Finite Group DatabasebetaFinite Group DatabasebetaOrder 6848.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic856 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C107⋊C32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | D16×C107 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | Dic428 | — | 2 | 1 | no | — |
| 5 | D8×C107 | — | 2 | 1 | no | — |
| 6 | C16×C107 | C22 | 1 | 1 | yes | characteristic |
| 7 | D4×C107 | — | 4 | 1 | no | — |
| 8 | Dic214 | — | 4 | 1 | no | — |
| 9 | C8×C107 | D4 | 1 | 1 | yes | characteristic |
| 10 | C22×C107 | — | 8 | 1 | no | — |
| 11 | C4×C107 | D8 | 1 | 1 | yes | characteristic |
| 12 | Dic107 | — | 8 | 1 | no | — |
| 13 | C2×C107 | — | 16 | 1 | no | — |
| 14 | C2×C107 | D16 | 1 | 1 | yes | characteristic |
| 15 | C107 | SD64 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | SD64 | — | 107 | 1 | no | maximal, Sylow |
| 17 | Q32 | — | 107 | 1 | no | — |
| 18 | D16 | D107 | 1 | 1 | yes | characteristic |
| 19 | C32 | — | 107 | 1 | no | — |
| 20 | Q16 | — | 214 | 1 | no | — |
| 21 | D8 | — | 2 | 1 | no | — |
| 22 | C16 | C2×D107 | 1 | 1 | yes | characteristic |
| 23 | Q8 | — | 428 | 1 | no | — |
| 24 | D4 | — | 4 | 1 | no | — |
| 25 | C8 | C22⋊D107 | 1 | 1 | yes | characteristic |
| 26 | C22 | — | 8 | 1 | no | — |
| 27 | C4 | — | 856 | 1 | no | — |
| 28 | C4 | #13133803 | 1 | 1 | yes | characteristic |
| 29 | C2 | — | 16 | 1 | no | — |
| 30 | C2 | #16812050 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16812050 |
| commutator | C16×C107 | C22 |
| Frattini | C16 | C2×D107 |
| Fitting | D16×C107 | C2 |
| socle | C2×C107 | D16 |
| hypercenter | D16 | D107 |
| hypocenter | C107 | SD64 |
| LCS | G ▷ C16×C107 ▷ C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Dic856 | C2 | semidirect |
| 2. C107⋊C32 | C2 | semidirect |
| 15. C107 | SD64 | semidirect |