Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
372 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 150 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 | #2968 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811675 | C2 | semidirect |
| 2. #16811653 | C2 | semidirect |
| 3. #16811653 | C2 | semidirect |
| 4. #16811651 | C2 | semidirect |
| 5. #16811648 | C2 | semidirect |
| 6. #16811641 | C2 | semidirect |
| 7. #16811588 | C2 | semidirect |
| 8. #16811582 | C2 | semidirect |
| 121. C53 | #2968 | semidirect |