← #22574920
Subgroups — #22574920
Order 8576.
1424 conjugacy classes of subgroups.
The subgroup lattice has 186 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #16864232 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #16864163 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #16864162 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #16864161 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | C2×#14524049 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #131×D67 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | C2×#131×C67 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #14524091 | C4 | 1 | 2 | yes | — |
| 9 | #14524090 | C4 | 1 | 4 | yes | — |
| 10 | #14524089 | C4 | 1 | 4 | yes | — |
| 11 | #14524088 | C4 | 1 | 6 | yes | — |
| 12 | C2×#11761628 | C22 | 1 | 6 | yes | — |
| 13 | #41×D67 | C22 | 1 | 8 | yes | — |
| 14 | #14524051 | C22 | 1 | 24 | yes | complemented |
| 15 | #14524050 | C22 | 1 | 8 | yes | complemented |
| 16 | #14524049 | C22 | 1 | 12 | yes | complemented |
| 17 | #14524049 | C22 | 1 | 4 | yes | complemented |
| 18 | #14524049 | C22 | 1 | 12 | yes | complemented |
| 19 | #14524049 | C22 | 1 | 12 | yes | complemented |
| 20 | C2×#11761612 | C22 | 1 | 6 | yes | complemented |
| 21 | C2×#11761611 | C22 | 1 | 12 | yes | complemented |
| 22 | M16×D67 | C22 | 1 | 24 | yes | complemented |
| 23 | C2×C8×D67 | C22 | 1 | 12 | yes | complemented |
| 24 | C2×#41×C67 | C22 | 1 | 1 | yes | characteristic |
| 25 | #131×C67 | C22 | 1 | 4 | yes | complemented |
| 26 | #131×C67 | C22 | 1 | 4 | yes | complemented |
| 27 | C2×M16×C67 | C22 | 1 | 3 | yes | complemented |
| 28 | C22×C8×C67 | C22 | 1 | 3 | yes | complemented |
| 29 | #11761629 | C2×C4 | 1 | 8 | yes | — |
| 30 | #11761628 | C2×C4 | 1 | 12 | yes | — |
| 31 | #11761628 | C2×C4 | 1 | 12 | yes | — |
| 32 | #11761628 | C23 | 1 | 4 | yes | — |
| 33 | #11761628 | C23 | 1 | 12 | yes | — |
| 34 | #11761627 | C2×C4 | 1 | 24 | yes | — |
| 35 | C2×Dic134 | C2×C4 | 1 | 6 | yes | — |
| 36 | Q8×D67 | C2×C4 | 1 | 8 | yes | — |
| 37 | C2×D268 | C2×C4 | 1 | 6 | yes | — |
| 38 | C2×C22⋊D67 | C2×C4 | 1 | 12 | yes | — |
| 39 | D4×D67 | C2×C4 | 1 | 24 | yes | — |
| 40 | C2×C4×D67 | C23 | 1 | 12 | yes | — |
| 41 | #11761612 | C23 | 1 | 12 | yes | — |
| 42 | #11761612 | C23 | 1 | 4 | yes | — |
| 43 | #11761611 | C23 | 1 | 24 | yes | — |
| 44 | #11761611 | C23 | 1 | 8 | yes | — |
| 45 | C2×C67⋊C8 | C23 | 1 | 12 | yes | — |
| 46 | C8×D67 | C23 | 1 | 8 | yes | — |
| 47 | C8×D67 | C23 | 1 | 24 | yes | — |
| 48 | #41×C67 | C2×C4 | 1 | 4 | yes | — |
| 49 | #41×C67 | C23 | 1 | 4 | yes | — |
| 50 | C2×Q8×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C2×D4×C67 | C2×C4 | 1 | 3 | yes | — |
| 52 | C22×C4×C67 | C23 | 1 | 3 | yes | — |
| 53 | M16×C67 | C23 | 1 | 12 | yes | — |
| 54 | C2×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 55 | C2×C8×C67 | C23 | 1 | 3 | yes | — |
| 56 | C2×C8×C67 | C23 | 1 | 6 | yes | — |
| 57 | C2×C8×C67 | C23 | 1 | 6 | yes | — |
| 58 | C23×C67 | C22×C4 | 1 | 3 | yes | — |
| 59 | C22×D67 | C22×C4 | 1 | 12 | yes | — |
| 60 | Q8×C67 | C22×C4 | 1 | 4 | yes | — |
| 61 | D4×C67 | C22×C4 | 1 | 12 | yes | — |
| 62 | C2×C4×C67 | C22×C4 | 1 | 3 | yes | — |
| 63 | C2×C4×C67 | C22×C4 | 1 | 6 | yes | — |
| 64 | C2×C4×C67 | C24 | 1 | 1 | yes | characteristic |
| 65 | C2×C4×C67 | C24 | 1 | 6 | yes | — |
| 66 | C22⋊D67 | C22×C4 | 1 | 24 | yes | — |
| 67 | C22⋊D67 | C22×C4 | 1 | 8 | yes | — |
| 68 | C2×Dic67 | C22×C4 | 1 | 12 | yes | — |
| 69 | D268 | C22×C4 | 1 | 12 | yes | — |
| 70 | D268 | C22×C4 | 1 | 4 | yes | — |
| 71 | C4×D67 | C22×C4 | 1 | 24 | yes | — |
| 72 | C4×D67 | C24 | 1 | 8 | yes | — |
| 73 | Dic134 | C22×C4 | 1 | 12 | yes | — |
| 74 | Dic134 | C22×C4 | 1 | 4 | yes | — |
| 75 | C8×C67 | C24 | 1 | 1 | yes | characteristic |
| 76 | C8×C67 | C24 | 1 | 6 | yes | — |
| 77 | C8×C67 | C24 | 1 | 1 | yes | characteristic |
| 78 | C67⋊C8 | C24 | 1 | 8 | yes | — |
| 79 | C22×C67 | — | 4 | 3 | no | — |
| 80 | C22×C67 | C23×C4 | 1 | 6 | yes | — |
| 81 | C22×C67 | C23×C4 | 1 | 1 | yes | characteristic |
| 82 | C2×D67 | — | 4 | 12 | no | — |
| 83 | C2×D67 | C23×C4 | 1 | 8 | yes | — |
| 84 | C4×C67 | C23×C4 | 1 | 1 | yes | characteristic |
| 85 | C4×C67 | C23×C4 | 1 | 6 | yes | — |
| 86 | C4×C67 | C25 | 1 | 1 | yes | characteristic |
| 87 | Dic67 | C23×C4 | 1 | 8 | yes | — |
| 88 | C2×C67 | — | 2 | 6 | no | — |
| 89 | C2×C67 | — | 2 | 1 | no | — |
| 90 | C2×C67 | C24×C4 | 1 | 1 | yes | characteristic |
| 91 | D67 | — | 2 | 8 | no | — |
| 92 | #3573 | — | 67 | 1 | no | maximal, Sylow |
| 93 | C67 | #3573 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 94 | #585 | — | 67 | 1 | no | — |
| 95 | #568 | — | 67 | 1 | no | — |
| 96 | #568 | — | 67 | 8 | no | — |
| 97 | #568 | — | 67 | 6 | no | — |
| 98 | C2×#131 | — | 67 | 8 | no | — |
| 99 | C2×#131 | — | 67 | 6 | no | — |
| 100 | C2×#131 | D67 | 1 | 1 | yes | characteristic, complemented |
| 101 | 2−1+4 | — | 67 | 4 | no | — |
| 102 | 2−1+4 | — | 67 | 2 | no | — |
| 103 | H(5,𝔽2) | — | 67 | 4 | no | — |
| 104 | H(5,𝔽2) | — | 67 | 6 | no | — |
| 105 | C2×#41 | — | 67 | 8 | no | — |
| 106 | C2×#41 | — | 67 | 6 | no | — |
| 107 | C2×#41 | C2×D67 | 1 | 1 | yes | characteristic |
| 108 | #131 | — | 67 | 8 | no | — |
| 109 | #131 | — | 67 | 24 | no | — |
| 110 | #131 | — | 67 | 12 | no | — |
| 111 | #131 | — | 67 | 4 | no | — |
| 112 | #131 | — | 67 | 12 | no | — |
| 113 | #131 | — | 67 | 12 | no | — |
| 114 | #131 | C2×D67 | 1 | 4 | yes | complemented |
| 115 | #131 | C2×D67 | 1 | 4 | yes | complemented |
| 116 | C2×M16 | — | 67 | 6 | no | — |
| 117 | C2×M16 | — | 67 | 24 | no | — |
| 118 | C2×M16 | — | 67 | 12 | no | — |
| 119 | C2×M16 | C2×D67 | 1 | 3 | yes | complemented |
| 120 | C22×C8 | — | 67 | 12 | no | — |
| 121 | C22×C8 | C2×D67 | 1 | 3 | yes | complemented |
| 122 | #41 | — | 67 | 8 | no | — |
| 123 | #41 | — | 67 | 24 | no | — |
| 124 | #41 | — | 67 | 12 | no | — |
| 125 | #41 | — | 67 | 12 | no | — |
| 126 | #41 | — | 67 | 12 | no | — |
| 127 | #41 | — | 67 | 4 | no | — |
| 128 | #41 | C4×D67 | 1 | 4 | yes | — |
| 129 | #41 | C22×D67 | 1 | 4 | yes | — |
| 130 | C2×Q8 | — | 67 | 8 | no | — |
| 131 | C2×Q8 | — | 67 | 6 | no | — |
| 132 | C2×Q8 | C4×D67 | 1 | 1 | yes | characteristic |
| 133 | C2×D4 | — | 67 | 12 | no | — |
| 134 | C2×D4 | — | 67 | 24 | no | — |
| 135 | C2×D4 | — | 67 | 6 | no | — |
| 136 | C2×D4 | C4×D67 | 1 | 3 | yes | — |
| 137 | C22×C4 | — | 67 | 12 | no | — |
| 138 | C22×C4 | C22×D67 | 1 | 3 | yes | — |
| 139 | M16 | — | 67 | 12 | no | — |
| 140 | M16 | — | 67 | 4 | no | — |
| 141 | M16 | — | 67 | 24 | no | — |
| 142 | M16 | — | 67 | 8 | no | — |
| 143 | M16 | C22×D67 | 1 | 12 | yes | — |
| 144 | C2×C8 | — | 67 | 12 | no | — |
| 145 | C2×C8 | — | 67 | 24 | no | — |
| 146 | C2×C8 | — | 67 | 8 | no | — |
| 147 | C2×C8 | C22×D67 | 1 | 6 | yes | — |
| 148 | C2×C8 | C22×D67 | 1 | 3 | yes | — |
| 149 | C2×C8 | C22×D67 | 1 | 6 | yes | — |
| 150 | C2×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 151 | C23 | — | 67 | 12 | no | — |
| 152 | C23 | C2×C4×D67 | 1 | 3 | yes | — |
| 153 | Q8 | — | 67 | 12 | no | — |
| 154 | Q8 | — | 67 | 4 | no | — |
| 155 | Q8 | C2×C4×D67 | 1 | 4 | yes | — |
| 156 | D4 | — | 67 | 24 | no | — |
| 157 | D4 | — | 67 | 8 | no | — |
| 158 | D4 | — | 67 | 12 | no | — |
| 159 | D4 | — | 67 | 4 | no | — |
| 160 | D4 | C2×C4×D67 | 1 | 12 | yes | — |
| 161 | C2×C4 | — | 67 | 12 | no | — |
| 162 | C2×C4 | — | 67 | 24 | no | — |
| 163 | C2×C4 | — | 67 | 8 | no | — |
| 164 | C2×C4 | C2×C4×D67 | 1 | 6 | yes | — |
| 165 | C2×C4 | C2×C4×D67 | 1 | 3 | yes | — |
| 166 | C2×C4 | C23×D67 | 1 | 6 | yes | — |
| 167 | C2×C4 | C23×D67 | 1 | 1 | yes | characteristic |
| 168 | C8 | — | 67 | 8 | no | — |
| 169 | C8 | C23×D67 | 1 | 6 | yes | — |
| 170 | C8 | C23×D67 | 1 | 1 | yes | characteristic |
| 171 | C8 | C23×D67 | 1 | 1 | yes | characteristic |
| 172 | C22 | — | 4 | 3 | no | — |
| 173 | C22 | — | 268 | 12 | no | — |
| 174 | C22 | — | 67 | 8 | no | — |
| 175 | C22 | C22×C4×D67 | 1 | 6 | yes | — |
| 176 | C22 | C22×C4×D67 | 1 | 1 | yes | characteristic |
| 177 | C4 | — | 67 | 8 | no | — |
| 178 | C4 | C22×C4×D67 | 1 | 6 | yes | — |
| 179 | C4 | C22×C4×D67 | 1 | 1 | yes | characteristic |
| 180 | C4 | C24×D67 | 1 | 1 | yes | characteristic |
| 181 | C2 | — | 2 | 1 | no | — |
| 182 | C2 | — | 2 | 6 | no | — |
| 183 | C2 | — | 134 | 8 | no | — |
| 184 | C2 | C23×C4×D67 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Finite Group Databasebeta