Finite Group DatabasebetaFinite Group DatabasebetaOrder 6834.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C17×C67 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D17×C67 | — | 3 | 1 | no | maximal, Hall |
| 3 | C17×C67 | D3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | D3×C67 | — | 17 | 1 | no | maximal, Hall |
| 5 | C3×C67 | D17 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C2×C67 | — | 51 | 1 | no | Hall |
| 7 | D51 | C67 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 8 | C67 | D51 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 9 | C3×C17 | C2×C67 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | D17 | — | 3 | 1 | no | Hall |
| 11 | C17 | D3×C67 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | D3 | — | 17 | 1 | no | Hall |
| 13 | C3 | D17×C67 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2 | — | 51 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C67 | D51 |
| commutator, hypocenter | C3×C17 | C2×C67 |
| Frattini | C1 | G |
| Fitting, socle | C3×C17×C67 | C2 |
| LCS | G ▷ C3×C17 | |
| UCS | C1 ◁ C67 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C17×C67 | C2 | semidirect |
| 3. C17×C67 | D3 | semidirect |
| 5. C3×C67 | D17 | semidirect |
| 7. D51 | C67 | direct (N × H) |
| 8. C67 | D51 | direct (N × H) |
| 9. C3×C17 | C2×C67 | semidirect |
| 11. C17 | D3×C67 | semidirect |
| 13. C3 | D17×C67 | semidirect |