Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
398 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 43 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 | #23775193 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #23765426 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #23761567 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #23761555 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C79×#1624 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #19099369 | — | 2 | 2 | no | — |
| 7 | #19098823 | — | 2 | 2 | no | — |
| 8 | #19098745 | C22 | 1 | 2 | yes | — |
| 9 | #19098735 | — | 2 | 2 | no | — |
| 10 | #19098730 | — | 2 | 1 | no | — |
| 11 | #19098681 | — | 2 | 1 | no | — |
| 12 | #19098362 | C22 | 1 | 2 | yes | complemented |
| 13 | #19098360 | — | 2 | 2 | no | — |
| 14 | #555×C79 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #492×C79 | C22 | 1 | 1 | yes | characteristic |
| 16 | #459×C79 | — | 2 | 1 | no | — |
| 17 | #333×C79 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | D4⋊C8×C79 | — | 2 | 1 | no | — |
| 19 | #14579209 | D4 | 1 | 2 | yes | — |
| 20 | #14579206 | — | 2 | 1 | no | — |
| 21 | #14579192 | — | 2 | 2 | no | — |
| 22 | #14579190 | — | 4 | 2 | no | — |
| 23 | #14579188 | — | 4 | 2 | no | — |
| 24 | #14579187 | — | 2 | 2 | no | — |
| 25 | #14579187 | — | 4 | 2 | no | — |
| 26 | #14579183 | — | 2 | 1 | no | — |
| 27 | #14579182 | — | 4 | 2 | no | — |
| 28 | SD16×D79 | — | 4 | 2 | no | — |
| 29 | #14579167 | — | 2 | 2 | no | — |
| 30 | #14579097 | — | 2 | 1 | no | — |
| 31 | #14579089 | — | 2 | 2 | no | — |
| 32 | #14579089 | — | 4 | 2 | no | — |
| 33 | #14579080 | — | 2 | 2 | no | — |
| 34 | #14579080 | D4 | 1 | 2 | yes | — |
| 35 | #14579078 | — | 2 | 1 | no | — |
| 36 | C2×SD16×C79 | — | 2 | 2 | no | — |
| 37 | C2×D8×C79 | — | 4 | 1 | no | — |
| 38 | C4⋊Q8×C79 | C23 | 1 | 1 | yes | characteristic |
| 39 | #127×C79 | D4 | 1 | 1 | yes | characteristic |
| 40 | #124×C79 | — | 2 | 2 | no | — |
| 41 | #121×C79 | — | 4 | 1 | no | — |
| 42 | C4×D4×C79 | — | 2 | 1 | no | — |
| 43 | #117×C79 | — | 4 | 1 | no | — |
| 44 | C4⋊C8×C79 | — | 2 | 1 | no | — |
| 45 | #102×C79 | — | 4 | 1 | no | — |
| 46 | C4×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 47 | #11929625 | — | 4 | 2 | no | — |
| 48 | #11929624 | — | 2 | 2 | no | — |
| 49 | #11929624 | — | 4 | 2 | no | — |
| 50 | #11929623 | — | 4 | 2 | no | — |
| 51 | Q8×D79 | — | 4 | 2 | no | — |
| 52 | C2×C22⋊D79 | — | 8 | 1 | no | — |
| 53 | D4×D79 | — | 4 | 2 | no | — |
| 54 | Dic316 | — | 2 | 1 | no | — |
| 55 | #11929614 | — | 4 | 2 | no | — |
| 56 | #11929613 | — | 2 | 2 | no | — |
| 57 | #11929612 | — | 4 | 2 | no | — |
| 58 | #11929612 | — | 8 | 2 | no | — |
| 59 | #11929611 | — | 4 | 2 | no | — |
| 60 | D632 | — | 2 | 1 | no | — |
| 61 | #11929609 | — | 4 | 2 | no | — |
| 62 | #11929609 | — | 8 | 2 | no | — |
| 63 | #11929608 | — | 2 | 2 | no | — |
| 64 | #11929608 | — | 4 | 2 | no | — |
| 65 | #11929607 | — | 4 | 2 | no | — |
| 66 | C8×D79 | — | 4 | 2 | no | — |
| 67 | C2×Q8×C79 | C2×D4 | 1 | 2 | yes | — |
| 68 | C2×D4×C79 | — | 2 | 1 | no | — |
| 69 | C2×D4×C79 | — | 2 | 2 | no | — |
| 70 | C2×D4×C79 | — | 2 | 1 | no | — |
| 71 | C2×D4×C79 | — | 4 | 1 | no | — |
| 72 | C22×C4×C79 | — | 4 | 1 | no | — |
| 73 | SD16×C79 | — | 4 | 2 | no | — |
| 74 | SD16×C79 | — | 4 | 2 | no | — |
| 75 | D8×C79 | — | 8 | 2 | no | — |
| 76 | C2×C8×C79 | — | 2 | 1 | no | — |
| 77 | C2×C8×C79 | — | 4 | 1 | no | — |
| 78 | C4⋊C4×C79 | — | 2 | 1 | no | — |
| 79 | C4⋊C4×C79 | — | 4 | 1 | no | — |
| 80 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 81 | C22⋊C4×C79 | — | 4 | 1 | no | — |
| 82 | C42×C79 | — | 2 | 2 | no | — |
| 83 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C23×C79 | — | 4 | 1 | no | — |
| 85 | C23×C79 | — | 4 | 1 | no | — |
| 86 | C22×D79 | — | 8 | 1 | no | — |
| 87 | Q8×C79 | — | 4 | 2 | no | — |
| 88 | D4×C79 | — | 4 | 2 | no | — |
| 89 | D4×C79 | — | 4 | 1 | no | — |
| 90 | D4×C79 | — | 4 | 1 | no | — |
| 91 | D4×C79 | — | 8 | 1 | no | — |
| 92 | D4×C79 | — | 8 | 1 | no | — |
| 93 | D4×C79 | — | 8 | 2 | no | — |
| 94 | C2×C4×C79 | — | 2 | 1 | no | — |
| 95 | C2×C4×C79 | — | 2 | 2 | no | — |
| 96 | C2×C4×C79 | — | 4 | 1 | no | — |
| 97 | C2×C4×C79 | — | 8 | 1 | no | — |
| 98 | C2×C4×C79 | — | 8 | 1 | no | — |
| 99 | C2×C4×C79 | #120 | 1 | 1 | yes | characteristic |
| 100 | C22⋊D79 | — | 4 | 2 | no | — |
| 101 | C22⋊D79 | — | 8 | 2 | no | — |
| 102 | C2×Dic79 | — | 8 | 1 | no | — |
| 103 | D316 | — | 2 | 2 | no | — |
| 104 | D316 | — | 4 | 2 | no | — |
| 105 | C4×D79 | — | 4 | 2 | no | — |
| 106 | C4×D79 | — | 8 | 2 | no | — |
| 107 | Dic158 | — | 2 | 2 | no | — |
| 108 | Dic158 | — | 4 | 2 | no | — |
| 109 | C8×C79 | — | 2 | 1 | no | — |
| 110 | C8×C79 | — | 2 | 1 | no | — |
| 111 | C8×C79 | — | 8 | 1 | no | — |
| 112 | C79⋊C8 | — | 4 | 2 | no | — |
| 113 | C79⋊C8 | — | 8 | 2 | no | — |
| 114 | C22×C79 | — | 8 | 1 | no | — |
| 115 | C22×C79 | — | 8 | 1 | no | — |
| 116 | C22×C79 | — | 8 | 1 | no | — |
| 117 | C22×C79 | — | 8 | 1 | no | — |
| 118 | C22×C79 | — | 16 | 1 | no | — |
| 119 | C22×C79 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 120 | C2×D79 | — | 4 | 2 | no | — |
| 121 | C2×D79 | — | 32 | 1 | no | — |
| 122 | C4×C79 | — | 2 | 1 | no | — |
| 123 | C4×C79 | — | 2 | 1 | no | — |
| 124 | C4×C79 | — | 2 | 1 | no | — |
| 125 | C4×C79 | — | 4 | 2 | no | — |
| 126 | C4×C79 | — | 8 | 1 | no | — |
| 127 | Dic79 | — | 4 | 2 | no | — |
| 128 | #13681 | — | 79 | 1 | no | maximal, Sylow |
| 129 | C2×C79 | — | 2 | 1 | no | — |
| 130 | C2×C79 | — | 16 | 1 | no | — |
| 131 | C2×C79 | — | 16 | 1 | no | — |
| 132 | C2×C79 | #2196 | 1 | 1 | yes | characteristic |
| 133 | D79 | — | 8 | 2 | no | — |
| 134 | #3291 | — | 79 | 1 | no | — |
| 135 | #2204 | — | 79 | 2 | no | — |
| 136 | #1624 | D79 | 1 | 1 | yes | characteristic, complemented |
| 137 | #1410 | — | 79 | 1 | no | — |
| 138 | #1405 | — | 79 | 2 | no | — |
| 139 | C79 | #13681 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 140 | #577 | — | 158 | 2 | no | — |
| 141 | #555 | C2×D79 | 1 | 1 | yes | characteristic, complemented |
| 142 | #492 | C2×D79 | 1 | 1 | yes | characteristic |
| 143 | #471 | — | 158 | 2 | no | — |
| 144 | #459 | — | 2 | 1 | no | — |
| 145 | #456 | — | 79 | 2 | no | — |
| 146 | #454 | — | 158 | 2 | no | — |
| 147 | #453 | — | 158 | 1 | no | — |
| 148 | #443 | — | 158 | 1 | no | — |
| 149 | #356 | — | 79 | 2 | no | — |
| 150 | #355 | — | 158 | 2 | no | — |
| 151 | #333 | C2×D79 | 1 | 1 | yes | characteristic, complemented |
| 152 | D4⋊C8 | — | 2 | 1 | no | — |
| 153 | 2−1+4 | — | 79 | 2 | no | — |
| 154 | H(5,𝔽2) | — | 158 | 1 | no | — |
| 155 | #137 | — | 316 | 2 | no | — |
| 156 | #137 | — | 158 | 2 | no | — |
| 157 | AGL(1,ℤ/8) | — | 316 | 2 | no | — |
| 158 | AGL(1,ℤ/8) | — | 316 | 2 | no | — |
| 159 | AGL(1,ℤ/8) | — | 158 | 2 | no | — |
| 160 | #135 | — | 316 | 2 | no | — |
| 161 | #135 | — | 158 | 1 | no | — |
| 162 | C2×SD16 | — | 2 | 2 | no | — |
| 163 | C2×SD16 | — | 316 | 2 | no | — |
| 164 | C2×D8 | — | 4 | 1 | no | — |
| 165 | #131 | — | 158 | 2 | no | — |
| 166 | C4⋊Q8 | C22×D79 | 1 | 1 | yes | characteristic |
| 167 | #127 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 168 | #124 | — | 2 | 2 | no | — |
| 169 | #121 | — | 4 | 1 | no | — |
| 170 | C4×D4 | — | 2 | 1 | no | — |
| 171 | #117 | — | 4 | 1 | no | — |
| 172 | #108 | — | 158 | 1 | no | — |
| 173 | C4⋊C8 | — | 2 | 1 | no | — |
| 174 | #104 | — | 316 | 2 | no | — |
| 175 | #104 | — | 158 | 2 | no | — |
| 176 | #102 | — | 4 | 1 | no | — |
| 177 | #101 | — | 79 | 2 | no | — |
| 178 | #101 | — | 158 | 2 | no | — |
| 179 | #100 | — | 158 | 1 | no | — |
| 180 | C4×C8 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 181 | #41 | — | 316 | 2 | no | — |
| 182 | #41 | — | 316 | 2 | no | — |
| 183 | #41 | — | 316 | 2 | no | — |
| 184 | #41 | — | 158 | 2 | no | — |
| 185 | C2×Q8 | — | 316 | 2 | no | — |
| 186 | C2×Q8 | D4×D79 | 1 | 2 | yes | — |
| 187 | C2×D4 | — | 2 | 1 | no | — |
| 188 | C2×D4 | — | 2 | 2 | no | — |
| 189 | C2×D4 | — | 2 | 1 | no | — |
| 190 | C2×D4 | — | 4 | 1 | no | — |
| 191 | C2×D4 | — | 316 | 2 | no | — |
| 192 | C2×D4 | — | 632 | 1 | no | — |
| 193 | C22×C4 | — | 4 | 1 | no | — |
| 194 | Q16 | — | 316 | 2 | no | — |
| 195 | Q16 | — | 158 | 1 | no | — |
| 196 | SD16 | — | 4 | 2 | no | — |
| 197 | SD16 | — | 4 | 2 | no | — |
| 198 | SD16 | — | 316 | 2 | no | — |
| 199 | SD16 | — | 316 | 2 | no | — |
| 200 | SD16 | — | 632 | 2 | no | — |
| 201 | SD16 | — | 158 | 2 | no | — |
| 202 | D8 | — | 8 | 2 | no | — |
| 203 | D8 | — | 316 | 2 | no | — |
| 204 | D8 | — | 632 | 2 | no | — |
| 205 | D8 | — | 158 | 1 | no | — |
| 206 | M16 | — | 316 | 2 | no | — |
| 207 | M16 | — | 316 | 2 | no | — |
| 208 | M16 | — | 158 | 2 | no | — |
| 209 | C2×C8 | — | 2 | 1 | no | — |
| 210 | C2×C8 | — | 4 | 1 | no | — |
| 211 | C2×C8 | — | 316 | 2 | no | — |
| 212 | C4⋊C4 | — | 2 | 1 | no | — |
| 213 | C4⋊C4 | — | 4 | 1 | no | — |
| 214 | C22⋊C4 | — | 4 | 2 | no | — |
| 215 | C22⋊C4 | — | 4 | 1 | no | — |
| 216 | C42 | — | 2 | 2 | no | — |
| 217 | C42 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 218 | C23 | — | 4 | 1 | no | — |
| 219 | C23 | — | 4 | 1 | no | — |
| 220 | C23 | — | 632 | 1 | no | — |
| 221 | Q8 | — | 4 | 2 | no | — |
| 222 | Q8 | — | 316 | 2 | no | — |
| 223 | Q8 | — | 158 | 2 | no | — |
| 224 | D4 | — | 4 | 1 | no | — |
| 225 | D4 | — | 4 | 2 | no | — |
| 226 | D4 | — | 4 | 1 | no | — |
| 227 | D4 | — | 8 | 1 | no | — |
| 228 | D4 | — | 8 | 1 | no | — |
| 229 | D4 | — | 8 | 2 | no | — |
| 230 | D4 | — | 316 | 2 | no | — |
| 231 | D4 | — | 316 | 2 | no | — |
| 232 | D4 | — | 632 | 2 | no | — |
| 233 | D4 | — | 158 | 2 | no | — |
| 234 | C2×C4 | — | 2 | 1 | no | — |
| 235 | C2×C4 | — | 2 | 2 | no | — |
| 236 | C2×C4 | — | 4 | 1 | no | — |
| 237 | C2×C4 | — | 8 | 1 | no | — |
| 238 | C2×C4 | — | 8 | 1 | no | — |
| 239 | C2×C4 | — | 316 | 2 | no | — |
| 240 | C2×C4 | — | 632 | 2 | no | — |
| 241 | C2×C4 | — | 632 | 1 | no | — |
| 242 | C2×C4 | #14579129 | 1 | 1 | yes | characteristic |
| 243 | C8 | — | 2 | 1 | no | — |
| 244 | C8 | — | 2 | 1 | no | — |
| 245 | C8 | — | 8 | 1 | no | — |
| 246 | C8 | — | 316 | 2 | no | — |
| 247 | C8 | — | 632 | 2 | no | — |
| 248 | C22 | — | 8 | 1 | no | — |
| 249 | C22 | — | 8 | 1 | no | — |
| 250 | C22 | — | 8 | 1 | no | — |
| 251 | C22 | — | 8 | 1 | no | — |
| 252 | C22 | — | 16 | 1 | no | — |
| 253 | C22 | — | 316 | 2 | no | — |
| 254 | C22 | — | 2528 | 1 | no | — |
| 255 | C22 | #19098749 | 1 | 1 | yes | characteristic |
| 256 | C4 | — | 2 | 1 | no | — |
| 257 | C4 | — | 2 | 1 | no | — |
| 258 | C4 | — | 2 | 1 | no | — |
| 259 | C4 | — | 4 | 2 | no | — |
| 260 | C4 | — | 8 | 1 | no | — |
| 261 | C4 | — | 316 | 2 | no | — |
| 262 | C2 | — | 2 | 1 | no | — |
| 263 | C2 | — | 16 | 1 | no | — |
| 264 | C2 | — | 16 | 1 | no | — |
| 265 | C2 | — | 632 | 2 | no | — |
| 266 | C2 | #23765383 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23775193 | C2 | semidirect |
| 2. #23765426 | C2 | semidirect |
| 3. #23761567 | C2 | semidirect |
| 4. #23761555 | C2 | semidirect |
| 5. C79×#1624 | C2 | semidirect |
| 12. #19098362 | C22 | semidirect |
| 14. #555×C79 | C22 | semidirect |
| 17. #333×C79 | C22 | semidirect |
| 136. #1624 | D79 | semidirect |
| 139. C79 | #13681 | semidirect |
| 141. #555 | C2×D79 | semidirect |
| 151. #333 | C2×D79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23775193 | 2 |
| #23765426 | 2 |
| #23761567 | 2 |
| #23761555 | 2 |
| C79×#1624 | 2 |
| #13681 | 79 |