Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
280 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #16811000 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C82×C53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×D424 | — | 2 | 12 | no | — |
| 4 | #13133214 | C22 | 1 | 4 | yes | — |
| 5 | C4×C8×C53 | C22 | 1 | 3 | yes | — |
| 6 | C2×D212 | — | 2 | 12 | no | — |
| 7 | C2×C8×C53 | D4 | 1 | 6 | yes | — |
| 8 | C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 9 | D424 | — | 4 | 24 | no | — |
| 10 | C22×D53 | — | 4 | 4 | no | — |
| 11 | C2×C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 12 | D212 | — | 4 | 24 | no | — |
| 13 | C8×C53 | D8 | 1 | 12 | yes | complemented |
| 14 | C22×C53 | #127 | 1 | 1 | yes | characteristic |
| 15 | C2×D53 | — | 8 | 12 | no | — |
| 16 | C4×C53 | C2×D8 | 1 | 6 | yes | — |
| 17 | #1712 | — | 53 | 1 | no | maximal, Sylow |
| 18 | C2×C53 | #493 | 1 | 3 | yes | — |
| 19 | D53 | — | 16 | 4 | no | — |
| 20 | #493 | — | 53 | 6 | no | — |
| 21 | C82 | D53 | 1 | 1 | yes | characteristic, complemented |
| 22 | C53 | #1712 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 23 | C2×D8 | — | 106 | 12 | no | — |
| 24 | #127 | — | 53 | 4 | no | — |
| 25 | C4×C8 | C2×D53 | 1 | 3 | yes | — |
| 26 | C2×D4 | — | 106 | 12 | no | — |
| 27 | D8 | — | 212 | 24 | no | — |
| 28 | C2×C8 | D212 | 1 | 6 | yes | — |
| 29 | C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 30 | C23 | — | 212 | 4 | no | — |
| 31 | D4 | — | 212 | 24 | no | — |
| 32 | C2×C4 | C2×D212 | 1 | 3 | yes | — |
| 33 | C8 | D424 | 1 | 12 | yes | complemented |
| 34 | C22 | — | 424 | 12 | no | — |
| 35 | C22 | #13133214 | 1 | 1 | yes | characteristic |
| 36 | C4 | C2×D424 | 1 | 6 | yes | — |
| 37 | C2 | — | 848 | 4 | no | — |
| 38 | C2 | #16811000 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133214 |
| commutator | C42×C53 | C23 |
| Frattini | C42 | C22×D53 |
| Fitting | C82×C53 | C2 |
| socle | C22×C53 | #127 |
| hypercenter | C82 | D53 |
| hypocenter | C53 | #1712 |
| LCS | G ▷ C42×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811000 | C2 | semidirect |
| 2. C82×C53 | C2 | semidirect |
| 13. C8×C53 | D8 | semidirect |
| 21. C82 | D53 | semidirect |
| 22. C53 | #1712 | semidirect |
| 33. C8 | D424 | semidirect |