← #34203691
Subgroups — #34203691
Order 20224.
318 conjugacy classes of subgroups.
The subgroup lattice has 248 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #23765186 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#19098385 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#19098341 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #23763284 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #23761508 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #23761507 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C79×#1776 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#14579101 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#14579101 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#14579097 | C22 | 1 | 1 | yes | characteristic |
| 11 | #19098601 | C22 | 1 | 1 | yes | characteristic |
| 12 | #19098518 | — | 2 | 1 | no | — |
| 13 | #19098390 | C4 | 1 | 2 | yes | complemented |
| 14 | #19098388 | C4 | 1 | 2 | yes | complemented |
| 15 | #19098385 | C4 | 1 | 4 | yes | complemented |
| 16 | #19098341 | C4 | 1 | 4 | yes | complemented |
| 17 | #19098323 | — | 2 | 1 | no | — |
| 18 | C2×#107×C79 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#106×C79 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#97×C79 | C22 | 1 | 1 | yes | characteristic |
| 21 | C4×C4⋊C4×C79 | — | 2 | 1 | no | — |
| 22 | #340×C79 | — | 2 | 1 | no | — |
| 23 | C2×#11929608 | — | 2 | 1 | no | — |
| 24 | #14579120 | — | 2 | 1 | no | — |
| 25 | #14579101 | — | 2 | 1 | no | — |
| 26 | #14579101 | C2×C4 | 1 | 2 | yes | — |
| 27 | #14579101 | C2×C4 | 1 | 2 | yes | — |
| 28 | #14579101 | C2×C4 | 1 | 2 | yes | — |
| 29 | C2×C79⋊C16 | — | 2 | 1 | no | — |
| 30 | C2×C79⋊C16 | C2×C4 | 1 | 2 | yes | — |
| 31 | #14579097 | — | 2 | 1 | no | — |
| 32 | #14579097 | C2×C4 | 1 | 2 | yes | — |
| 33 | #14579095 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | #14579095 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | #14579093 | — | 2 | 1 | no | — |
| 36 | #14579071 | — | 4 | 1 | no | — |
| 37 | C8×Dic79 | — | 4 | 1 | no | — |
| 38 | C22×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 39 | C22×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×C4⋊C4×C79 | — | 2 | 1 | no | — |
| 41 | C2×C4⋊C4×C79 | — | 2 | 1 | no | — |
| 42 | C2×C42×C79 | — | 4 | 1 | no | — |
| 43 | C2×C42×C79 | D4 | 1 | 1 | yes | characteristic |
| 44 | #107×C79 | — | 2 | 1 | no | — |
| 45 | #107×C79 | C2×C4 | 1 | 2 | yes | — |
| 46 | #106×C79 | — | 2 | 1 | no | — |
| 47 | #106×C79 | C2×C4 | 1 | 2 | yes | — |
| 48 | #97×C79 | C2×C4 | 1 | 2 | yes | — |
| 49 | #97×C79 | Q8 | 1 | 2 | yes | — |
| 50 | #95×C79 | — | 4 | 1 | no | — |
| 51 | #11929608 | — | 2 | 2 | no | — |
| 52 | #11929608 | — | 4 | 1 | no | — |
| 53 | C2×C79⋊C8 | — | 4 | 1 | no | — |
| 54 | C4⋊Dic79 | — | 2 | 1 | no | — |
| 55 | C4⋊Dic79 | — | 2 | 1 | no | — |
| 56 | C22⋊Dic79 | — | 4 | 1 | no | — |
| 57 | C4×Dic79 | — | 4 | 1 | no | — |
| 58 | C79⋊C16 | — | 2 | 2 | no | — |
| 59 | C79⋊C16 | C42 | 1 | 4 | yes | complemented |
| 60 | C22×C4×C79 | — | 2 | 1 | no | — |
| 61 | C22×C4×C79 | — | 4 | 1 | no | — |
| 62 | C22×C4×C79 | — | 4 | 1 | no | — |
| 63 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C2×C8×C79 | — | 2 | 1 | no | — |
| 65 | C2×C8×C79 | — | 2 | 1 | no | — |
| 66 | C2×C8×C79 | C4⋊C4 | 1 | 2 | yes | — |
| 67 | C2×C8×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C2×C8×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 69 | C2×C8×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 70 | C2×C8×C79 | C2×Q8 | 1 | 1 | yes | characteristic |
| 71 | C2×C8×C79 | #41 | 1 | 1 | yes | characteristic |
| 72 | C2×C8×C79 | #41 | 1 | 1 | yes | characteristic |
| 73 | C4⋊C4×C79 | — | 2 | 2 | no | — |
| 74 | C4⋊C4×C79 | — | 2 | 2 | no | — |
| 75 | C4⋊C4×C79 | — | 4 | 1 | no | — |
| 76 | C4⋊C4×C79 | — | 4 | 1 | no | — |
| 77 | C42×C79 | — | 2 | 1 | no | — |
| 78 | C42×C79 | — | 8 | 2 | no | — |
| 79 | C42×C79 | C4⋊C4 | 1 | 2 | yes | — |
| 80 | C23×C79 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 81 | C2×C4×C79 | — | 2 | 2 | no | — |
| 82 | C2×C4×C79 | — | 4 | 1 | no | — |
| 83 | C2×C4×C79 | — | 4 | 1 | no | — |
| 84 | C2×C4×C79 | — | 4 | 2 | no | — |
| 85 | C2×C4×C79 | — | 4 | 2 | no | — |
| 86 | C2×C4×C79 | — | 8 | 1 | no | — |
| 87 | C2×C4×C79 | — | 8 | 1 | no | — |
| 88 | C2×C4×C79 | — | 8 | 1 | no | — |
| 89 | C2×C4×C79 | — | 8 | 1 | no | — |
| 90 | C2×C4×C79 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×C79 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 94 | C2×C4×C79 | #137 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C79 | #137 | 1 | 1 | yes | characteristic |
| 96 | C2×Dic79 | — | 4 | 1 | no | — |
| 97 | C2×Dic79 | — | 4 | 1 | no | — |
| 98 | C8×C79 | — | 2 | 2 | no | — |
| 99 | C8×C79 | C2×C42 | 1 | 1 | yes | characteristic |
| 100 | C8×C79 | #117 | 1 | 1 | yes | characteristic |
| 101 | C8×C79 | C4×Q8 | 1 | 1 | yes | characteristic |
| 102 | C8×C79 | C4×Q8 | 1 | 1 | yes | characteristic |
| 103 | C79⋊C8 | — | 4 | 2 | no | — |
| 104 | C22×C79 | — | 2 | 1 | no | — |
| 105 | C22×C79 | — | 2 | 1 | no | — |
| 106 | C22×C79 | #428 | 1 | 1 | yes | characteristic |
| 107 | C22×C79 | #440 | 1 | 1 | yes | characteristic |
| 108 | C22×C79 | #442 | 1 | 1 | yes | characteristic |
| 109 | C4×C79 | — | 4 | 2 | no | — |
| 110 | C4×C79 | — | 8 | 2 | no | — |
| 111 | C4×C79 | — | 8 | 2 | no | — |
| 112 | C4×C79 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 113 | C4×C79 | #428 | 1 | 1 | yes | characteristic |
| 114 | C4×C79 | #440 | 1 | 1 | yes | characteristic |
| 115 | C4×C79 | #441 | 1 | 1 | yes | characteristic |
| 116 | Dic79 | — | 8 | 1 | no | — |
| 117 | Dic79 | — | 8 | 1 | no | — |
| 118 | #12677 | — | 79 | 1 | no | maximal, Sylow |
| 119 | C2×C79 | — | 2 | 1 | no | — |
| 120 | C2×C79 | — | 2 | 1 | no | — |
| 121 | C2×C79 | #1776 | 1 | 1 | yes | characteristic |
| 122 | C2×C79 | #2180 | 1 | 1 | yes | characteristic |
| 123 | C2×C79 | AGL(1,ℤ/16) | 1 | 1 | yes | characteristic |
| 124 | #2159 | — | 79 | 1 | no | — |
| 125 | C2×#365 | — | 79 | 1 | no | — |
| 126 | C2×#347 | — | 79 | 1 | no | — |
| 127 | #1776 | D79 | 1 | 1 | yes | characteristic |
| 128 | #1773 | — | 79 | 1 | no | — |
| 129 | #1387 | — | 79 | 1 | no | — |
| 130 | #1386 | — | 79 | 1 | no | — |
| 131 | C79 | #12677 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 132 | C2×M32 | — | 79 | 1 | no | — |
| 133 | C2×M32 | — | 79 | 1 | no | — |
| 134 | C2×#108 | — | 79 | 1 | no | — |
| 135 | #427 | — | 79 | 1 | no | — |
| 136 | C2×#107 | C2×D79 | 1 | 1 | yes | characteristic |
| 137 | C2×#106 | C2×D79 | 1 | 1 | yes | characteristic |
| 138 | #405 | — | 158 | 1 | no | — |
| 139 | C2×#97 | C2×D79 | 1 | 1 | yes | characteristic |
| 140 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 141 | #367 | — | 79 | 2 | no | — |
| 142 | #366 | — | 79 | 2 | no | — |
| 143 | #365 | — | 79 | 4 | no | — |
| 144 | #347 | — | 79 | 4 | no | — |
| 145 | #340 | — | 2 | 1 | no | — |
| 146 | #339 | — | 158 | 1 | no | — |
| 147 | C2×M16 | — | 158 | 1 | no | — |
| 148 | C22×C8 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 149 | C22×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 150 | #117 | — | 158 | 1 | no | — |
| 151 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 152 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 153 | C2×C42 | — | 4 | 1 | no | — |
| 154 | C2×C42 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 155 | M32 | — | 79 | 2 | no | — |
| 156 | M32 | — | 79 | 2 | no | — |
| 157 | M32 | — | 79 | 2 | no | — |
| 158 | M32 | — | 158 | 1 | no | — |
| 159 | C2×C16 | — | 79 | 2 | no | — |
| 160 | C2×C16 | — | 158 | 1 | no | — |
| 161 | #108 | — | 79 | 2 | no | — |
| 162 | #108 | — | 158 | 1 | no | — |
| 163 | #107 | — | 2 | 1 | no | — |
| 164 | #107 | — | 79 | 1 | no | — |
| 165 | #107 | — | 79 | 1 | no | — |
| 166 | #107 | C4×D79 | 1 | 2 | yes | — |
| 167 | #106 | — | 2 | 1 | no | — |
| 168 | #106 | — | 158 | 1 | no | — |
| 169 | #106 | C4×D79 | 1 | 2 | yes | — |
| 170 | #97 | — | 316 | 1 | no | — |
| 171 | #97 | Dic158 | 1 | 2 | yes | — |
| 172 | #97 | C4×D79 | 1 | 2 | yes | — |
| 173 | C4×C8 | — | 316 | 1 | no | — |
| 174 | #95 | — | 4 | 1 | no | — |
| 175 | C22×C4 | — | 2 | 1 | no | — |
| 176 | C22×C4 | — | 4 | 1 | no | — |
| 177 | C22×C4 | — | 4 | 1 | no | — |
| 178 | C22×C4 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 179 | M16 | — | 316 | 1 | no | — |
| 180 | M16 | — | 158 | 2 | no | — |
| 181 | C2×C8 | — | 2 | 1 | no | — |
| 182 | C2×C8 | — | 2 | 1 | no | — |
| 183 | C2×C8 | — | 316 | 1 | no | — |
| 184 | C2×C8 | Dic79⋊C4 | 1 | 2 | yes | — |
| 185 | C2×C8 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 186 | C2×C8 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 187 | C2×C8 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 188 | C2×C8 | C2×Dic158 | 1 | 1 | yes | characteristic |
| 189 | C2×C8 | #11929624 | 1 | 1 | yes | characteristic |
| 190 | C2×C8 | #11929624 | 1 | 1 | yes | characteristic |
| 191 | C4⋊C4 | — | 2 | 2 | no | — |
| 192 | C4⋊C4 | — | 2 | 2 | no | — |
| 193 | C4⋊C4 | — | 4 | 1 | no | — |
| 194 | C4⋊C4 | — | 4 | 1 | no | — |
| 195 | C4⋊C4 | — | 158 | 1 | no | — |
| 196 | C4⋊C4 | — | 158 | 1 | no | — |
| 197 | C22⋊C4 | — | 316 | 1 | no | — |
| 198 | C42 | — | 2 | 1 | no | — |
| 199 | C42 | — | 8 | 2 | no | — |
| 200 | C42 | — | 316 | 1 | no | — |
| 201 | C42 | Dic79⋊C4 | 1 | 2 | yes | — |
| 202 | C16 | — | 79 | 4 | no | — |
| 203 | C16 | — | 158 | 2 | no | — |
| 204 | C23 | #14579187 | 1 | 1 | yes | characteristic |
| 205 | C2×C4 | — | 2 | 2 | no | — |
| 206 | C2×C4 | — | 4 | 1 | no | — |
| 207 | C2×C4 | — | 4 | 1 | no | — |
| 208 | C2×C4 | — | 4 | 2 | no | — |
| 209 | C2×C4 | — | 4 | 2 | no | — |
| 210 | C2×C4 | — | 8 | 1 | no | — |
| 211 | C2×C4 | — | 8 | 1 | no | — |
| 212 | C2×C4 | — | 8 | 1 | no | — |
| 213 | C2×C4 | — | 8 | 1 | no | — |
| 214 | C2×C4 | — | 316 | 1 | no | — |
| 215 | C2×C4 | — | 316 | 1 | no | — |
| 216 | C2×C4 | C2×Dic79⋊C4 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | C4×C22⋊D79 | 1 | 1 | yes | characteristic |
| 218 | C2×C4 | C4×C22⋊D79 | 1 | 1 | yes | characteristic |
| 219 | C2×C4 | #14579186 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | #14579191 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #14579192 | 1 | 1 | yes | characteristic |
| 222 | C8 | — | 2 | 2 | no | — |
| 223 | C8 | — | 316 | 2 | no | — |
| 224 | C8 | C42×D79 | 1 | 1 | yes | characteristic |
| 225 | C8 | #14579118 | 1 | 1 | yes | characteristic |
| 226 | C8 | C4×Dic158 | 1 | 1 | yes | characteristic |
| 227 | C8 | C4×Dic158 | 1 | 1 | yes | characteristic |
| 228 | C22 | — | 2 | 1 | no | — |
| 229 | C22 | — | 2 | 1 | no | — |
| 230 | C22 | #19098603 | 1 | 1 | yes | characteristic |
| 231 | C22 | #19098663 | 1 | 1 | yes | characteristic |
| 232 | C22 | #19098673 | 1 | 1 | yes | characteristic |
| 233 | C4 | — | 4 | 2 | no | — |
| 234 | C4 | — | 8 | 2 | no | — |
| 235 | C4 | — | 8 | 2 | no | — |
| 236 | C4 | — | 632 | 1 | no | — |
| 237 | C4 | — | 632 | 1 | no | — |
| 238 | C4 | C4×Dic79⋊C4 | 1 | 1 | yes | characteristic |
| 239 | C4 | #19098606 | 1 | 1 | yes | characteristic |
| 240 | C4 | #19098661 | 1 | 1 | yes | characteristic |
| 241 | C4 | #19098668 | 1 | 1 | yes | characteristic |
| 242 | C2 | — | 2 | 1 | no | — |
| 243 | C2 | — | 2 | 1 | no | — |
| 244 | C2 | #23763298 | 1 | 1 | yes | characteristic |
| 245 | C2 | #23765289 | 1 | 1 | yes | characteristic |
| 246 | C2 | #23765296 | 1 | 1 | yes | characteristic |
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