Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
740 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 740 conjugacy classes of subgroups — too many to draw. Only its 170 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 | #127×C53 | C4 |
| hypercenter | #127 | #6157 |
| hypocenter | C53 | #2328 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×#6157 | C2 | semidirect |
| 2. #16811575 | C2 | semidirect |
| 3. #16811571 | C2 | semidirect |
| 4. #16811567 | C2 | semidirect |
| 5. #16811559 | C2 | semidirect |
| 18. #13133214 | C4 | semidirect |
| 19. #13133213 | C4 | semidirect |
| 20. #13133213 | C4 | semidirect |
| 22. #127×C53 | C4 | semidirect |
| 29. C2×D4×C53 | C2×C4 | semidirect |
| 30. C2×D4×C53 | C2×C4 | semidirect |
| 31. C2×C22⋊D53 | C2×C4 | semidirect |
| 32. C2×C22⋊D53 | C2×C4 | semidirect |
| 36. C2×D212 | C2×C4 | semidirect |
| 37. C2×D212 | C2×C4 | semidirect |
| 44. C4×#6157 | D4 | semidirect |
| 45. C42×C53 | C2×C4 | semidirect |
| 46. C4×Dic53 | C2×C4 | semidirect |
| 47. C4×Dic53 | C2×C4 | semidirect |
| 106. C53 | #2328 | semidirect |
| 113. #127 | #6157 | semidirect |
| 131. C2×D4 | C2×#6157 | semidirect |
| 132. C2×D4 | C2×#6157 | semidirect |
| 147. C42 | C2×#6157 | semidirect |