 | no | — |
| 145 | C4×C79 | — | 8 | 1 | no | — |
| 146 | Dic79 | — | 16 | 2 | no | — |
| 147 | #7394 | — | 79 | 1 | no | maximal, Sylow |
| 148 | C2×C79 | — | 2 | 1 | no | — |
| 149 | C2×C79 | — | 2 | 1 | no | — |
| 150 | C2×C79 | — | 2 | 1 | no | — |
| 151 | C2×C79 | — | 4 | 2 | no | — |
| 152 | C2×C79 | — | 4 | 1 | no | — |
| 153 | C2×C79 | — | 4 | 1 | no | — |
| 154 | C2×C79 | — | 8 | 1 | no | — |
| 155 | C2×C79 | — | 16 | 1 | no | — |
| 156 | C2×C79 | #1343 | 1 | 1 | yes | characteristic |
| 157 | #2021 | D79 | 1 | 1 | yes | characteristic |
| 158 | #1316 | — | 79 | 1 | no | — |
| 159 | #1304 | — | 79 | 1 | no | — |
| 160 | C79 | #7394 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 161 | #534 | Dic79 | 1 | 1 | yes | characteristic, complemented |
| 162 | C2×#120 | — | 2 | 1 | no | — |
| 163 | C2×#99 | — | 2 | 1 | no | — |
| 164 | #379 | — | 158 | 1 | no | — |
| 165 | #379 | C2×D79 | 1 | 1 | yes | characteristic |
| 166 | #342 | Dic79 | 1 | 1 | yes | characteristic, complemented |
| 167 | #323 | — | 158 | 1 | no | — |
| 168 | C25 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 169 | C22×D4 | — | 2 | 1 | no | — |
| 170 | C22×D4 | — | 4 | 1 | no | — |
| 171 | #123 | — | 4 | 1 | no | — |
| 172 | #121 | — | 4 | 1 | no | — |
| 173 | #120 | — | 4 | 2 | no | — |
| 174 | #120 | — | 4 | 1 | no | — |
| 175 | #120 | — | 4 | 1 | no | — |
| 176 | #120 | — | 4 | 1 | no | — |
| 177 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 178 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 179 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 180 | C2×C22⋊C4 | — | 316 | 1 | no | — |
| 181 | C2×C22⋊C4 | — | 316 | 1 | no | — |
| 182 | C2×C22⋊C4 | — | 158 | 2 | no | — |
| 183 | C2×C22⋊C4 | C2×Dic79 | 1 | 1 | yes | characteristic |
| 184 | C2×C22⋊C4 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 185 | #99 | — | 4 | 2 | no | — |
| 186 | C22⋊C8 | — | 316 | 1 | no | — |
| 187 | C24 | — | 2 | 1 | no | — |
| 188 | C24 | — | 2 | 1 | no | — |
| 189 | C24 | — | 2 | 1 | no | — |
| 190 | C24 | — | 4 | 1 | no | — |
| 191 | C24 | — | 4 | 1 | no | — |
| 192 | C24 | — | 8 | 2 | no | — |
| 193 | C24 | C22⋊Dic79 | 1 | 1 | yes | characteristic |
| 194 | C24 | #11929609 | 1 | 1 | yes | characteristic |
| 195 | C24 | #11929612 | 1 | 1 | yes | characteristic |
| 196 | C2×D4 | — | 4 | 1 | no | — |
| 197 | C2×D4 | — | 4 | 1 | no | — |
| 198 | C2×D4 | — | 4 | 1 | no | — |
| 199 | C2×D4 | — | 4 | 2 | no | — |
| 200 | C2×D4 | — | 4 | 2 | no | — |
| 201 | C2×D4 | — | 4 | 1 | no | — |
| 202 | C2×D4 | — | 4 | 1 | no | — |
| 203 | C2×D4 | — | 4 | 1 | no | — |
| 204 | C2×D4 | — | 4 | 1 | no | — |
| 205 | C2×D4 | — | 8 | 4 | no | — |
| 206 | C22×C4 | — | 2 | 1 | no | — |
| 207 | C22×C4 | — | 2 | 1 | no | — |
| 208 | C22×C4 | — | 2 | 1 | no | — |
| 209 | C22×C4 | — | 4 | 1 | no | — |
| 210 | C22×C4 | — | 316 | 2 | no | — |
| 211 | C2×C8 | — | 632 | 1 | no | — |
| 212 | C4⋊C4 | — | 4 | 1 | no | — |
| 213 | C22⋊C4 | — | 2 | 2 | no | — |
| 214 | C22⋊C4 | — | 4 | 2 | no | — |
| 215 | C22⋊C4 | — | 4 | 2 | no | — |
| 216 | C22⋊C4 | — | 4 | 2 | no | — |
| 217 | C22⋊C4 | — | 4 | 1 | no | — |
| 218 | C22⋊C4 | — | 4 | 1 | no | — |
| 219 | C22⋊C4 | — | 4 | 1 | no | — |
| 220 | C22⋊C4 | — | 8 | 2 | no | — |
| 221 | C22⋊C4 | — | 632 | 2 | no | — |
| 222 | C22⋊C4 | — | 632 | 1 | no | — |
| 223 | C22⋊C4 | — | 632 | 1 | no | — |
| 224 | C22⋊C4 | — | 632 | 2 | no | — |
| 225 | C23 | — | 2 | 1 | no | — |
| 226 | C23 | — | 2 | 1 | no | — |
| 227 | C23 | — | 2 | 1 | no | — |
| 228 | C23 | — | 2 | 1 | no | — |
| 229 | C23 | — | 4 | 1 | no | — |
| 230 | C23 | — | 4 | 1 | no | — |
| 231 | C23 | — | 4 | 2 | no | — |
| 232 | C23 | — | 4 | 2 | no | — |
| 233 | C23 | — | 4 | 2 | no | — |
| 234 | C23 | — | 4 | 1 | no | — |
| 235 | C23 | — | 4 | 1 | no | — |
| 236 | C23 | — | 4 | 4 | no | — |
| 237 | C23 | — | 4 | 1 | no | — |
| 238 | C23 | — | 4 | 1 | no | — |
| 239 | C23 | — | 4 | 1 | no | — |
| 240 | C23 | — | 8 | 1 | no | — |
| 241 | C23 | — | 8 | 1 | no | — |
| 242 | C23 | — | 8 | 1 | no | — |
| 243 | C23 | — | 8 | 1 | no | — |
| 244 | C23 | — | 8 | 1 | no | — |
| 245 | C23 | — | 8 | 2 | no | — |
| 246 | C23 | — | 8 | 2 | no | — |
| 247 | C23 | — | 8 | 2 | no | — |
| 248 | C23 | — | 8 | 2 | no | — |
| 249 | C23 | #14579076 | 1 | 1 | yes | characteristic |
| 250 | C23 | #14579082 | 1 | 1 | yes | characteristic |
| 251 | C23 | #14579088 | 1 | 1 | yes | characteristic |
| 252 | D4 | — | 8 | 1 | no | — |
| 253 | D4 | — | 8 | 1 | no | — |
| 254 | D4 | — | 8 | 4 | no | — |
| 255 | D4 | — | 8 | 4 | no | — |
| 256 | D4 | — | 8 | 1 | no | — |
| 257 | D4 | — | 8 | 1 | no | — |
| 258 | D4 | — | 16 | 1 | no | — |
| 259 | C2×C4 | — | 4 | 2 | no | — |
| 260 | C2×C4 | — | 4 | 1 | no | — |
| 261 | C2×C4 | — | 4 | 1 | no | — |
| 262 | C2×C4 | — | 4 | 1 | no | — |
| 263 | C2×C4 | — | 4 | 1 | no | — |
| 264 | C2×C4 | — | 4 | 1 | no | — |
| 265 | C2×C4 | — | 8 | 2 | no | — |
| 266 | C2×C4 | — | 8 | 1 | no | — |
| 267 | C2×C4 | — | 8 | 1 | no | — |
| 268 | C2×C4 | — | 632 | 2 | no | — |
| 269 | C2×C4 | — | 632 | 2 | no | — |
| 270 | C2×C4 | — | 632 | 2 | no | — |
| 271 | C8 | — | 1264 | 1 | no | — |
| 272 | C22 | — | 2 | 2 | no | — |
| 273 | C22 | — | 2 | 1 | no | — |
| 274 | C22 | — | 2 | 1 | no | — |
| 275 | C22 | — | 4 | 1 | no | — |
| 276 | C22 | — | 4 | 1 | no | — |
| 277 | C22 | — | 4 | 1 | no | — |
| 278 | C22 | — | 4 | 2 | no | — |
| 279 | C22 | — | 4 | 1 | no | — |
| 280 | C22 | — | 4 | 1 | no | — |
| 281 | C22 | — | 4 | 1 | no | — |
| 282 | C22 | — | 4 | 2 | no | — |
| 283 | C22 | — | 4 | 1 | no | — |
| 284 | C22 | — | 4 | 2 | no | — |
| 285 | C22 | — | 4 | 1 | no | — |
| 286 | C22 | — | 8 | 1 | no | — |
| 287 | C22 | — | 8 | 2 | no | — |
| 288 | C22 | — | 8 | 2 | no | — |
| 289 | C22 | — | 8 | 2 | no | — |
| 290 | C22 | — | 8 | 1 | no | — |
| 291 | C22 | — | 8 | 4 | no | — |
| 292 | C22 | — | 8 | 1 | no | — |
| 293 | C22 | — | 8 | 2 | no | — |
| 294 | C22 | — | 8 | 1 | no | — |
| 295 | C22 | — | 16 | 1 | no | — |
| 296 | C22 | #19098297 | 1 | 1 | yes | characteristic |
| 297 | C22 | #19098350 | 1 | 1 | yes | characteristic |
| 298 | C22 | #19098356 | 1 | 1 | yes | characteristic |
| 299 | C4 | — | 8 | 1 | no | — |
| 300 | C4 | — | 8 | 1 | no | — |
| 301 | C4 | — | 8 | 2 | no | — |
| 302 | C4 | — | 8 | 1 | no | — |
| 303 | C4 | — | 1264 | 2 | no | — |
| 304 | C2 | — | 2 | 1 | no | — |
| 305 | C2 | — | 2 | 1 | no | — |
| 306 | C2 | — | 2 | 1 | no | — |
| 307 | C2 | — | 4 | 1 | no | — |
| 308 | C2 | — | 4 | 2 | no | — |
| 309 | C2 | — | 4 | 1 | no | — |
| 310 | C2 | — | 8 | 1 | no | — |
| 311 | C2 | — | 16 | 1 | no | — |
| 312 | C2 | #23761397 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23761328 | C2 | semidirect |
| 2. C79⋊#1304 | C2 | semidirect |
| 6. #534×C79 | C4 | semidirect |
| 10. #342×C79 | C4 | semidirect |
| 160. C79 | #7394 | semidirect |
| 161. #534 | Dic79 | semidirect |
| 166. #342 | Dic79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23761328 | 2 |
| C79⋊#1304 | 2 |
| C79×#2021 | 2 |
| #7394 | 79 |