Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1144 conjugacy classes of subgroups; only the 269 normal subgroups are shown.
This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 269 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×#381 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #381 | D53 |
| hypocenter | C53 | C2×#381 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #381 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810486 | C2 | semidirect |
| 4. #16810485 | C2 | semidirect |
| 5. #16810484 | C2 | semidirect |
| 6. #95×D53 | C2 | semidirect |
| 7. C53×#381 | C2 | semidirect |
| 11. C22×C4×D53 | C4 | semidirect |
| 18. #13133124 | C22 | semidirect |
| 19. #13133124 | C22 | semidirect |
| 22. #95×C53 | C22 | semidirect |
| 30. C22×C4×C53 | C2×C4 | semidirect |
| 34. C22×Dic53 | C2×C4 | semidirect |
| 43. C2×C4×D53 | C2×C4 | semidirect |
| 96. D53 | #381 | direct (N × H) |
| 102. #381 | D53 | direct (N × H) |
| 104. C53 | C2×#381 | semidirect |
| 119. #95 | C2×D53 | semidirect |
| 138. C22×C4 | C4×D53 | semidirect |