Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
670 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 75 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 | #23774295 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #23772779 | C2 | 1 | 2 | yes | maximal |
| 3 | #23765053 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23761523 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C79×#1794 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C2×#14579191 | — | 2 | 2 | no | — |
| 7 | #19099134 | C22 | 1 | 2 | yes | — |
| 8 | #19098905 | — | 2 | 2 | no | — |
| 9 | #19098821 | — | 2 | 1 | no | — |
| 10 | #19098804 | — | 2 | 1 | no | — |
| 11 | #19098746 | — | 2 | 4 | no | — |
| 12 | #19098741 | — | 2 | 4 | no | — |
| 13 | #19098721 | — | 2 | 2 | no | — |
| 14 | #19098580 | — | 2 | 2 | no | — |
| 15 | #19098547 | C22 | 1 | 2 | yes | — |
| 16 | #19098352 | — | 2 | 4 | no | — |
| 17 | #585×C79 | C22 | 1 | 1 | yes | characteristic |
| 18 | #421×C79 | — | 2 | 1 | no | — |
| 19 | #416×C79 | — | 2 | 1 | no | — |
| 20 | #410×C79 | C22 | 1 | 1 | yes | characteristic |
| 21 | #408×C79 | C22 | 1 | 1 | yes | characteristic |
| 22 | C22×Dic158 | D4 | 1 | 2 | yes | — |
| 23 | #14579191 | — | 2 | 8 | no | — |
| 24 | #14579191 | — | 4 | 4 | no | — |
| 25 | C2×Dic316 | — | 2 | 1 | no | — |
| 26 | C2×Dic316 | — | 2 | 1 | no | — |
| 27 | C2×#11929614 | — | 4 | 2 | no | — |
| 28 | C2×#11929612 | — | 4 | 2 | no | — |
| 29 | C2×#11929608 | — | 2 | 2 | no | — |
| 30 | #14579159 | — | 2 | 2 | no | — |
| 31 | #14579148 | — | 2 | 2 | no | — |
| 32 | #14579142 | — | 2 | 2 | no | — |
| 33 | #14579142 | — | 2 | 2 | no | — |
| 34 | #14579140 | — | 2 | 2 | no | — |
| 35 | #14579140 | — | 2 | 2 | no | — |
| 36 | #14579120 | D4 | 1 | 2 | yes | — |
| 37 | #14579095 | — | 2 | 1 | no | — |
| 38 | #14579093 | — | 2 | 1 | no | — |
| 39 | #14579088 | — | 2 | 8 | no | — |
| 40 | #14579086 | — | 2 | 2 | no | — |
| 41 | #14579086 | — | 2 | 2 | no | — |
| 42 | #14579080 | — | 2 | 2 | no | — |
| 43 | #14579078 | D4 | 1 | 4 | yes | — |
| 44 | 2−1+4×C79 | — | 4 | 1 | no | — |
| 45 | 2−1+4×C79 | D4 | 1 | 2 | yes | — |
| 46 | H(5,𝔽2)×C79 | — | 2 | 2 | no | — |
| 47 | H(5,𝔽2)×C79 | — | 2 | 1 | no | — |
| 48 | H(5,𝔽2)×C79 | — | 4 | 1 | no | — |
| 49 | C2×#41×C79 | — | 2 | 1 | no | — |
| 50 | C2×#41×C79 | — | 2 | 2 | no | — |
| 51 | C2×#41×C79 | — | 4 | 2 | no | — |
| 52 | C2×#41×C79 | C23 | 1 | 1 | yes | characteristic |
| 53 | C2×M16×C79 | D4 | 1 | 1 | yes | characteristic |
| 54 | C22×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 55 | #117×C79 | — | 2 | 1 | no | — |
| 56 | #104×C79 | — | 4 | 2 | no | — |
| 57 | #103×C79 | — | 4 | 1 | no | — |
| 58 | #102×C79 | — | 4 | 1 | no | — |
| 59 | #99×C79 | — | 2 | 2 | no | — |
| 60 | C22⋊C8×C79 | — | 2 | 2 | no | — |
| 61 | C2×Dic158 | — | 2 | 2 | no | — |
| 62 | C2×Dic158 | — | 2 | 2 | no | — |
| 63 | C2×Dic158 | — | 4 | 2 | no | — |
| 64 | C2×Dic158 | — | 4 | 2 | no | — |
| 65 | C22×Dic79 | — | 2 | 2 | no | — |
| 66 | Dic316 | — | 8 | 1 | no | — |
| 67 | Dic316 | — | 8 | 1 | no | — |
| 68 | #11929614 | — | 4 | 8 | no | — |
| 69 | #11929612 | — | 4 | 8 | no | — |
| 70 | #11929608 | — | 2 | 4 | no | — |
| 71 | #11929608 | — | 4 | 2 | no | — |
| 72 | C2×C79⋊C8 | — | 4 | 2 | no | — |
| 73 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 74 | Dic79⋊C4 | — | 4 | 2 | no | — |
| 75 | Dic79⋊C4 | — | 4 | 2 | no | — |
| 76 | C22⋊Dic79 | — | 2 | 2 | no | — |
| 77 | C22⋊Dic79 | — | 4 | 2 | no | — |
| 78 | C22⋊Dic79 | — | 4 | 2 | no | — |
| 79 | C4×Dic79 | — | 2 | 2 | no | — |
| 80 | #41×C79 | — | 2 | 2 | no | — |
| 81 | #41×C79 | — | 2 | 4 | no | — |
| 82 | #41×C79 | — | 2 | 4 | no | — |
| 83 | #41×C79 | — | 2 | 2 | no | — |
| 84 | #41×C79 | — | 4 | 1 | no | — |
| 85 | #41×C79 | — | 4 | 1 | no | — |
| 86 | #41×C79 | — | 4 | 2 | no | — |
| 87 | #41×C79 | — | 4 | 1 | no | — |
| 88 | #41×C79 | — | 4 | 2 | no | — |
| 89 | #41×C79 | — | 4 | 4 | no | — |
| 90 | #41×C79 | — | 4 | 2 | no | — |
| 91 | #41×C79 | — | 4 | 1 | no | — |
| 92 | C2×Q8×C79 | — | 2 | 1 | no | — |
| 93 | C2×Q8×C79 | — | 2 | 2 | no | — |
| 94 | C2×Q8×C79 | — | 4 | 2 | no | — |
| 95 | C2×Q8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×D4×C79 | — | 2 | 1 | no | — |
| 97 | C2×D4×C79 | — | 2 | 2 | no | — |
| 98 | C2×D4×C79 | — | 4 | 4 | no | — |
| 99 | C2×D4×C79 | — | 4 | 2 | no | — |
| 100 | C2×D4×C79 | — | 4 | 2 | no | — |
| 101 | C2×D4×C79 | — | 4 | 1 | no | — |
| 102 | C2×D4×C79 | C2×D4 | 1 | 2 | yes | — |
| 103 | C2×D4×C79 | #41 | 1 | 1 | yes | characteristic |
| 104 | C22×C4×C79 | — | 4 | 2 | no | — |
| 105 | C22×C4×C79 | — | 4 | 1 | no | — |
| 106 | C22×C4×C79 | C2×D4 | 1 | 2 | yes | — |
| 107 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 108 | M16×C79 | — | 4 | 1 | no | — |
| 109 | C2×C8×C79 | — | 2 | 1 | no | — |
| 110 | C2×C8×C79 | — | 2 | 1 | no | — |
| 111 | C2×C8×C79 | — | 4 | 1 | no | — |
| 112 | C4⋊C4×C79 | — | 4 | 1 | no | — |
| 113 | C22⋊C4×C79 | — | 4 | 1 | no | — |
| 114 | C42×C79 | — | 4 | 1 | no | — |
| 115 | C23×C79 | — | 4 | 1 | no | — |
| 116 | C23×C79 | — | 4 | 2 | no | — |
| 117 | C23×C79 | #121 | 1 | 2 | yes | — |
| 118 | C23×C79 | #127 | 1 | 1 | yes | characteristic |
| 119 | Q8×C79 | — | 2 | 4 | no | — |
| 120 | Q8×C79 | — | 2 | 2 | no | — |
| 121 | Q8×C79 | — | 4 | 1 | no | — |
| 122 | Q8×C79 | — | 4 | 1 | no | — |
| 123 | D4×C79 | — | 2 | 4 | no | — |
| 124 | D4×C79 | — | 2 | 2 | no | — |
| 125 | D4×C79 | — | 4 | 2 | no | — |
| 126 | D4×C79 | — | 4 | 1 | no | — |
| 127 | D4×C79 | — | 4 | 2 | no | — |
| 128 | D4×C79 | — | 4 | 4 | no | — |
| 129 | D4×C79 | — | 4 | 2 | no | — |
| 130 | D4×C79 | — | 4 | 1 | no | — |
| 131 | C2×C4×C79 | — | 2 | 2 | no | — |
| 132 | C2×C4×C79 | — | 2 | 2 | no | — |
| 133 | C2×C4×C79 | — | 2 | 1 | no | — |
| 134 | C2×C4×C79 | — | 2 | 1 | no | — |
| 135 | C2×C4×C79 | — | 4 | 4 | no | — |
| 136 | C2×C4×C79 | — | 4 | 2 | no | — |
| 137 | C2×C4×C79 | — | 4 | 1 | no | — |
| 138 | C2×C4×C79 | — | 4 | 2 | no | — |
| 139 | C2×C4×C79 | — | 4 | 2 | no | — |
| 140 | C2×C4×C79 | — | 8 | 1 | no | — |
| 141 | C2×C4×C79 | #120 | 1 | 1 | yes | characteristic |
| 142 | C2×C4×C79 | #120 | 1 | 2 | yes | — |
| 143 | C2×C4×C79 | #121 | 1 | 1 | yes | characteristic |
| 144 | C2×Dic79 | — | 4 | 2 | no | — |
| 145 | C2×Dic79 | — | 4 | 2 | no | — |
| 146 | C2×Dic79 | — | 8 | 2 | no | — |
| 147 | Dic158 | — | 4 | 4 | no | — |
| 148 | Dic158 | — | 4 | 2 | no | — |
| 149 | Dic158 | — | 4 | 2 | no | — |
| 150 | C8×C79 | — | 4 | 1 | no | — |
| 151 | C8×C79 | — | 4 | 1 | no | — |
| 152 | C79⋊C8 | — | 4 | 4 | no | — |
| 153 | C22×C79 | — | 2 | 1 | no | — |
| 154 | C22×C79 | — | 2 | 2 | no | — |
| 155 | C22×C79 | — | 4 | 2 | no | — |
| 156 | C22×C79 | — | 4 | 1 | no | — |
| 157 | C22×C79 | — | 4 | 2 | no | — |
| 158 | C22×C79 | — | 16 | 1 | no | — |
| 159 | C22×C79 | — | 16 | 2 | no | — |
| 160 | C22×C79 | #392 | 1 | 1 | yes | characteristic |
| 161 | C4×C79 | — | 2 | 2 | no | — |
| 162 | C4×C79 | — | 2 | 1 | no | — |
| 163 | C4×C79 | — | 4 | 2 | no | — |
| 164 | C4×C79 | — | 16 | 1 | no | — |
| 165 | C4×C79 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 166 | C4×C79 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 167 | Dic79 | — | 8 | 2 | no | — |
| 168 | Dic79 | — | 8 | 2 | no | — |
| 169 | #13362 | — | 79 | 1 | no | maximal, Sylow |
| 170 | C2×C79 | — | 2 | 1 | no | — |
| 171 | C2×C79 | — | 4 | 1 | no | — |
| 172 | C2×C79 | — | 4 | 2 | no | — |
| 173 | C2×C79 | — | 8 | 2 | no | — |
| 174 | C2×C79 | #2021 | 1 | 1 | yes | characteristic |
| 175 | #3202 | — | 79 | 1 | no | — |
| 176 | #3022 | — | 79 | 2 | no | — |
| 177 | #2123 | — | 79 | 2 | no | — |
| 178 | #1794 | D79 | 1 | 1 | yes | characteristic |
| 179 | #1392 | — | 79 | 1 | no | — |
| 180 | C79 | #13362 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 181 | #585 | C2×D79 | 1 | 1 | yes | characteristic |
| 182 | C2×#137 | — | 158 | 2 | no | — |
| 183 | #536 | — | 79 | 2 | no | — |
| 184 | #485 | — | 158 | 2 | no | — |
| 185 | #470 | — | 158 | 1 | no | — |
| 186 | #467 | — | 158 | 1 | no | — |
| 187 | #456 | — | 158 | 4 | no | — |
| 188 | #455 | — | 158 | 4 | no | — |
| 189 | #451 | — | 158 | 2 | no | — |
| 190 | #421 | — | 2 | 1 | no | — |
| 191 | #421 | — | 158 | 2 | no | — |
| 192 | #416 | — | 2 | 1 | no | — |
| 193 | #413 | — | 79 | 2 | no | — |
| 194 | #410 | C2×D79 | 1 | 1 | yes | characteristic |
| 195 | #408 | C2×D79 | 1 | 1 | yes | characteristic |
| 196 | #352 | — | 158 | 4 | no | — |
| 197 | 2−1+4 | — | 4 | 1 | no | — |
| 198 | 2−1+4 | D316 | 1 | 2 | yes | — |
| 199 | H(5,𝔽2) | — | 2 | 2 | no | — |
| 200 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 201 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 202 | C2×#41 | — | 2 | 2 | no | — |
| 203 | C2×#41 | — | 2 | 1 | no | — |
| 204 | C2×#41 | — | 4 | 2 | no | — |
| 205 | C2×#41 | C22×D79 | 1 | 1 | yes | characteristic |
| 206 | C22×Q8 | — | 79 | 2 | no | — |
| 207 | #137 | — | 316 | 4 | no | — |
| 208 | #137 | — | 158 | 8 | no | — |
| 209 | C2×Q16 | — | 316 | 2 | no | — |
| 210 | C2×Q16 | — | 158 | 1 | no | — |
| 211 | C2×Q16 | — | 158 | 1 | no | — |
| 212 | C2×SD16 | — | 316 | 2 | no | — |
| 213 | C2×M16 | — | 158 | 2 | no | — |
| 214 | C2×M16 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 215 | C22×C8 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 216 | C4⋊Q8 | — | 158 | 2 | no | — |
| 217 | #124 | — | 158 | 2 | no | — |
| 218 | #123 | — | 158 | 2 | no | — |
| 219 | #123 | — | 158 | 2 | no | — |
| 220 | C22⋊Q8 | — | 158 | 2 | no | — |
| 221 | C22⋊Q8 | — | 158 | 2 | no | — |
| 222 | #117 | — | 2 | 1 | no | — |
| 223 | #117 | — | 79 | 2 | no | — |
| 224 | #107 | — | 158 | 1 | no | — |
| 225 | #106 | — | 158 | 1 | no | — |
| 226 | #104 | — | 4 | 2 | no | — |
| 227 | #104 | — | 158 | 8 | no | — |
| 228 | #103 | — | 4 | 1 | no | — |
| 229 | #103 | — | 158 | 2 | no | — |
| 230 | #103 | — | 158 | 2 | no | — |
| 231 | #102 | — | 4 | 1 | no | — |
| 232 | #101 | — | 158 | 2 | no | — |
| 233 | #100 | — | 79 | 4 | no | — |
| 234 | #99 | — | 2 | 2 | no | — |
| 235 | C22⋊C8 | — | 2 | 2 | no | — |
| 236 | #41 | — | 2 | 2 | no | — |
| 237 | #41 | — | 2 | 4 | no | — |
| 238 | #41 | — | 2 | 4 | no | — |
| 239 | #41 | — | 2 | 2 | no | — |
| 240 | #41 | — | 4 | 1 | no | — |
| 241 | #41 | — | 4 | 1 | no | — |
| 242 | #41 | — | 4 | 1 | no | — |
| 243 | #41 | — | 4 | 2 | no | — |
| 244 | #41 | — | 4 | 2 | no | — |
| 245 | #41 | — | 4 | 4 | no | — |
| 246 | #41 | — | 4 | 2 | no | — |
| 247 | #41 | — | 4 | 1 | no | — |
| 248 | C2×Q8 | — | 2 | 1 | no | — |
| 249 | C2×Q8 | — | 2 | 2 | no | — |
| 250 | C2×Q8 | — | 4 | 2 | no | — |
| 251 | C2×Q8 | — | 316 | 2 | no | — |
| 252 | C2×Q8 | — | 316 | 2 | no | — |
| 253 | C2×Q8 | — | 158 | 2 | no | — |
| 254 | C2×Q8 | — | 158 | 2 | no | — |
| 255 | C2×Q8 | C2×D316 | 1 | 1 | yes | characteristic |
| 256 | C2×D4 | — | 2 | 1 | no | — |
| 257 | C2×D4 | — | 2 | 2 | no | — |
| 258 | C2×D4 | — | 4 | 4 | no | — |
| 259 | C2×D4 | — | 4 | 2 | no | — |
| 260 | C2×D4 | — | 4 | 2 | no | — |
| 261 | C2×D4 | — | 4 | 1 | no | — |
| 262 | C2×D4 | D4×D79 | 1 | 2 | yes | — |
| 263 | C2×D4 | #11929624 | 1 | 1 | yes | characteristic |
| 264 | C22×C4 | — | 4 | 1 | no | — |
| 265 | C22×C4 | — | 4 | 2 | no | — |
| 266 | C22×C4 | — | 158 | 2 | no | — |
| 267 | C22×C4 | D4×D79 | 1 | 2 | yes | — |
| 268 | C22×C4 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 269 | Q16 | — | 316 | 8 | no | — |
| 270 | Q16 | — | 632 | 1 | no | — |
| 271 | Q16 | — | 632 | 1 | no | — |
| 272 | SD16 | — | 316 | 8 | no | — |
| 273 | M16 | — | 4 | 1 | no | — |
| 274 | M16 | — | 316 | 2 | no | — |
| 275 | M16 | — | 158 | 4 | no | — |
| 276 | C2×C8 | — | 2 | 1 | no | — |
| 277 | C2×C8 | — | 2 | 1 | no | — |
| 278 | C2×C8 | — | 4 | 1 | no | — |
| 279 | C2×C8 | — | 316 | 2 | no | — |
| 280 | C4⋊C4 | — | 4 | 1 | no | — |
| 281 | C4⋊C4 | — | 316 | 2 | no | — |
| 282 | C4⋊C4 | — | 316 | 2 | no | — |
| 283 | C4⋊C4 | — | 158 | 2 | no | — |
| 284 | C22⋊C4 | — | 4 | 1 | no | — |
| 285 | C22⋊C4 | — | 316 | 2 | no | — |
| 286 | C22⋊C4 | — | 316 | 2 | no | — |
| 287 | C22⋊C4 | — | 158 | 2 | no | — |
| 288 | C42 | — | 4 | 1 | no | — |
| 289 | C42 | — | 158 | 2 | no | — |
| 290 | C23 | — | 4 | 1 | no | — |
| 291 | C23 | — | 4 | 2 | no | — |
| 292 | C23 | #14579134 | 1 | 2 | yes | — |
| 293 | C23 | #14579157 | 1 | 1 | yes | characteristic |
| 294 | Q8 | — | 2 | 4 | no | — |
| 295 | Q8 | — | 2 | 2 | no | — |
| 296 | Q8 | — | 4 | 1 | no | — |
| 297 | Q8 | — | 4 | 1 | no | — |
| 298 | Q8 | — | 316 | 4 | no | — |
| 299 | Q8 | — | 316 | 2 | no | — |
| 300 | Q8 | — | 316 | 2 | no | — |
| 301 | D4 | — | 2 | 4 | no | — |
| 302 | D4 | — | 2 | 2 | no | — |
| 303 | D4 | — | 4 | 2 | no | — |
| 304 | D4 | — | 4 | 1 | no | — |
| 305 | D4 | — | 4 | 2 | no | — |
| 306 | D4 | — | 4 | 4 | no | — |
| 307 | D4 | — | 4 | 2 | no | — |
| 308 | D4 | — | 4 | 1 | no | — |
| 309 | C2×C4 | — | 2 | 1 | no | — |
| 310 | C2×C4 | — | 2 | 1 | no | — |
| 311 | C2×C4 | — | 2 | 2 | no | — |
| 312 | C2×C4 | — | 2 | 2 | no | — |
| 313 | C2×C4 | — | 4 | 4 | no | — |
| 314 | C2×C4 | — | 4 | 2 | no | — |
| 315 | C2×C4 | — | 4 | 1 | no | — |
| 316 | C2×C4 | — | 4 | 2 | no | — |
| 317 | C2×C4 | — | 4 | 2 | no | — |
| 318 | C2×C4 | — | 8 | 1 | no | — |
| 319 | C2×C4 | — | 316 | 2 | no | — |
| 320 | C2×C4 | — | 316 | 2 | no | — |
| 321 | C2×C4 | — | 632 | 2 | no | — |
| 322 | C2×C4 | #14579129 | 1 | 1 | yes | characteristic |
| 323 | C2×C4 | #14579131 | 1 | 2 | yes | — |
| 324 | C2×C4 | #14579135 | 1 | 1 | yes | characteristic |
| 325 | C8 | — | 4 | 1 | no | — |
| 326 | C8 | — | 4 | 1 | no | — |
| 327 | C8 | — | 316 | 4 | no | — |
| 328 | C22 | — | 2 | 1 | no | — |
| 329 | C22 | — | 2 | 2 | no | — |
| 330 | C22 | — | 4 | 2 | no | — |
| 331 | C22 | — | 4 | 1 | no | — |
| 332 | C22 | — | 4 | 2 | no | — |
| 333 | C22 | — | 16 | 1 | no | — |
| 334 | C22 | — | 16 | 2 | no | — |
| 335 | C22 | #19098472 | 1 | 1 | yes | characteristic |
| 336 | C4 | — | 2 | 2 | no | — |
| 337 | C4 | — | 2 | 1 | no | — |
| 338 | C4 | — | 4 | 2 | no | — |
| 339 | C4 | — | 16 | 1 | no | — |
| 340 | C4 | — | 632 | 2 | no | — |
| 341 | C4 | — | 632 | 2 | no | — |
| 342 | C4 | #19098749 | 1 | 1 | yes | characteristic |
| 343 | C4 | #19098749 | 1 | 1 | yes | characteristic |
| 344 | C2 | — | 2 | 1 | no | — |
| 345 | C2 | — | 4 | 1 | no | — |
| 346 | C2 | — | 4 | 2 | no | — |
| 347 | C2 | — | 8 | 2 | no | — |
| 348 | C2 | #23764526 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23774295 | C2 | semidirect |
| 3. #23765053 | C2 | semidirect |
| 4. #23761523 | C2 | semidirect |
| 180. C79 | #13362 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23774295 | 2 |
| #23772779 | 2 |
| #23765053 | 2 |
| #23761523 | 2 |
| C79×#1794 | 2 |
| #13362 | 79 |