Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
262 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #16811365 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #16811013 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #16811012 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #16810362 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C53×#329 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#11719913 | — | 2 | 2 | no | — |
| 7 | C2×#11719924 | — | 2 | 1 | no | — |
| 8 | C2×D424 | — | 2 | 2 | no | — |
| 9 | C2×#11719922 | — | 2 | 1 | no | — |
| 10 | #13133214 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #13133206 | — | 2 | 1 | no | — |
| 12 | #13133205 | C22 | 1 | 2 | yes | — |
| 13 | #13133203 | — | 2 | 2 | no | — |
| 14 | #13133187 | — | 2 | 2 | no | — |
| 15 | #13133185 | — | 2 | 1 | no | — |
| 16 | #13133128 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | #124×C53 | C22 | 1 | 1 | yes | characteristic |
| 18 | #97×C53 | C22 | 1 | 2 | yes | complemented |
| 19 | C23×D53 | — | 2 | 1 | no | — |
| 20 | C2×Q8×C53 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×D4×C53 | D4 | 1 | 1 | yes | characteristic |
| 22 | C2×C22⋊D53 | — | 2 | 2 | no | — |
| 23 | C2×D212 | — | 2 | 1 | no | — |
| 24 | C2×D212 | — | 2 | 2 | no | — |
| 25 | C2×D212 | — | 2 | 1 | no | — |
| 26 | C2×D212 | D4 | 1 | 2 | yes | — |
| 27 | C2×Dic106 | D4 | 1 | 2 | yes | — |
| 28 | C2×C8×C53 | — | 2 | 2 | no | — |
| 29 | C22⋊C4×C53 | — | 2 | 2 | no | — |
| 30 | C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 31 | C22⋊Dic53 | — | 2 | 1 | no | — |
| 32 | #11719924 | — | 4 | 2 | no | — |
| 33 | #11719922 | — | 4 | 2 | no | — |
| 34 | #11719921 | — | 2 | 2 | no | — |
| 35 | #11719921 | — | 2 | 2 | no | — |
| 36 | #11719921 | — | 2 | 2 | no | — |
| 37 | C4×Dic53 | — | 2 | 1 | no | — |
| 38 | C2×#88322 | — | 2 | 1 | no | — |
| 39 | D424 | — | 4 | 4 | no | — |
| 40 | #11719913 | — | 4 | 4 | no | — |
| 41 | C23×C53 | — | 2 | 1 | no | — |
| 42 | C22×D53 | — | 2 | 2 | no | — |
| 43 | C22×D53 | — | 4 | 1 | no | — |
| 44 | C22×D53 | — | 8 | 2 | no | — |
| 45 | C22×D53 | — | 8 | 1 | no | — |
| 46 | Q8×C53 | — | 4 | 1 | no | — |
| 47 | D4×C53 | — | 4 | 1 | no | — |
| 48 | C2×C4×C53 | — | 2 | 1 | no | — |
| 49 | C2×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 50 | C2×C4×C53 | C2×D4 | 1 | 2 | yes | — |
| 51 | C22⋊D53 | — | 8 | 2 | no | — |
| 52 | C2×Dic53 | — | 2 | 2 | no | — |
| 53 | D212 | — | 4 | 2 | no | — |
| 54 | D212 | — | 4 | 2 | no | — |
| 55 | D212 | — | 4 | 2 | no | — |
| 56 | D212 | — | 4 | 1 | no | — |
| 57 | D212 | — | 4 | 1 | no | — |
| 58 | D212 | — | 8 | 1 | no | — |
| 59 | Dic106 | — | 4 | 2 | no | — |
| 60 | C8×C53 | — | 2 | 4 | no | — |
| 61 | #88322 | — | 2 | 2 | no | — |
| 62 | C22×C53 | — | 4 | 2 | no | — |
| 63 | C22×C53 | — | 4 | 1 | no | — |
| 64 | C22×C53 | #120 | 1 | 1 | yes | characteristic |
| 65 | C2×D53 | — | 4 | 2 | no | — |
| 66 | C2×D53 | — | 4 | 2 | no | — |
| 67 | C2×D53 | — | 4 | 2 | no | — |
| 68 | C2×D53 | — | 8 | 4 | no | — |
| 69 | C2×D53 | — | 8 | 2 | no | — |
| 70 | C2×D53 | — | 8 | 1 | no | — |
| 71 | C4×C53 | — | 2 | 1 | no | — |
| 72 | C4×C53 | — | 2 | 2 | no | — |
| 73 | C4×C53 | — | 4 | 1 | no | — |
| 74 | Dic53 | — | 4 | 2 | no | — |
| 75 | #1655 | — | 53 | 1 | no | maximal, Sylow |
| 76 | C2×C53 | — | 8 | 1 | no | — |
| 77 | C2×C53 | #453 | 1 | 2 | yes | — |
| 78 | C2×C53 | #453 | 1 | 1 | yes | characteristic |
| 79 | D53 | — | 8 | 2 | no | — |
| 80 | D53 | — | 16 | 1 | no | — |
| 81 | H(3,𝔽4) | — | 53 | 1 | no | — |
| 82 | #496 | — | 53 | 1 | no | — |
| 83 | #496 | — | 53 | 2 | no | — |
| 84 | #329 | — | 53 | 2 | no | — |
| 85 | #329 | D53 | 1 | 1 | yes | characteristic, complemented |
| 86 | C53 | #1655 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 87 | C2×SD16 | — | 106 | 2 | no | — |
| 88 | C2×SD16 | — | 106 | 1 | no | — |
| 89 | C2×D8 | — | 106 | 2 | no | — |
| 90 | C2×D8 | — | 106 | 1 | no | — |
| 91 | #127 | — | 53 | 1 | no | — |
| 92 | #124 | — | 53 | 2 | no | — |
| 93 | #124 | — | 106 | 2 | no | — |
| 94 | #124 | — | 106 | 1 | no | — |
| 95 | #124 | C2×D53 | 1 | 1 | yes | characteristic |
| 96 | #120 | — | 106 | 2 | no | — |
| 97 | #120 | — | 106 | 1 | no | — |
| 98 | #97 | — | 53 | 1 | no | — |
| 99 | #97 | C2×D53 | 1 | 2 | yes | complemented |
| 100 | C24 | — | 106 | 1 | no | — |
| 101 | C2×Q8 | — | 53 | 2 | no | — |
| 102 | C2×Q8 | D212 | 1 | 1 | yes | characteristic |
| 103 | C2×D4 | — | 53 | 2 | no | — |
| 104 | C2×D4 | — | 106 | 1 | no | — |
| 105 | C2×D4 | — | 106 | 2 | no | — |
| 106 | C2×D4 | — | 106 | 1 | no | — |
| 107 | C2×D4 | — | 106 | 2 | no | — |
| 108 | C2×D4 | D212 | 1 | 1 | yes | characteristic |
| 109 | SD16 | — | 212 | 4 | no | — |
| 110 | SD16 | — | 212 | 2 | no | — |
| 111 | D8 | — | 212 | 4 | no | — |
| 112 | D8 | — | 212 | 2 | no | — |
| 113 | C2×C8 | — | 2 | 2 | no | — |
| 114 | C2×C8 | — | 106 | 1 | no | — |
| 115 | C22⋊C4 | — | 2 | 2 | no | — |
| 116 | C22⋊C4 | — | 106 | 2 | no | — |
| 117 | C22⋊C4 | — | 106 | 1 | no | — |
| 118 | C22⋊C4 | — | 106 | 2 | no | — |
| 119 | C22⋊C4 | — | 106 | 2 | no | — |
| 120 | C42 | — | 106 | 1 | no | — |
| 121 | C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 122 | C23 | — | 2 | 1 | no | — |
| 123 | C23 | — | 106 | 2 | no | — |
| 124 | C23 | — | 424 | 2 | no | — |
| 125 | C23 | — | 424 | 1 | no | — |
| 126 | C23 | — | 212 | 1 | no | — |
| 127 | Q8 | — | 4 | 1 | no | — |
| 128 | Q8 | — | 212 | 2 | no | — |
| 129 | D4 | — | 4 | 1 | no | — |
| 130 | D4 | — | 424 | 1 | no | — |
| 131 | D4 | — | 424 | 2 | no | — |
| 132 | D4 | — | 212 | 2 | no | — |
| 133 | D4 | — | 212 | 1 | no | — |
| 134 | D4 | — | 212 | 2 | no | — |
| 135 | D4 | — | 212 | 1 | no | — |
| 136 | D4 | — | 212 | 2 | no | — |
| 137 | C2×C4 | — | 2 | 1 | no | — |
| 138 | C2×C4 | — | 106 | 2 | no | — |
| 139 | C2×C4 | C2×D212 | 1 | 1 | yes | characteristic |
| 140 | C2×C4 | D4×D53 | 1 | 2 | yes | — |
| 141 | C8 | — | 2 | 4 | no | — |
| 142 | C8 | — | 106 | 2 | no | — |
| 143 | C22 | — | 4 | 2 | no | — |
| 144 | C22 | — | 4 | 1 | no | — |
| 145 | C22 | — | 424 | 4 | no | — |
| 146 | C22 | — | 424 | 2 | no | — |
| 147 | C22 | — | 424 | 1 | no | — |
| 148 | C22 | — | 212 | 2 | no | — |
| 149 | C22 | — | 212 | 2 | no | — |
| 150 | C22 | — | 212 | 2 | no | — |
| 151 | C22 | #13133187 | 1 | 1 | yes | characteristic |
| 152 | C4 | — | 2 | 2 | no | — |
| 153 | C4 | — | 2 | 1 | no | — |
| 154 | C4 | — | 4 | 1 | no | — |
| 155 | C4 | — | 212 | 2 | no | — |
| 156 | C2 | — | 8 | 1 | no | — |
| 157 | C2 | — | 848 | 1 | no | — |
| 158 | C2 | — | 424 | 2 | no | — |
| 159 | C2 | #16810792 | 1 | 2 | yes | — |
| 160 | C2 | #16810793 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133187 |
| commutator | C42×C53 | C23 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×#329 | C2 |
| socle | C22×C53 | #120 |
| hypercenter | #329 | D53 |
| hypocenter | C53 | #1655 |
| LCS | G ▷ C42×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #329 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811365 | C2 | semidirect |
| 2. #16811013 | C2 | semidirect |
| 3. #16811012 | C2 | semidirect |
| 4. #16810362 | C2 | semidirect |
| 5. C53×#329 | C2 | semidirect |
| 10. #13133214 | C22 | semidirect |
| 16. #13133128 | C22 | semidirect |
| 18. #97×C53 | C22 | semidirect |
| 85. #329 | D53 | semidirect |
| 86. C53 | #1655 | semidirect |
| 99. #97 | C2×D53 | semidirect |