← #34186987
Subgroups — #34186987
Order 20224.
322 conjugacy classes of subgroups.
The subgroup lattice has 252 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #23763571 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4⋊C4×C79⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #23762176 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #23762175 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #23762146 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #23761303 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C79×#1303 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #19098616 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#14579096 | — | 2 | 1 | no | — |
| 10 | C2×#14579096 | — | 2 | 1 | no | — |
| 11 | #19098587 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C4⋊(C79⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C4×C79⋊C8 | — | 2 | 1 | no | — |
| 14 | C2×C4×C79⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 15 | #19098318 | — | 2 | 1 | no | — |
| 16 | #19098316 | — | 2 | 2 | no | — |
| 17 | #19098315 | — | 2 | 2 | no | — |
| 18 | #19098315 | C4 | 1 | 2 | yes | complemented |
| 19 | #19098313 | C4 | 1 | 2 | yes | complemented |
| 20 | #431×C79 | C22 | 1 | 1 | yes | characteristic |
| 21 | #423×C79 | C22 | 1 | 1 | yes | characteristic |
| 22 | C4×C4⋊C4×C79 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#11929608 | — | 2 | 1 | no | — |
| 24 | C22×C79⋊C8 | — | 2 | 1 | no | — |
| 25 | C22×C79⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 26 | C22×C79⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 27 | #14579096 | — | 4 | 2 | no | — |
| 28 | #14579096 | — | 4 | 2 | no | — |
| 29 | C4⋊(C79⋊C8) | — | 2 | 1 | no | — |
| 30 | C4⋊(C79⋊C8) | — | 2 | 1 | no | — |
| 31 | C4⋊(C79⋊C8) | — | 2 | 1 | no | — |
| 32 | C4⋊(C79⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 33 | C4⋊(C79⋊C8) | D4 | 1 | 2 | yes | — |
| 34 | C22⋊(C79⋊C8) | — | 2 | 1 | no | — |
| 35 | C4×C79⋊C8 | — | 2 | 1 | no | — |
| 36 | C4×C79⋊C8 | — | 4 | 2 | no | — |
| 37 | C4×C79⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C4×C79⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C4×C79⋊C8 | Q8 | 1 | 1 | yes | characteristic |
| 40 | C4×C79⋊C8 | Q8 | 1 | 1 | yes | characteristic |
| 41 | C2×M16×C79 | — | 2 | 1 | no | — |
| 42 | C2×C4⋊C4×C79 | — | 2 | 1 | no | — |
| 43 | C2×C42×C79 | — | 2 | 1 | no | — |
| 44 | C2×C42×C79 | C23 | 1 | 1 | yes | characteristic |
| 45 | C4⋊C8×C79 | — | 2 | 1 | no | — |
| 46 | C4⋊C8×C79 | — | 2 | 1 | no | — |
| 47 | C4⋊C8×C79 | C2×C4 | 1 | 2 | yes | — |
| 48 | C22⋊C8×C79 | — | 2 | 1 | no | — |
| 49 | C4×C8×C79 | — | 2 | 1 | no | — |
| 50 | #95×C79 | Q8 | 1 | 2 | yes | — |
| 51 | #11929608 | — | 4 | 1 | no | — |
| 52 | #11929608 | — | 4 | 1 | no | — |
| 53 | C2×C79⋊C8 | — | 2 | 2 | no | — |
| 54 | C2×C79⋊C8 | — | 2 | 2 | no | — |
| 55 | C2×C79⋊C8 | — | 2 | 1 | no | — |
| 56 | C2×C79⋊C8 | — | 2 | 1 | no | — |
| 57 | C2×C79⋊C8 | — | 4 | 1 | no | — |
| 58 | C2×C79⋊C8 | — | 4 | 1 | no | — |
| 59 | C2×C79⋊C8 | — | 4 | 1 | no | — |
| 60 | C2×C79⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C2×C79⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 62 | C2×C79⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 63 | C2×C79⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 64 | C2×C79⋊C8 | D8 | 1 | 2 | yes | — |
| 65 | C2×C79⋊C8 | Q16 | 1 | 2 | yes | — |
| 66 | C22×C4×C79 | — | 2 | 2 | no | — |
| 67 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | C22×C4×C79 | C2×Q8 | 1 | 1 | yes | characteristic |
| 69 | C22×C4×C79 | #41 | 1 | 1 | yes | characteristic |
| 70 | M16×C79 | — | 4 | 1 | no | — |
| 71 | M16×C79 | — | 4 | 1 | no | — |
| 72 | C2×C8×C79 | — | 2 | 2 | no | — |
| 73 | C2×C8×C79 | — | 4 | 1 | no | — |
| 74 | C4⋊C4×C79 | — | 2 | 4 | no | — |
| 75 | C42×C79 | — | 4 | 1 | no | — |
| 76 | C42×C79 | — | 4 | 1 | no | — |
| 77 | C42×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | C42×C79 | C2×Q8 | 1 | 1 | yes | characteristic |
| 80 | C42×C79 | #41 | 1 | 1 | yes | characteristic |
| 81 | C23×C79 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 82 | C2×C4×C79 | — | 2 | 2 | no | — |
| 83 | C2×C4×C79 | — | 2 | 4 | no | — |
| 84 | C2×C4×C79 | — | 4 | 1 | no | — |
| 85 | C2×C4×C79 | — | 4 | 2 | no | — |
| 86 | C2×C4×C79 | — | 4 | 2 | no | — |
| 87 | C2×C4×C79 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C79 | C4×Q8 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C79 | #121 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C79 | #125 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×C79 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 94 | C2×C4×C79 | C2×Q16 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C79 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 96 | C2×C4×C79 | #137 | 1 | 1 | yes | characteristic |
| 97 | C8×C79 | — | 4 | 2 | no | — |
| 98 | C8×C79 | — | 8 | 1 | no | — |
| 99 | C79⋊C8 | — | 2 | 1 | no | — |
| 100 | C79⋊C8 | — | 2 | 1 | no | — |
| 101 | C79⋊C8 | — | 4 | 2 | no | — |
| 102 | C79⋊C8 | — | 4 | 2 | no | — |
| 103 | C79⋊C8 | — | 8 | 1 | no | — |
| 104 | C79⋊C8 | #107 | 1 | 2 | yes | — |
| 105 | C79⋊C8 | #107 | 1 | 2 | yes | — |
| 106 | C22×C79 | — | 2 | 1 | no | — |
| 107 | C22×C79 | — | 2 | 1 | no | — |
| 108 | C22×C79 | #389 | 1 | 1 | yes | characteristic |
| 109 | C22×C79 | #474 | 1 | 1 | yes | characteristic |
| 110 | C22×C79 | #477 | 1 | 1 | yes | characteristic |
| 111 | C4×C79 | — | 2 | 1 | no | — |
| 112 | C4×C79 | — | 2 | 1 | no | — |
| 113 | C4×C79 | — | 2 | 1 | no | — |
| 114 | C4×C79 | — | 4 | 4 | no | — |
| 115 | C4×C79 | C2×#107 | 1 | 1 | yes | characteristic |
| 116 | C4×C79 | #428 | 1 | 1 | yes | characteristic |
| 117 | C4×C79 | #443 | 1 | 2 | yes | — |
| 118 | C4×C79 | #459 | 1 | 1 | yes | characteristic |
| 119 | C4×C79 | #462 | 1 | 1 | yes | characteristic |
| 120 | #9627 | — | 79 | 1 | no | maximal, Sylow |
| 121 | C2×C79 | — | 2 | 1 | no | — |
| 122 | C2×C79 | — | 2 | 1 | no | — |
| 123 | C2×C79 | #1945 | 1 | 1 | yes | characteristic |
| 124 | C2×C79 | #1947 | 1 | 1 | yes | characteristic |
| 125 | C2×C79 | #1953 | 1 | 1 | yes | characteristic |
| 126 | #1848 | — | 79 | 1 | no | — |
| 127 | C8×C4⋊C4 | — | 79 | 1 | no | — |
| 128 | #1571 | — | 79 | 1 | no | — |
| 129 | #1571 | — | 79 | 1 | no | — |
| 130 | #1564 | — | 79 | 1 | no | — |
| 131 | #1303 | — | 79 | 1 | no | — |
| 132 | #1303 | D79 | 1 | 1 | yes | characteristic |
| 133 | C79 | #9627 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 134 | #431 | — | 79 | 1 | no | — |
| 135 | #431 | C2×D79 | 1 | 1 | yes | characteristic |
| 136 | C2×#108 | — | 158 | 1 | no | — |
| 137 | C2×#108 | — | 158 | 1 | no | — |
| 138 | #423 | — | 79 | 1 | no | — |
| 139 | #423 | C2×D79 | 1 | 1 | yes | characteristic |
| 140 | C2×C4⋊C8 | — | 79 | 1 | no | — |
| 141 | C2×C4×C8 | — | 79 | 1 | no | — |
| 142 | C2×C4×C8 | — | 158 | 1 | no | — |
| 143 | C4×C4⋊C4 | C2×D79 | 1 | 1 | yes | characteristic |
| 144 | #336 | — | 158 | 1 | no | — |
| 145 | #335 | — | 79 | 2 | no | — |
| 146 | #335 | — | 158 | 2 | no | — |
| 147 | #335 | — | 158 | 2 | no | — |
| 148 | #334 | — | 79 | 2 | no | — |
| 149 | C2×M16 | — | 2 | 1 | no | — |
| 150 | C2×M16 | — | 158 | 1 | no | — |
| 151 | C22×C8 | — | 79 | 1 | no | — |
| 152 | C22×C8 | — | 79 | 1 | no | — |
| 153 | C22×C8 | — | 158 | 1 | no | — |
| 154 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 155 | C2×C42 | — | 2 | 1 | no | — |
| 156 | C2×C42 | C22×D79 | 1 | 1 | yes | characteristic |
| 157 | #108 | — | 316 | 2 | no | — |
| 158 | #108 | — | 316 | 2 | no | — |
| 159 | C4⋊C8 | — | 2 | 1 | no | — |
| 160 | C4⋊C8 | — | 2 | 1 | no | — |
| 161 | C4⋊C8 | — | 79 | 2 | no | — |
| 162 | C4⋊C8 | — | 79 | 2 | no | — |
| 163 | C4⋊C8 | — | 158 | 1 | no | — |
| 164 | C4⋊C8 | — | 158 | 1 | no | — |
| 165 | C4⋊C8 | — | 158 | 1 | no | — |
| 166 | C4⋊C8 | C4×D79 | 1 | 2 | yes | — |
| 167 | C22⋊C8 | — | 2 | 1 | no | — |
| 168 | C22⋊C8 | — | 158 | 1 | no | — |
| 169 | C4×C8 | — | 2 | 1 | no | — |
| 170 | C4×C8 | — | 316 | 2 | no | — |
| 171 | C4×C8 | — | 79 | 1 | no | — |
| 172 | C4×C8 | — | 79 | 1 | no | — |
| 173 | C4×C8 | — | 79 | 1 | no | — |
| 174 | C4×C8 | — | 79 | 1 | no | — |
| 175 | C4×C8 | — | 158 | 1 | no | — |
| 176 | #95 | Dic158 | 1 | 2 | yes | — |
| 177 | C22×C4 | — | 2 | 2 | no | — |
| 178 | C22×C4 | D4×D79 | 1 | 1 | yes | characteristic |
| 179 | C22×C4 | C2×Dic158 | 1 | 1 | yes | characteristic |
| 180 | C22×C4 | #11929623 | 1 | 1 | yes | characteristic |
| 181 | M16 | — | 4 | 1 | no | — |
| 182 | M16 | — | 4 | 1 | no | — |
| 183 | M16 | — | 316 | 1 | no | — |
| 184 | M16 | — | 316 | 1 | no | — |
| 185 | C2×C8 | — | 2 | 2 | no | — |
| 186 | C2×C8 | — | 4 | 1 | no | — |
| 187 | C2×C8 | — | 316 | 1 | no | — |
| 188 | C2×C8 | — | 316 | 1 | no | — |
| 189 | C2×C8 | — | 316 | 1 | no | — |
| 190 | C2×C8 | — | 79 | 1 | no | — |
| 191 | C2×C8 | — | 79 | 1 | no | — |
| 192 | C2×C8 | — | 79 | 2 | no | — |
| 193 | C2×C8 | — | 79 | 2 | no | — |
| 194 | C2×C8 | — | 79 | 1 | no | — |
| 195 | C2×C8 | — | 79 | 1 | no | — |
| 196 | C2×C8 | — | 158 | 2 | no | — |
| 197 | C2×C8 | — | 158 | 2 | no | — |
| 198 | C2×C8 | — | 158 | 1 | no | — |
| 199 | C2×C8 | — | 158 | 1 | no | — |
| 200 | C4⋊C4 | — | 2 | 4 | no | — |
| 201 | C42 | — | 4 | 1 | no | — |
| 202 | C42 | — | 4 | 1 | no | — |
| 203 | C42 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 204 | C42 | D4×D79 | 1 | 1 | yes | characteristic |
| 205 | C42 | Q8×D79 | 1 | 1 | yes | characteristic |
| 206 | C42 | #11929624 | 1 | 1 | yes | characteristic |
| 207 | C23 | #14579138 | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | — | 2 | 2 | no | — |
| 209 | C2×C4 | — | 2 | 4 | no | — |
| 210 | C2×C4 | — | 4 | 1 | no | — |
| 211 | C2×C4 | — | 4 | 2 | no | — |
| 212 | C2×C4 | — | 4 | 2 | no | — |
| 213 | C2×C4 | C4⋊C4×D79 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | (C22⋊D79)⋊C4 | 1 | 1 | yes | characteristic |
| 215 | C2×C4 | C4×Dic158 | 1 | 1 | yes | characteristic |
| 216 | C2×C4 | #14579134 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | #14579151 | 1 | 1 | yes | characteristic |
| 218 | C2×C4 | #14579160 | 1 | 1 | yes | characteristic |
| 219 | C2×C4 | D8×D79 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | Q16×D79 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #14579188 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #14579190 | 1 | 1 | yes | characteristic |
| 223 | C8 | — | 4 | 2 | no | — |
| 224 | C8 | — | 8 | 1 | no | — |
| 225 | C8 | — | 316 | 2 | no | — |
| 226 | C8 | — | 316 | 2 | no | — |
| 227 | C8 | — | 79 | 2 | no | — |
| 228 | C8 | — | 79 | 2 | no | — |
| 229 | C8 | — | 632 | 1 | no | — |
| 230 | C8 | — | 158 | 1 | no | — |
| 231 | C8 | — | 158 | 1 | no | — |
| 232 | C22 | — | 2 | 1 | no | — |
| 233 | C22 | — | 2 | 1 | no | — |
| 234 | C22 | #19098460 | 1 | 1 | yes | characteristic |
| 235 | C22 | #19098841 | 1 | 1 | yes | characteristic |
| 236 | C22 | #19098856 | 1 | 1 | yes | characteristic |
| 237 | C4 | — | 2 | 1 | no | — |
| 238 | C4 | — | 2 | 1 | no | — |
| 239 | C4 | — | 2 | 1 | no | — |
| 240 | C4 | — | 4 | 4 | no | — |
| 241 | C4 | #107×D79 | 1 | 1 | yes | characteristic |
| 242 | C4 | #19098602 | 1 | 1 | yes | characteristic |
| 243 | C4 | #19098677 | 1 | 2 | yes | — |
| 244 | C4 | #19098757 | 1 | 1 | yes | characteristic |
| 245 | C4 | #19098772 | 1 | 1 | yes | characteristic |
| 246 | C2 | — | 2 | 1 | no | — |
| 247 | C2 | — | 2 | 1 | no | — |
| 248 | C2 | #23764147 | 1 | 1 | yes | characteristic |
| 249 | C2 | #23764157 | 1 | 1 | yes | characteristic |
| 250 | C2 | #23764189 | 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