Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
596 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 596 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×#121×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#121 | D53 |
| hypocenter | C53 | #2333 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16810526 | C2 | semidirect |
| 2. #16810516 | C2 | semidirect |
| 3. #16810509 | C2 | semidirect |
| 4. #16810503 | C2 | semidirect |
| 5. #16810503 | C2 | semidirect |
| 6. #16810486 | C2 | semidirect |
| 7. C22⋊C4×Dic53 | C2 | semidirect |
| 8. C4×C22⋊Dic53 | C2 | semidirect |
| 29. #121×C53 | C4 | semidirect |
| 39. C2×D4×C53 | C2×C4 | semidirect |
| 42. C22×C4×C53 | C2×C4 | semidirect |
| 53. C4⋊C4×C53 | C2×C4 | semidirect |
| 54. C22⋊C4×C53 | C2×C4 | semidirect |
| 144. C53 | #2333 | semidirect |
| 149. #121 | Dic53 | semidirect |
| 174. C2×D4 | C2×Dic53 | semidirect |
| 184. C22×C4 | C2×Dic53 | semidirect |
| 189. C4⋊C4 | C2×Dic53 | semidirect |
| 196. C22⋊C4 | C2×Dic53 | semidirect |