Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
548 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#459 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #459 | D53 |
| hypocenter | C53 | #3036 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133241 | C2 | semidirect |
| 2. #16811171 | C2 | semidirect |
| 3. C4×D4×D53 | C2 | semidirect |
| 4. #16810837 | C2 | semidirect |
| 5. #16810836 | C2 | semidirect |
| 6. #16810829 | C2 | semidirect |
| 7. #16810828 | C2 | semidirect |
| 8. #16810815 | C2 | semidirect |
| 9. #16810645 | C2 | semidirect |
| 10. #16810621 | C2 | semidirect |
| 11. C53×#459 | C2 | semidirect |
| 20. #13133217 | C22 | semidirect |
| 21. #13133204 | C22 | semidirect |
| 34. C4⋊#88322 | C22 | semidirect |
| 35. #13133146 | C22 | semidirect |
| 36. #13133142 | C22 | semidirect |
| 37. #13133140 | C22 | semidirect |
| 38. #13133138 | C22 | semidirect |
| 40. #127×C53 | C22 | semidirect |
| 42. C4⋊C8×C53 | C22 | semidirect |
| 43. #102×C53 | C22 | semidirect |
| 57. D4×D53 | D4 | semidirect |
| 145. C4×C53 | AGL(1,ℤ/8) | semidirect |
| 169. #459 | D53 | semidirect |
| 172. C53 | #3036 | semidirect |
| 183. #127 | C2×D53 | semidirect |
| 200. C4⋊C8 | C2×D53 | semidirect |
| 204. #102 | C2×D53 | semidirect |
| 311. C4 | #13133241 | semidirect |