Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
540 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 217 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C2×#124×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#124 | D53 |
| hypocenter | C53 | #2337 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16810530 | C2 | semidirect |
| 2. #16810516 | C2 | semidirect |
| 3. #16810516 | C2 | semidirect |
| 5. #16810498 | C2 | semidirect |
| 6. #16810491 | C2 | semidirect |
| 7. C4×C4⋊Dic53 | C2 | semidirect |
| 8. C22⋊C4×Dic53 | C2 | semidirect |
| 28. #124×C53 | C4 | semidirect |
| 34. C2×Q8×C53 | C2×C4 | semidirect |
| 48. C22⋊C4×C53 | C2×C4 | semidirect |
| 49. C22⋊C4×C53 | C2×C4 | semidirect |
| 50. C42×C53 | C2×C4 | semidirect |
| 128. C53 | #2337 | semidirect |
| 132. #124 | Dic53 | semidirect |
| 153. C2×Q8 | C2×Dic53 | semidirect |
| 172. C22⋊C4 | C2×Dic53 | semidirect |
| 173. C22⋊C4 | C2×Dic53 | semidirect |
| 177. C42 | C2×Dic53 | semidirect |