Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
520 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | #124×C53 | C4 |
| hypercenter | #124 | #6157 |
| hypocenter | C53 | #2379 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. (C2×#6157)⋊Q8 | C2 | semidirect |
| 2. #16811573 | C2 | semidirect |
| 4. #16811570 | C2 | semidirect |
| 5. #16811567 | C2 | semidirect |
| 19. #13133206 | C4 | semidirect |
| 20. #13133205 | C4 | semidirect |
| 21. #13133204 | C4 | semidirect |
| 22. #13133203 | C4 | semidirect |
| 25. #124×C53 | C4 | semidirect |
| 31. C2×Q8×C53 | C2×C4 | semidirect |
| 42. C2×Dic106 | C2×C4 | semidirect |
| 43. C2×Dic106 | C2×C4 | semidirect |
| 50. C22⋊C4×C53 | C2×C4 | semidirect |
| 51. C42×C53 | C2×C4 | semidirect |
| 52. C22⋊Dic53 | C2×C4 | semidirect |
| 53. #11719921 | C2×C4 | semidirect |
| 54. #11719921 | C2×C4 | semidirect |
| 55. C4×Dic53 | C2×C4 | semidirect |
| 56. C4×Dic53 | C2×C4 | semidirect |
| 123. C53 | #2379 | semidirect |
| 131. #124 | #6157 | semidirect |
| 148. C2×Q8 | C2×#6157 | semidirect |
| 169. C22⋊C4 | C2×#6157 | semidirect |
| 173. C42 | C2×#6157 | semidirect |