Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
456 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 173 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 | C53×#384 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #384 | D53 |
| hypocenter | C53 | #2383 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16810518 | C2 | semidirect |
| 4. #16810504 | C2 | semidirect |
| 6. #16810491 | C2 | semidirect |
| 7. C53×#384 | C2 | semidirect |
| 9. #13133211 | C4 | semidirect |
| 32. C42×C53 | C2×C4 | semidirect |
| 36. #11719919 | C2×C4 | semidirect |
| 78. #384 | D53 | semidirect |
| 80. C53 | #2383 | semidirect |
| 110. C42 | C4×D53 | semidirect |