Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
484 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 182 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 | #125×C53 | C4 |
| hypercenter | #125 | #6157 |
| hypocenter | C53 | C4×#125 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 18. #13133209 | C4 | semidirect |
| 19. #13133208 | C4 | semidirect |
| 20. #13133207 | C4 | semidirect |
| 24. #125×C53 | C4 | semidirect |
| 80. #6157 | #125 | direct (N × H) |
| 100. C53 | C4×#125 | semidirect |
| 104. #125 | #6157 | direct (N × H) |