← #31913822
Subgroups — #31913822
Order 18176.
302 conjugacy classes of subgroups.
The subgroup lattice has 238 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | M32×Dic71 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22588510 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22588495 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #22586440 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #22586439 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #22586438 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C71×#1383 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#14556068 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#14556064 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#14556063 | — | 2 | 1 | no | — |
| 11 | C2×#14556060 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#14556059 | — | 2 | 1 | no | — |
| 13 | #17986022 | — | 2 | 1 | no | — |
| 14 | C2×C8×Dic71 | C22 | 1 | 1 | yes | characteristic |
| 15 | #17985868 | — | 2 | 1 | no | — |
| 16 | #17985774 | C4 | 1 | 2 | yes | complemented |
| 17 | C16×Dic71 | C4 | 1 | 2 | yes | complemented |
| 18 | C2×M32×C71 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#108×C71 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#106×C71 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#11765849 | — | 2 | 1 | no | — |
| 22 | C22×C71⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 23 | C2×C4⋊Dic71 | — | 2 | 1 | no | — |
| 24 | C2×Dic71⋊C4 | — | 4 | 2 | no | — |
| 25 | C2×C4×Dic71 | D4 | 1 | 1 | yes | characteristic |
| 26 | #14556068 | — | 2 | 1 | no | — |
| 27 | #14556068 | — | 2 | 1 | no | — |
| 28 | C2×C71⋊C16 | C2×C4 | 1 | 2 | yes | — |
| 29 | #14556064 | — | 2 | 1 | no | — |
| 30 | #14556064 | — | 2 | 1 | no | — |
| 31 | #14556063 | — | 4 | 2 | no | — |
| 32 | #14556060 | — | 2 | 1 | no | — |
| 33 | #14556060 | — | 2 | 1 | no | — |
| 34 | #14556059 | — | 4 | 2 | no | — |
| 35 | Dic71⋊C8 | — | 4 | 2 | no | — |
| 36 | C8×Dic71 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C8×Dic71 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C8×Dic71 | Q8 | 1 | 1 | yes | characteristic |
| 39 | C8×Dic71 | Q8 | 1 | 1 | yes | characteristic |
| 40 | C2×M16×C71 | — | 2 | 1 | no | — |
| 41 | C22×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 42 | C2×C4⋊C4×C71 | — | 2 | 1 | no | — |
| 43 | M32×C71 | — | 2 | 1 | no | — |
| 44 | M32×C71 | — | 2 | 1 | no | — |
| 45 | C2×C16×C71 | C2×C4 | 1 | 2 | yes | — |
| 46 | #108×C71 | — | 2 | 1 | no | — |
| 47 | #108×C71 | — | 2 | 1 | no | — |
| 48 | #106×C71 | — | 2 | 1 | no | — |
| 49 | #106×C71 | — | 2 | 1 | no | — |
| 50 | C22×Dic71 | — | 4 | 1 | no | — |
| 51 | C22×Dic71 | SD16 | 1 | 2 | yes | — |
| 52 | #11765849 | — | 4 | 1 | no | — |
| 53 | #11765849 | — | 4 | 1 | no | — |
| 54 | C2×C71⋊C8 | — | 4 | 1 | no | — |
| 55 | C2×C71⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 56 | C2×C71⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 57 | C2×C71⋊C8 | D8 | 1 | 2 | yes | — |
| 58 | C2×C71⋊C8 | Q16 | 1 | 2 | yes | — |
| 59 | C4⋊Dic71 | — | 4 | 1 | no | — |
| 60 | C4⋊Dic71 | — | 4 | 1 | no | — |
| 61 | Dic71⋊C4 | — | 8 | 4 | no | — |
| 62 | C4×Dic71 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 63 | C4×Dic71 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 64 | C4×Dic71 | SD16 | 1 | 2 | yes | — |
| 65 | C71⋊C16 | — | 2 | 2 | no | — |
| 66 | C22×C4×C71 | — | 4 | 1 | no | — |
| 67 | C22×C4×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | M16×C71 | — | 4 | 1 | no | — |
| 69 | M16×C71 | — | 4 | 1 | no | — |
| 70 | C2×C8×C71 | — | 4 | 1 | no | — |
| 71 | C2×C8×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 72 | C2×C8×C71 | C2×Q8 | 1 | 1 | yes | characteristic |
| 73 | C2×C8×C71 | #41 | 1 | 1 | yes | characteristic |
| 74 | C2×C8×C71 | #41 | 1 | 1 | yes | characteristic |
| 75 | C2×C8×C71 | #41 | 1 | 1 | yes | characteristic |
| 76 | C2×C8×C71 | #41 | 1 | 1 | yes | characteristic |
| 77 | C4⋊C4×C71 | — | 4 | 1 | no | — |
| 78 | C4⋊C4×C71 | — | 4 | 1 | no | — |
| 79 | C16×C71 | — | 2 | 2 | no | — |
| 80 | C23×C71 | C2×SD16 | 1 | 1 | yes | characteristic |
| 81 | C2×C4×C71 | — | 8 | 2 | no | — |
| 82 | C2×C4×C71 | — | 8 | 1 | no | — |
| 83 | C2×C4×C71 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 84 | C2×C4×C71 | #124 | 1 | 1 | yes | characteristic |
| 85 | C2×C4×C71 | #124 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C71 | C2×D8 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C71 | C2×SD16 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C71 | C2×Q16 | 1 | 1 | yes | characteristic |
| 89 | C2×Dic71 | — | 2 | 4 | no | — |
| 90 | C2×Dic71 | — | 8 | 2 | no | — |
| 91 | C2×Dic71 | — | 8 | 1 | no | — |
| 92 | C2×Dic71 | #106 | 1 | 2 | yes | — |
| 93 | C2×Dic71 | #106 | 1 | 2 | yes | — |
| 94 | C8×C71 | — | 8 | 1 | no | — |
| 95 | C8×C71 | #117 | 1 | 1 | yes | characteristic |
| 96 | C8×C71 | #117 | 1 | 1 | yes | characteristic |
| 97 | C8×C71 | #125 | 1 | 1 | yes | characteristic |
| 98 | C8×C71 | #125 | 1 | 1 | yes | characteristic |
| 99 | C71⋊C8 | — | 8 | 1 | no | — |
| 100 | C71⋊C8 | #107 | 1 | 2 | yes | — |
| 101 | C71⋊C8 | #107 | 1 | 2 | yes | — |
| 102 | C22×C71 | — | 2 | 2 | no | — |
| 103 | C22×C71 | C2×#106 | 1 | 1 | yes | characteristic |
| 104 | C22×C71 | #486 | 1 | 1 | yes | characteristic |
| 105 | C22×C71 | #487 | 1 | 1 | yes | characteristic |
| 106 | C4×C71 | — | 16 | 1 | no | — |
| 107 | C4×C71 | #382 | 1 | 1 | yes | characteristic |
| 108 | C4×C71 | C2×#107 | 1 | 1 | yes | characteristic |
| 109 | C4×C71 | #486 | 1 | 1 | yes | characteristic |
| 110 | C4×C71 | #487 | 1 | 1 | yes | characteristic |
| 111 | Dic71 | — | 2 | 2 | no | — |
| 112 | Dic71 | — | 16 | 1 | no | — |
| 113 | Dic71 | #365 | 1 | 4 | yes | — |
| 114 | #12886 | — | 71 | 1 | no | maximal, Sylow |
| 115 | C2×C71 | — | 2 | 1 | no | — |
| 116 | C2×C71 | — | 2 | 1 | no | — |
| 117 | C2×C71 | #1834 | 1 | 1 | yes | characteristic |
| 118 | C2×C71 | C2×#365 | 1 | 2 | yes | — |
| 119 | C4×M32 | — | 71 | 1 | no | — |
| 120 | #1848 | — | 71 | 1 | no | — |
| 121 | #1844 | — | 71 | 1 | no | — |
| 122 | #1383 | — | 71 | 1 | no | — |
| 123 | #1383 | — | 71 | 1 | no | — |
| 124 | #1383 | — | 71 | 1 | no | — |
| 125 | #1383 | D71 | 1 | 1 | yes | characteristic |
| 126 | C71 | #12886 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 127 | C2×M32 | — | 71 | 1 | no | — |
| 128 | C2×M32 | C2×D71 | 1 | 1 | yes | characteristic |
| 129 | C2×#108 | — | 71 | 1 | no | — |
| 130 | C2×#108 | — | 142 | 1 | no | — |
| 131 | C2×#108 | C2×D71 | 1 | 1 | yes | characteristic |
| 132 | C2×#106 | — | 71 | 1 | no | — |
| 133 | C2×#106 | — | 142 | 1 | no | — |
| 134 | C2×#106 | C2×D71 | 1 | 1 | yes | characteristic |
| 135 | #423 | — | 142 | 1 | no | — |
| 136 | C2×C4×C8 | — | 71 | 1 | no | — |
| 137 | #384 | — | 142 | 1 | no | — |
| 138 | #346 | — | 71 | 2 | no | — |
| 139 | C4×C16 | — | 71 | 2 | no | — |
| 140 | C2×M16 | — | 2 | 1 | no | — |
| 141 | C2×M16 | — | 142 | 1 | no | — |
| 142 | C22×C8 | — | 71 | 1 | no | — |
| 143 | C22×C8 | C22×D71 | 1 | 1 | yes | characteristic |
| 144 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 145 | C2×C4⋊C4 | — | 284 | 2 | no | — |
| 146 | C2×C4⋊C4 | — | 142 | 1 | no | — |
| 147 | C2×C42 | — | 71 | 1 | no | — |
| 148 | M32 | — | 2 | 1 | no | — |
| 149 | M32 | — | 2 | 1 | no | — |
| 150 | M32 | — | 142 | 1 | no | — |
| 151 | M32 | — | 142 | 1 | no | — |
| 152 | C2×C16 | — | 71 | 2 | no | — |
| 153 | C2×C16 | C4×D71 | 1 | 2 | yes | — |
| 154 | #108 | — | 2 | 1 | no | — |
| 155 | #108 | — | 2 | 1 | no | — |
| 156 | #108 | — | 284 | 2 | no | — |
| 157 | #108 | — | 142 | 1 | no | — |
| 158 | #108 | — | 142 | 1 | no | — |
| 159 | #106 | — | 2 | 1 | no | — |
| 160 | #106 | — | 2 | 1 | no | — |
| 161 | #106 | — | 284 | 2 | no | — |
| 162 | #106 | — | 142 | 1 | no | — |
| 163 | #106 | — | 142 | 1 | no | — |
| 164 | C4⋊C8 | — | 284 | 2 | no | — |
| 165 | C4×C8 | — | 71 | 1 | no | — |
| 166 | C4×C8 | — | 71 | 1 | no | — |
| 167 | C4×C8 | — | 71 | 1 | no | — |
| 168 | C4×C8 | — | 71 | 1 | no | — |
| 169 | C22×C4 | — | 4 | 1 | no | — |
| 170 | C22×C4 | — | 284 | 1 | no | — |
| 171 | C22×C4 | — | 71 | 2 | no | — |
| 172 | C22×C4 | D4×D71 | 1 | 1 | yes | characteristic |
| 173 | M16 | — | 4 | 1 | no | — |
| 174 | M16 | — | 4 | 1 | no | — |
| 175 | M16 | — | 284 | 1 | no | — |
| 176 | M16 | — | 284 | 1 | no | — |
| 177 | C2×C8 | — | 4 | 1 | no | — |
| 178 | C2×C8 | — | 284 | 1 | no | — |
| 179 | C2×C8 | — | 71 | 1 | no | — |
| 180 | C2×C8 | — | 71 | 1 | no | — |
| 181 | C2×C8 | — | 71 | 2 | no | — |
| 182 | C2×C8 | — | 71 | 2 | no | — |
| 183 | C2×C8 | C2×C4×D71 | 1 | 1 | yes | characteristic |
| 184 | C2×C8 | Q8×D71 | 1 | 1 | yes | characteristic |
| 185 | C2×C8 | #11765864 | 1 | 1 | yes | characteristic |
| 186 | C2×C8 | #11765864 | 1 | 1 | yes | characteristic |
| 187 | C2×C8 | #11765865 | 1 | 1 | yes | characteristic |
| 188 | C2×C8 | #11765865 | 1 | 1 | yes | characteristic |
| 189 | C4⋊C4 | — | 4 | 1 | no | — |
| 190 | C4⋊C4 | — | 4 | 1 | no | — |
| 191 | C4⋊C4 | — | 284 | 1 | no | — |
| 192 | C4⋊C4 | — | 284 | 1 | no | — |
| 193 | C4⋊C4 | — | 568 | 4 | no | — |
| 194 | C42 | — | 71 | 1 | no | — |
| 195 | C42 | — | 71 | 1 | no | — |
| 196 | C42 | — | 71 | 2 | no | — |
| 197 | C16 | — | 2 | 2 | no | — |
| 198 | C16 | — | 142 | 2 | no | — |
| 199 | C23 | SD16×D71 | 1 | 1 | yes | characteristic |
| 200 | C2×C4 | — | 8 | 2 | no | — |
| 201 | C2×C4 | — | 8 | 1 | no | — |
| 202 | C2×C4 | — | 568 | 2 | no | — |
| 203 | C2×C4 | — | 568 | 1 | no | — |
| 204 | C2×C4 | — | 71 | 2 | no | — |
| 205 | C2×C4 | — | 71 | 2 | no | — |
| 206 | C2×C4 | — | 142 | 4 | no | — |
| 207 | C2×C4 | C4⋊C4×D71 | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | #14556114 | 1 | 1 | yes | characteristic |
| 209 | C2×C4 | #14556114 | 1 | 1 | yes | characteristic |
| 210 | C2×C4 | D8×D71 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | SD16×D71 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | Q16×D71 | 1 | 1 | yes | characteristic |
| 213 | C8 | — | 8 | 1 | no | — |
| 214 | C8 | — | 568 | 1 | no | — |
| 215 | C8 | — | 71 | 2 | no | — |
| 216 | C8 | — | 71 | 2 | no | — |
| 217 | C8 | #14556084 | 1 | 1 | yes | characteristic |
| 218 | C8 | #14556085 | 1 | 1 | yes | characteristic |
| 219 | C8 | #14556118 | 1 | 1 | yes | characteristic |
| 220 | C8 | #14556118 | 1 | 1 | yes | characteristic |
| 221 | C22 | — | 2 | 2 | no | — |
| 222 | C22 | #106×D71 | 1 | 1 | yes | characteristic |
| 223 | C22 | #17986344 | 1 | 1 | yes | characteristic |
| 224 | C22 | #17986348 | 1 | 1 | yes | characteristic |
| 225 | C4 | — | 16 | 1 | no | — |
| 226 | C4 | — | 71 | 4 | no | — |
| 227 | C4 | — | 1136 | 1 | no | — |
| 228 | C4 | — | 142 | 2 | no | — |
| 229 | C4 | #17985863 | 1 | 1 | yes | characteristic |
| 230 | C4 | #107×D71 | 1 | 1 | yes | characteristic |
| 231 | C4 | #17986344 | 1 | 1 | yes | characteristic |
| 232 | C4 | #17986348 | 1 | 1 | yes | characteristic |
| 233 | C2 | — | 2 | 1 | no | — |
| 234 | C2 | — | 2 | 1 | no | — |
| 235 | C2 | #22588466 | 1 | 1 | yes | characteristic |
| 236 | C2 | #365×D71 | 1 | 2 | yes | — |
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