Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
520 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 185 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D53 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C53×#424 | C2 |
| hypercenter | #424 | D53 |
| hypocenter | C53 | #2929 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ #424 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133225 | C2 | semidirect |
| 2. C22×C8×D53 | C2 | semidirect |
| 4. #16810693 | C2 | semidirect |
| 5. #16810688 | C2 | semidirect |
| 6. #16810589 | C2 | semidirect |
| 7. #16810586 | C2 | semidirect |
| 8. #16810579 | C2 | semidirect |
| 9. C53×#424 | C2 | semidirect |
| 14. C2×#11719912 | C22 | semidirect |
| 19. C2×C8×D53 | C22 | semidirect |
| 26. #13133129 | C22 | semidirect |
| 27. #13133128 | C22 | semidirect |
| 28. C4×#88322 | C22 | semidirect |
| 32. C4⋊C8×C53 | C22 | semidirect |
| 58. C2×#88322 | D4 | semidirect |
| 101. C2×D53 | #131 | semidirect |
| 119. #424 | D53 | semidirect |
| 123. C53 | #2929 | semidirect |
| 142. C4⋊C8 | C2×D53 | semidirect |