← #31762459
Subgroups — #31762459
Order 17152.
11222 conjugacy classes of subgroups.
The subgroup lattice has 174 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | 2−1+4×Dic67 | C2 | 1 | 12 | yes | maximal, complemented |
| 2 | H(5,𝔽2)×Dic67 | C2 | 1 | 20 | yes | maximal, complemented |
| 3 | #22573412 | C2 | 1 | 15 | yes | maximal, complemented |
| 4 | C2×#16863879 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | C2×#585×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #16863879 | C22 | 1 | 60 | yes | complemented |
| 7 | #16863879 | C22 | 1 | 120 | yes | complemented |
| 8 | #16863879 | C22 | 1 | 20 | yes | complemented |
| 9 | #16863874 | C22 | 1 | 30 | yes | complemented |
| 10 | #16863871 | C22 | 1 | 90 | yes | complemented |
| 11 | #41×Dic67 | C22 | 1 | 120 | yes | complemented |
| 12 | C2×Q8×Dic67 | C22 | 1 | 30 | yes | complemented |
| 13 | C2×D4×Dic67 | C22 | 1 | 90 | yes | complemented |
| 14 | C2×#14524002 | C22 | 1 | 45 | yes | complemented |
| 15 | C22×C4⋊Dic67 | C22 | 1 | 15 | yes | complemented |
| 16 | #585×C67 | C4 | 1 | 32 | yes | complemented |
| 17 | C2×2−1+4×C67 | C22 | 1 | 6 | yes | — |
| 18 | C2×H(5,𝔽2)×C67 | C22 | 1 | 10 | yes | — |
| 19 | C22×#41×C67 | C22 | 1 | 15 | yes | — |
| 20 | C23×Dic67 | C23 | 1 | 30 | yes | — |
| 21 | Q8×Dic67 | C23 | 1 | 120 | yes | — |
| 22 | Q8×Dic67 | C23 | 1 | 40 | yes | — |
| 23 | D4×Dic67 | C23 | 1 | 360 | yes | — |
| 24 | D4×Dic67 | C23 | 1 | 120 | yes | — |
| 25 | #14524002 | C23 | 1 | 60 | yes | — |
| 26 | #14524002 | C23 | 1 | 180 | yes | — |
| 27 | C2×C4⋊Dic67 | C23 | 1 | 45 | yes | — |
| 28 | C2×C4⋊Dic67 | C23 | 1 | 90 | yes | — |
| 29 | C2×C4⋊Dic67 | C23 | 1 | 15 | yes | — |
| 30 | C2×C22⋊Dic67 | C23 | 1 | 90 | yes | — |
| 31 | C2×C4×Dic67 | C23 | 1 | 90 | yes | — |
| 32 | 2−1+4×C67 | C2×C4 | 1 | 96 | yes | complemented |
| 33 | H(5,𝔽2)×C67 | C2×C4 | 1 | 160 | yes | complemented |
| 34 | C2×#41×C67 | C2×C4 | 1 | 240 | yes | complemented |
| 35 | C2×#41×C67 | C23 | 1 | 60 | yes | — |
| 36 | C2×#41×C67 | C23 | 1 | 20 | yes | — |
| 37 | C22×Q8×C67 | C23 | 1 | 15 | yes | — |
| 38 | C22×D4×C67 | C23 | 1 | 45 | yes | — |
| 39 | C23×C4×C67 | C23 | 1 | 15 | yes | — |
| 40 | C22×Dic67 | — | 8 | 30 | no | — |
| 41 | C22×Dic67 | C24 | 1 | 30 | yes | — |
| 42 | C22×Dic67 | C24 | 1 | 90 | yes | — |
| 43 | C4⋊Dic67 | C24 | 1 | 10 | yes | — |
| 44 | C4⋊Dic67 | C24 | 1 | 6 | yes | — |
| 45 | C4⋊Dic67 | C24 | 1 | 90 | yes | — |
| 46 | C4⋊Dic67 | C24 | 1 | 30 | yes | — |
| 47 | C22⋊Dic67 | C24 | 1 | 120 | yes | — |
| 48 | C4×Dic67 | C24 | 1 | 120 | yes | — |
| 49 | C24×C67 | C24 | 1 | 15 | yes | — |
| 50 | #41×C67 | C22×C4 | 1 | 160 | yes | complemented |
| 51 | #41×C67 | C22×C4 | 1 | 480 | yes | complemented |
| 52 | C2×Q8×C67 | C22×C4 | 1 | 120 | yes | complemented |
| 53 | C2×Q8×C67 | C24 | 1 | 20 | yes | — |
| 54 | C2×D4×C67 | C22×C4 | 1 | 360 | yes | complemented |
| 55 | C2×D4×C67 | C24 | 1 | 60 | yes | — |
| 56 | C22×C4×C67 | C22×C4 | 1 | 120 | yes | complemented |
| 57 | C22×C4×C67 | C24 | 1 | 15 | yes | — |
| 58 | C22×C4×C67 | C24 | 1 | 45 | yes | — |
| 59 | C23×C67 | — | 4 | 15 | no | — |
| 60 | C23×C67 | — | 4 | 15 | no | — |
| 61 | C23×C67 | C23×C4 | 1 | 60 | yes | — |
| 62 | C23×C67 | C25 | 1 | 15 | yes | — |
| 63 | Q8×C67 | C23×C4 | 1 | 80 | yes | — |
| 64 | D4×C67 | C23×C4 | 1 | 240 | yes | — |
| 65 | C2×C4×C67 | C23×C4 | 1 | 60 | yes | — |
| 66 | C2×C4×C67 | C23×C4 | 1 | 180 | yes | — |
| 67 | C2×C4×C67 | C25 | 1 | 1 | yes | characteristic |
| 68 | C2×C4×C67 | C25 | 1 | 15 | yes | — |
| 69 | C2×Dic67 | — | 4 | 90 | no | — |
| 70 | C2×Dic67 | — | 4 | 30 | no | — |
| 71 | C2×Dic67 | C25 | 1 | 12 | yes | — |
| 72 | C2×Dic67 | C25 | 1 | 20 | yes | — |
| 73 | C22×C67 | — | 2 | 15 | no | — |
| 74 | C22×C67 | — | 2 | 15 | no | — |
| 75 | C22×C67 | — | 4 | 60 | no | — |
| 76 | C22×C67 | C24×C4 | 1 | 30 | yes | — |
| 77 | C22×C67 | C26 | 1 | 1 | yes | characteristic |
| 78 | C4×C67 | C24×C4 | 1 | 2 | yes | — |
| 79 | C4×C67 | C24×C4 | 1 | 30 | yes | — |
| 80 | Dic67 | — | 2 | 20 | no | — |
| 81 | Dic67 | — | 2 | 12 | no | — |
| 82 | #62654 | — | 67 | 1 | no | maximal, Sylow |
| 83 | C2×C67 | — | 2 | 30 | no | — |
| 84 | C2×C67 | C25×C4 | 1 | 1 | yes | characteristic |
| 85 | C2×C67 | H(7,𝔽2) | 1 | 1 | yes | characteristic |
| 86 | C2×C67 | 2−1+6 | 1 | 1 | yes | characteristic |
| 87 | C2×#585 | D67 | 1 | 1 | yes | characteristic |
| 88 | C4×2−1+4 | — | 67 | 12 | no | — |
| 89 | C4×H(5,𝔽2) | — | 67 | 20 | no | — |
| 90 | #3428 | — | 67 | 15 | no | — |
| 91 | C2×#520 | — | 67 | 15 | no | — |
| 92 | C67 | #62654 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 93 | #585 | Dic67 | 1 | 32 | yes | complemented |
| 94 | C2×2−1+4 | C2×D67 | 1 | 6 | yes | — |
| 95 | C2×H(5,𝔽2) | C2×D67 | 1 | 10 | yes | — |
| 96 | C22×#41 | C2×D67 | 1 | 15 | yes | — |
| 97 | #520 | — | 67 | 60 | no | — |
| 98 | #520 | — | 67 | 120 | no | — |
| 99 | #520 | — | 67 | 20 | no | — |
| 100 | #519 | — | 67 | 30 | no | — |
| 101 | #518 | — | 67 | 90 | no | — |
| 102 | C4×#41 | — | 67 | 120 | no | — |
| 103 | C2×C4×Q8 | — | 67 | 30 | no | — |
| 104 | C2×C4×D4 | — | 67 | 90 | no | — |
| 105 | C2×#117 | — | 67 | 45 | no | — |
| 106 | C22×C4⋊C4 | — | 67 | 15 | no | — |
| 107 | 2−1+4 | C2×Dic67 | 1 | 96 | yes | complemented |
| 108 | H(5,𝔽2) | C2×Dic67 | 1 | 160 | yes | complemented |
| 109 | C2×#41 | C2×Dic67 | 1 | 240 | yes | complemented |
| 110 | C2×#41 | C22×D67 | 1 | 60 | yes | — |
| 111 | C2×#41 | C22×D67 | 1 | 20 | yes | — |
| 112 | C22×Q8 | C22×D67 | 1 | 15 | yes | — |
| 113 | C22×D4 | C22×D67 | 1 | 45 | yes | — |
| 114 | C23×C4 | — | 67 | 30 | no | — |
| 115 | C23×C4 | C22×D67 | 1 | 15 | yes | — |
| 116 | C4×Q8 | — | 67 | 120 | no | — |
| 117 | C4×Q8 | — | 67 | 40 | no | — |
| 118 | C4×D4 | — | 67 | 360 | no | — |
| 119 | C4×D4 | — | 67 | 120 | no | — |
| 120 | #117 | — | 67 | 60 | no | — |
| 121 | #117 | — | 67 | 180 | no | — |
| 122 | C2×C4⋊C4 | — | 67 | 45 | no | — |
| 123 | C2×C4⋊C4 | — | 67 | 90 | no | — |
| 124 | C2×C4⋊C4 | — | 67 | 15 | no | — |
| 125 | C2×C22⋊C4 | — | 67 | 90 | no | — |
| 126 | C2×C42 | — | 67 | 90 | no | — |
| 127 | C24 | C23×D67 | 1 | 15 | yes | — |
| 128 | #41 | C22×Dic67 | 1 | 160 | yes | complemented |
| 129 | #41 | C22×Dic67 | 1 | 480 | yes | complemented |
| 130 | C2×Q8 | C22×Dic67 | 1 | 120 | yes | complemented |
| 131 | C2×Q8 | C23×D67 | 1 | 20 | yes | — |
| 132 | C2×D4 | C22×Dic67 | 1 | 360 | yes | complemented |
| 133 | C2×D4 | C23×D67 | 1 | 60 | yes | — |
| 134 | C22×C4 | — | 536 | 30 | no | — |
| 135 | C22×C4 | — | 67 | 30 | no | — |
| 136 | C22×C4 | — | 67 | 90 | no | — |
| 137 | C22×C4 | C22×Dic67 | 1 | 120 | yes | complemented |
| 138 | C22×C4 | C23×D67 | 1 | 45 | yes | — |
| 139 | C22×C4 | C23×D67 | 1 | 15 | yes | — |
| 140 | C4⋊C4 | — | 67 | 10 | no | — |
| 141 | C4⋊C4 | — | 67 | 6 | no | — |
| 142 | C4⋊C4 | — | 67 | 90 | no | — |
| 143 | C4⋊C4 | — | 67 | 30 | no | — |
| 144 | C22⋊C4 | — | 67 | 120 | no | — |
| 145 | C42 | — | 67 | 120 | no | — |
| 146 | C23 | — | 4 | 15 | no | — |
| 147 | C23 | — | 4 | 15 | no | — |
| 148 | C23 | C23×Dic67 | 1 | 60 | yes | — |
| 149 | C23 | C24×D67 | 1 | 15 | yes | — |
| 150 | Q8 | C23×Dic67 | 1 | 80 | yes | — |
| 151 | D4 | C23×Dic67 | 1 | 240 | yes | — |
| 152 | C2×C4 | — | 268 | 90 | no | — |
| 153 | C2×C4 | — | 268 | 30 | no | — |
| 154 | C2×C4 | — | 67 | 12 | no | — |
| 155 | C2×C4 | — | 67 | 20 | no | — |
| 156 | C2×C4 | C23×Dic67 | 1 | 60 | yes | — |
| 157 | C2×C4 | C23×Dic67 | 1 | 180 | yes | — |
| 158 | C2×C4 | C24×D67 | 1 | 15 | yes | — |
| 159 | C2×C4 | C24×D67 | 1 | 1 | yes | characteristic |
| 160 | C22 | — | 2 | 15 | no | — |
| 161 | C22 | — | 2 | 15 | no | — |
| 162 | C22 | — | 4 | 60 | no | — |
| 163 | C22 | C24×Dic67 | 1 | 30 | yes | — |
| 164 | C22 | C25×D67 | 1 | 1 | yes | characteristic |
| 165 | C4 | — | 134 | 20 | no | — |
| 166 | C4 | — | 134 | 12 | no | — |
| 167 | C4 | C24×Dic67 | 1 | 2 | yes | — |
| 168 | C4 | C24×Dic67 | 1 | 30 | yes | — |
| 169 | C2 | — | 2 | 30 | no | — |
| 170 | C2 | C25×Dic67 | 1 | 1 | yes | characteristic |
| 171 | C2 | #22575020 | 1 | 1 | yes | characteristic |
| 172 | C2 | #22575022 | 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