Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D212 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#431 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #431 | D53 |
| hypocenter | C53 | #3211 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #431 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811157 | C2 | semidirect |
| 2. C2×#13133205 | C2 | semidirect |
| 3. #16811007 | C2 | semidirect |
| 4. #16810978 | C2 | semidirect |
| 5. #16810846 | C2 | semidirect |
| 6. #16810839 | C2 | semidirect |
| 7. #16810781 | C2 | semidirect |
| 8. #16810776 | C2 | semidirect |
| 9. #16810764 | C2 | semidirect |
| 11. C53×#431 | C2 | semidirect |
| 14. C2×Dic212 | C22 | semidirect |
| 16. C2×#11719913 | C22 | semidirect |
| 19. #13133209 | C22 | semidirect |
| 24. #13133205 | C22 | semidirect |
| 28. C4×Dic106 | C22 | semidirect |
| 29. C4×D212 | C22 | semidirect |
| 34. #13133142 | C22 | semidirect |
| 35. #13133142 | C22 | semidirect |
| 39. C4⋊C8×C53 | C22 | semidirect |
| 42. C4×C8×C53 | C22 | semidirect |
| 154. #431 | D53 | semidirect |
| 155. C53 | #3211 | semidirect |
| 179. C4⋊C8 | C2×D53 | semidirect |
| 186. C4×C8 | C2×D53 | semidirect |