← C2×C67×#2276 #30713249
Subgroups — C2×C67×#2276 #30713249
Order 17152.
3654 conjugacy classes of subgroups.
The subgroup lattice has 144 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×C4×Q8×C67 | C2 | 1 | 3 | yes | maximal |
| 2 | C67×#2276 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 3 | C2×C4×C4⋊C4×C67 | C2 | 1 | 9 | yes | maximal |
| 4 | C2×#376×C67 | C2 | 1 | 3 | yes | maximal |
| 5 | C23×Q8×C67 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×C4×Q8×C67 | C4 | 1 | 48 | yes | complemented |
| 7 | C2×C4×Q8×C67 | C22 | 1 | 24 | yes | — |
| 8 | C22×C4⋊C4×C67 | C22 | 1 | 9 | yes | — |
| 9 | C22×C42×C67 | C22 | 1 | 6 | yes | — |
| 10 | C22×C42×C67 | C22 | 1 | 9 | yes | — |
| 11 | C4×C4⋊C4×C67 | C22 | 1 | 72 | yes | — |
| 12 | #376×C67 | C22 | 1 | 24 | yes | — |
| 13 | C2×#95×C67 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#95×C67 | C22 | 1 | 9 | yes | — |
| 15 | C22×Q8×C67 | C2×C4 | 1 | 24 | yes | — |
| 16 | C22×Q8×C67 | C23 | 1 | 4 | yes | — |
| 17 | C23×C4×C67 | C23 | 1 | 9 | yes | — |
| 18 | C23×C4×C67 | C23 | 1 | 3 | yes | — |
| 19 | C23×C4×C67 | C23 | 1 | 3 | yes | — |
| 20 | C4×Q8×C67 | C2×C4 | 1 | 192 | yes | complemented |
| 21 | C2×C4⋊C4×C67 | C2×C4 | 1 | 72 | yes | — |
| 22 | C2×C4⋊C4×C67 | C23 | 1 | 36 | yes | — |
| 23 | C2×C42×C67 | — | 4 | 12 | no | — |
| 24 | C2×C42×C67 | C2×C4 | 1 | 72 | yes | — |
| 25 | C2×C42×C67 | C23 | 1 | 24 | yes | — |
| 26 | C2×C42×C67 | C23 | 1 | 36 | yes | — |
| 27 | #95×C67 | C23 | 1 | 4 | yes | — |
| 28 | #95×C67 | C23 | 1 | 36 | yes | — |
| 29 | C24×C67 | C24 | 1 | 1 | yes | characteristic |
| 30 | C2×Q8×C67 | C42 | 1 | 64 | yes | complemented |
| 31 | C2×Q8×C67 | C22×C4 | 1 | 48 | yes | — |
| 32 | C22×C4×C67 | — | 2 | 18 | no | — |
| 33 | C22×C4×C67 | — | 2 | 12 | no | — |
| 34 | C22×C4×C67 | — | 2 | 18 | no | — |
| 35 | C22×C4×C67 | C22×C4 | 1 | 12 | yes | — |
| 36 | C22×C4×C67 | C22×C4 | 1 | 36 | yes | — |
| 37 | C22×C4×C67 | C22×C4 | 1 | 36 | yes | — |
| 38 | C22×C4×C67 | C24 | 1 | 18 | yes | — |
| 39 | C22×C4×C67 | C24 | 1 | 6 | yes | — |
| 40 | C22×C4×C67 | C24 | 1 | 6 | yes | — |
| 41 | C4⋊C4×C67 | C22×C4 | 1 | 144 | yes | — |
| 42 | C42×C67 | — | 4 | 24 | no | — |
| 43 | C42×C67 | C22×C4 | 1 | 144 | yes | — |
| 44 | C23×C67 | C23×C4 | 1 | 6 | yes | — |
| 45 | C23×C67 | H(5,𝔽2) | 1 | 2 | yes | — |
| 46 | C23×C67 | 2−1+4 | 1 | 6 | yes | — |
| 47 | C23×C67 | C25 | 1 | 1 | yes | characteristic |
| 48 | Q8×C67 | C2×C42 | 1 | 64 | yes | complemented |
| 49 | C2×C4×C67 | — | 2 | 24 | no | — |
| 50 | C2×C4×C67 | — | 2 | 72 | no | — |
| 51 | C2×C4×C67 | — | 2 | 18 | no | — |
| 52 | C2×C4×C67 | — | 2 | 12 | no | — |
| 53 | C2×C4×C67 | — | 2 | 18 | no | — |
| 54 | C2×C4×C67 | C2×C42 | 1 | 48 | yes | — |
| 55 | C2×C4×C67 | C23×C4 | 1 | 12 | yes | — |
| 56 | C2×C4×C67 | C23×C4 | 1 | 36 | yes | — |
| 57 | C2×C4×C67 | C23×C4 | 1 | 36 | yes | — |
| 58 | C22×C67 | C22×C42 | 1 | 4 | yes | — |
| 59 | C22×C67 | #519 | 1 | 12 | yes | — |
| 60 | C22×C67 | #520 | 1 | 12 | yes | — |
| 61 | C22×C67 | C24×C4 | 1 | 3 | yes | — |
| 62 | C22×C67 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 63 | C22×C67 | C2×2−1+4 | 1 | 3 | yes | — |
| 64 | C4×C67 | — | 2 | 12 | no | — |
| 65 | C4×C67 | — | 2 | 36 | no | — |
| 66 | C4×C67 | C22×C42 | 1 | 24 | yes | — |
| 67 | C2×#2276 | C67 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 68 | C2×C67 | #2276 | 1 | 8 | yes | direct product factor |
| 69 | C2×C67 | C23×C42 | 1 | 1 | yes | characteristic |
| 70 | C2×C67 | C2×#519 | 1 | 3 | yes | — |
| 71 | C2×C67 | C2×#520 | 1 | 3 | yes | — |
| 72 | C22×C4×Q8 | C2×C67 | 1 | 3 | yes | — |
| 73 | #2276 | C2×C67 | 1 | 16 | yes | direct product factor |
| 74 | C2×C4×C4⋊C4 | C2×C67 | 1 | 9 | yes | — |
| 75 | C2×#376 | C2×C67 | 1 | 3 | yes | — |
| 76 | C67 | C2×#2276 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 77 | C23×Q8 | C22×C67 | 1 | 1 | yes | characteristic |
| 78 | C2×C4×Q8 | C4×C67 | 1 | 48 | yes | complemented |
| 79 | C2×C4×Q8 | C22×C67 | 1 | 24 | yes | — |
| 80 | C22×C4⋊C4 | C22×C67 | 1 | 9 | yes | — |
| 81 | C22×C42 | C22×C67 | 1 | 6 | yes | — |
| 82 | C22×C42 | C22×C67 | 1 | 9 | yes | — |
| 83 | C4×C4⋊C4 | C22×C67 | 1 | 72 | yes | — |
| 84 | #376 | C22×C67 | 1 | 24 | yes | — |
| 85 | C2×#95 | C22×C67 | 1 | 1 | yes | characteristic |
| 86 | C2×#95 | C22×C67 | 1 | 9 | yes | — |
| 87 | C22×Q8 | C2×C4×C67 | 1 | 24 | yes | — |
| 88 | C22×Q8 | C23×C67 | 1 | 4 | yes | — |
| 89 | C23×C4 | C23×C67 | 1 | 9 | yes | — |
| 90 | C23×C4 | C23×C67 | 1 | 3 | yes | — |
| 91 | C23×C4 | C23×C67 | 1 | 3 | yes | — |
| 92 | C4×Q8 | C2×C4×C67 | 1 | 192 | yes | complemented |
| 93 | C2×C4⋊C4 | C2×C4×C67 | 1 | 72 | yes | — |
| 94 | C2×C4⋊C4 | C23×C67 | 1 | 36 | yes | — |
| 95 | C2×C42 | — | 4 | 12 | no | — |
| 96 | C2×C42 | C2×C4×C67 | 1 | 72 | yes | — |
| 97 | C2×C42 | C23×C67 | 1 | 24 | yes | — |
| 98 | C2×C42 | C23×C67 | 1 | 36 | yes | — |
| 99 | #95 | C23×C67 | 1 | 4 | yes | — |
| 100 | #95 | C23×C67 | 1 | 36 | yes | — |
| 101 | C24 | C24×C67 | 1 | 1 | yes | characteristic |
| 102 | C2×Q8 | C42×C67 | 1 | 64 | yes | complemented |
| 103 | C2×Q8 | C22×C4×C67 | 1 | 48 | yes | — |
| 104 | C22×C4 | — | 2 | 18 | no | — |
| 105 | C22×C4 | — | 2 | 12 | no | — |
| 106 | C22×C4 | — | 2 | 18 | no | — |
| 107 | C22×C4 | C22×C4×C67 | 1 | 12 | yes | — |
| 108 | C22×C4 | C22×C4×C67 | 1 | 36 | yes | — |
| 109 | C22×C4 | C22×C4×C67 | 1 | 36 | yes | — |
| 110 | C22×C4 | C24×C67 | 1 | 6 | yes | — |
| 111 | C22×C4 | C24×C67 | 1 | 18 | yes | — |
| 112 | C22×C4 | C24×C67 | 1 | 6 | yes | — |
| 113 | C4⋊C4 | C22×C4×C67 | 1 | 144 | yes | — |
| 114 | C42 | — | 4 | 24 | no | — |
| 115 | C42 | C22×C4×C67 | 1 | 144 | yes | — |
| 116 | C23 | C23×C4×C67 | 1 | 6 | yes | — |
| 117 | C23 | H(5,𝔽2)×C67 | 1 | 2 | yes | — |
| 118 | C23 | 2−1+4×C67 | 1 | 6 | yes | — |
| 119 | C23 | C25×C67 | 1 | 1 | yes | characteristic |
| 120 | Q8 | C2×C42×C67 | 1 | 64 | yes | complemented |
| 121 | C2×C4 | — | 2 | 24 | no | — |
| 122 | C2×C4 | — | 2 | 72 | no | — |
| 123 | C2×C4 | — | 2 | 18 | no | — |
| 124 | C2×C4 | — | 2 | 12 | no | — |
| 125 | C2×C4 | — | 2 | 18 | no | — |
| 126 | C2×C4 | C2×C42×C67 | 1 | 48 | yes | — |
| 127 | C2×C4 | C23×C4×C67 | 1 | 12 | yes | — |
| 128 | C2×C4 | C23×C4×C67 | 1 | 36 | yes | — |
| 129 | C2×C4 | C23×C4×C67 | 1 | 36 | yes | — |
| 130 | C22 | C22×C42×C67 | 1 | 4 | yes | — |
| 131 | C22 | #519×C67 | 1 | 12 | yes | — |
| 132 | C22 | #520×C67 | 1 | 12 | yes | — |
| 133 | C22 | C24×C4×C67 | 1 | 3 | yes | — |
| 134 | C22 | C2×H(5,𝔽2)×C67 | 1 | 1 | yes | characteristic |
| 135 | C22 | C2×2−1+4×C67 | 1 | 3 | yes | — |
| 136 | C4 | — | 2 | 12 | no | — |
| 137 | C4 | — | 2 | 36 | no | — |
| 138 | C4 | C22×C42×C67 | 1 | 24 | yes | — |
| 139 | C2 | C67×#2276 | 1 | 8 | yes | direct product factor |
| 140 | C2 | C23×C42×C67 | 1 | 1 | yes | characteristic |
| 141 | C2 | C2×#519×C67 | 1 | 3 | yes | — |
| 142 | C2 | C2×#520×C67 | 1 | 3 | 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