Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
402 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 144 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 | #123×C53 | C4 |
| hypercenter | #123 | #6157 |
| hypocenter | C53 | #2978 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811657 | C2 | semidirect |
| 2. #16811656 | C2 | semidirect |
| 3. #16811653 | C2 | semidirect |
| 4. #16811652 | C2 | semidirect |
| 5. #16811647 | C2 | semidirect |
| 6. #16811597 | C2 | semidirect |
| 7. #16811596 | C2 | semidirect |
| 8. #16811594 | C2 | semidirect |
| 12. C2×#11719936 | C22 | semidirect |
| 13. C2×#11719940 | C22 | semidirect |
| 14. C2×D53⋊C8 | C22 | semidirect |
| 15. C22×#88324 | C22 | semidirect |
| 16. #13133284 | C22 | semidirect |
| 17. C4⋊#88324 | C22 | semidirect |
| 20. #13133274 | C22 | semidirect |
| 23. C22⋊#88324 | C22 | semidirect |
| 24. #13133272 | C22 | semidirect |
| 25. C4×#88324 | C22 | semidirect |
| 142. C53 | #2978 | semidirect |