← C71×#12911 #31833783
Subgroups — C71×#12911 #31833783
Order 18176.
266 conjugacy classes of subgroups.
The subgroup lattice has 188 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#368×C71 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C71×#2158 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C71×#2155 | C2 | 1 | 2 | yes | maximal |
| 4 | C71×#2108 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C71×#1833 | C2 | 1 | 2 | yes | maximal |
| 6 | C2×M32×C71 | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×C16×C71 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#108×C71 | C22 | 1 | 2 | yes | — |
| 9 | #427×C71 | C22 | 1 | 2 | yes | — |
| 10 | #405×C71 | C22 | 1 | 1 | yes | characteristic |
| 11 | #368×C71 | — | 2 | 1 | no | — |
| 12 | #368×C71 | — | 2 | 1 | no | — |
| 13 | #367×C71 | — | 2 | 1 | no | — |
| 14 | #366×C71 | — | 2 | 1 | no | — |
| 15 | #365×C71 | — | 2 | 4 | no | — |
| 16 | #346×C71 | C4 | 1 | 2 | yes | complemented |
| 17 | C4×C16×C71 | C4 | 1 | 2 | yes | complemented |
| 18 | #344×C71 | — | 2 | 2 | no | — |
| 19 | H(3,ℤ/4)×C71 | — | 2 | 2 | no | — |
| 20 | C2×M16×C71 | — | 2 | 2 | no | — |
| 21 | C2×M16×C71 | D4 | 1 | 1 | yes | characteristic |
| 22 | C22×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 23 | #117×C71 | — | 2 | 2 | no | — |
| 24 | #117×C71 | D4 | 1 | 1 | yes | characteristic |
| 25 | M32×C71 | D4 | 1 | 2 | yes | — |
| 26 | M32×C71 | Q8 | 1 | 2 | yes | — |
| 27 | C2×C16×C71 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C2×C16×C71 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C2×C16×C71 | C2×C4 | 1 | 2 | yes | — |
| 30 | C2×C16×C71 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×C16×C71 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×C16×C71 | Q8 | 1 | 1 | yes | characteristic |
| 33 | C2×C16×C71 | Q8 | 1 | 1 | yes | characteristic |
| 34 | #108×C71 | — | 2 | 2 | no | — |
| 35 | #108×C71 | — | 2 | 2 | no | — |
| 36 | #107×C71 | — | 2 | 2 | no | — |
| 37 | #106×C71 | — | 2 | 2 | no | — |
| 38 | #97×C71 | Q8 | 1 | 2 | yes | — |
| 39 | C4×C8×C71 | C2×C4 | 1 | 2 | yes | — |
| 40 | C22×C4×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 41 | M16×C71 | — | 2 | 1 | no | — |
| 42 | M16×C71 | — | 2 | 1 | no | — |
| 43 | M16×C71 | — | 4 | 2 | no | — |
| 44 | M16×C71 | — | 4 | 2 | no | — |
| 45 | C2×C8×C71 | — | 4 | 2 | no | — |
| 46 | C2×C8×C71 | C4⋊C4 | 1 | 2 | yes | — |
| 47 | C2×C8×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 48 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 49 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 50 | C2×C8×C71 | C2×Q8 | 1 | 1 | yes | characteristic |
| 51 | C2×C8×C71 | C2×Q8 | 1 | 1 | yes | characteristic |
| 52 | C2×C8×C71 | C2×Q8 | 1 | 1 | yes | characteristic |
| 53 | C4⋊C4×C71 | — | 2 | 1 | no | — |
| 54 | C4⋊C4×C71 | — | 4 | 2 | no | — |
| 55 | C22⋊C4×C71 | — | 2 | 1 | no | — |
| 56 | C22⋊C4×C71 | — | 4 | 2 | no | — |
| 57 | C42×C71 | — | 4 | 2 | no | — |
| 58 | C42×C71 | C4⋊C4 | 1 | 2 | yes | — |
| 59 | C16×C71 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 60 | C16×C71 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C16×C71 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 62 | C16×C71 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 63 | C16×C71 | C4⋊C4 | 1 | 4 | yes | — |
| 64 | C23×C71 | #135 | 1 | 1 | yes | characteristic |
| 65 | C2×C4×C71 | — | 2 | 2 | no | — |
| 66 | C2×C4×C71 | — | 8 | 2 | no | — |
| 67 | C2×C4×C71 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 68 | C2×C4×C71 | #127 | 1 | 1 | yes | characteristic |
| 69 | C2×C4×C71 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 70 | C2×C4×C71 | #135 | 1 | 1 | yes | characteristic |
| 71 | C2×C4×C71 | #135 | 1 | 1 | yes | characteristic |
| 72 | C2×C4×C71 | #135 | 1 | 1 | yes | characteristic |
| 73 | C8×C71 | — | 2 | 2 | no | — |
| 74 | C8×C71 | — | 8 | 2 | no | — |
| 75 | C8×C71 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 76 | C8×C71 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 77 | C8×C71 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 78 | C8×C71 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 79 | C22×C71 | — | 4 | 1 | no | — |
| 80 | C22×C71 | #427 | 1 | 1 | yes | characteristic |
| 81 | C22×C71 | #495 | 1 | 1 | yes | characteristic |
| 82 | C22×C71 | #499 | 1 | 1 | yes | characteristic |
| 83 | C4×C71 | — | 4 | 2 | no | — |
| 84 | C4×C71 | — | 16 | 2 | no | — |
| 85 | C4×C71 | #384 | 1 | 1 | yes | characteristic |
| 86 | C4×C71 | #427 | 1 | 1 | yes | characteristic |
| 87 | C4×C71 | #495 | 1 | 1 | yes | characteristic |
| 88 | C4×C71 | #499 | 1 | 1 | yes | characteristic |
| 89 | #12911 | C71 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 90 | C2×C71 | — | 2 | 1 | no | — |
| 91 | C2×C71 | — | 2 | 1 | no | — |
| 92 | C2×C71 | — | 2 | 1 | no | — |
| 93 | C2×C71 | #1846 | 1 | 1 | yes | characteristic |
| 94 | C2×#368 | C2×C71 | 1 | 1 | yes | characteristic |
| 95 | #2158 | C2×C71 | 1 | 1 | yes | characteristic |
| 96 | #2155 | C2×C71 | 1 | 2 | yes | — |
| 97 | #2108 | C2×C71 | 1 | 1 | yes | characteristic |
| 98 | #1833 | C2×C71 | 1 | 2 | yes | — |
| 99 | C71 | #12911 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 100 | C2×M32 | C22×C71 | 1 | 1 | yes | characteristic |
| 101 | C22×C16 | C22×C71 | 1 | 1 | yes | characteristic |
| 102 | C2×#108 | C22×C71 | 1 | 2 | yes | — |
| 103 | #427 | C22×C71 | 1 | 2 | yes | — |
| 104 | #405 | C22×C71 | 1 | 1 | yes | characteristic |
| 105 | #368 | — | 2 | 1 | no | — |
| 106 | #368 | — | 2 | 1 | no | — |
| 107 | #367 | — | 2 | 1 | no | — |
| 108 | #366 | — | 2 | 1 | no | — |
| 109 | #365 | — | 2 | 4 | no | — |
| 110 | #346 | C4×C71 | 1 | 2 | yes | complemented |
| 111 | C4×C16 | C4×C71 | 1 | 2 | yes | complemented |
| 112 | #344 | — | 2 | 2 | no | — |
| 113 | H(3,ℤ/4) | — | 2 | 2 | no | — |
| 114 | C2×M16 | — | 2 | 2 | no | — |
| 115 | C2×M16 | D4×C71 | 1 | 1 | yes | characteristic |
| 116 | C22×C8 | C23×C71 | 1 | 1 | yes | characteristic |
| 117 | #117 | — | 2 | 2 | no | — |
| 118 | #117 | D4×C71 | 1 | 1 | yes | characteristic |
| 119 | M32 | D4×C71 | 1 | 2 | yes | — |
| 120 | M32 | Q8×C71 | 1 | 2 | yes | — |
| 121 | C2×C16 | C2×C4×C71 | 1 | 1 | yes | characteristic |
| 122 | C2×C16 | C2×C4×C71 | 1 | 2 | yes | — |
| 123 | C2×C16 | C2×C4×C71 | 1 | 1 | yes | characteristic |
| 124 | C2×C16 | D4×C71 | 1 | 1 | yes | characteristic |
| 125 | C2×C16 | D4×C71 | 1 | 1 | yes | characteristic |
| 126 | C2×C16 | Q8×C71 | 1 | 1 | yes | characteristic |
| 127 | C2×C16 | Q8×C71 | 1 | 1 | yes | characteristic |
| 128 | #108 | — | 2 | 2 | no | — |
| 129 | #108 | — | 2 | 2 | no | — |
| 130 | #107 | — | 2 | 2 | no | — |
| 131 | #106 | — | 2 | 2 | no | — |
| 132 | #97 | Q8×C71 | 1 | 2 | yes | — |
| 133 | C4×C8 | C2×C4×C71 | 1 | 2 | yes | — |
| 134 | C22×C4 | C2×D4×C71 | 1 | 1 | yes | characteristic |
| 135 | M16 | — | 2 | 1 | no | — |
| 136 | M16 | — | 2 | 1 | no | — |
| 137 | M16 | — | 4 | 2 | no | — |
| 138 | M16 | — | 4 | 2 | no | — |
| 139 | C2×C8 | — | 4 | 2 | no | — |
| 140 | C2×C8 | C4⋊C4×C71 | 1 | 2 | yes | — |
| 141 | C2×C8 | C22×C4×C71 | 1 | 1 | yes | characteristic |
| 142 | C2×C8 | C2×D4×C71 | 1 | 1 | yes | characteristic |
| 143 | C2×C8 | C2×D4×C71 | 1 | 1 | yes | characteristic |
| 144 | C2×C8 | C2×Q8×C71 | 1 | 1 | yes | characteristic |
| 145 | C2×C8 | C2×Q8×C71 | 1 | 1 | yes | characteristic |
| 146 | C2×C8 | C2×Q8×C71 | 1 | 1 | yes | characteristic |
| 147 | C4⋊C4 | — | 2 | 1 | no | — |
| 148 | C4⋊C4 | — | 4 | 2 | no | — |
| 149 | C22⋊C4 | — | 2 | 1 | no | — |
| 150 | C22⋊C4 | — | 4 | 2 | no | — |
| 151 | C42 | — | 4 | 2 | no | — |
| 152 | C42 | C4⋊C4×C71 | 1 | 2 | yes | — |
| 153 | C16 | C4⋊C4×C71 | 1 | 1 | yes | characteristic |
| 154 | C16 | C4⋊C4×C71 | 1 | 1 | yes | characteristic |
| 155 | C16 | C4⋊C4×C71 | 1 | 1 | yes | characteristic |
| 156 | C16 | C4⋊C4×C71 | 1 | 1 | yes | characteristic |
| 157 | C16 | C4⋊C4×C71 | 1 | 4 | yes | — |
| 158 | C23 | #135×C71 | 1 | 1 | yes | characteristic |
| 159 | C2×C4 | — | 2 | 2 | no | — |
| 160 | C2×C4 | — | 8 | 2 | no | — |
| 161 | C2×C4 | C2×C4⋊C4×C71 | 1 | 1 | yes | characteristic |
| 162 | C2×C4 | #127×C71 | 1 | 1 | yes | characteristic |
| 163 | C2×C4 | C4⋊Q8×C71 | 1 | 1 | yes | characteristic |
| 164 | C2×C4 | #135×C71 | 1 | 1 | yes | characteristic |
| 165 | C2×C4 | #135×C71 | 1 | 1 | yes | characteristic |
| 166 | C2×C4 | #135×C71 | 1 | 1 | yes | characteristic |
| 167 | C8 | — | 2 | 2 | no | — |
| 168 | C8 | — | 8 | 2 | no | — |
| 169 | C8 | C2×C4⋊C4×C71 | 1 | 1 | yes | characteristic |
| 170 | C8 | C2×C4⋊C4×C71 | 1 | 1 | yes | characteristic |
| 171 | C8 | C4⋊Q8×C71 | 1 | 1 | yes | characteristic |
| 172 | C8 | C4⋊Q8×C71 | 1 | 1 | yes | characteristic |
| 173 | C22 | — | 4 | 1 | no | — |
| 174 | C22 | #427×C71 | 1 | 1 | yes | characteristic |
| 175 | C22 | #495×C71 | 1 | 1 | yes | characteristic |
| 176 | C22 | #499×C71 | 1 | 1 | yes | characteristic |
| 177 | C4 | — | 4 | 2 | no | — |
| 178 | C4 | — | 16 | 2 | no | — |
| 179 | C4 | #384×C71 | 1 | 1 | yes | characteristic |
| 180 | C4 | #427×C71 | 1 | 1 | yes | characteristic |
| 181 | C4 | #495×C71 | 1 | 1 | yes | characteristic |
| 182 | C4 | #499×C71 | 1 | 1 | yes | characteristic |
| 183 | C2 | — | 2 | 1 | no | — |
| 184 | C2 | — | 2 | 1 | no | — |
| 185 | C2 | — | 2 | 1 | no | — |
| 186 | C2 | C71×#1846 | 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