Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
560 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 560 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 | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C53×#391 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #391 | D53 |
| hypocenter | C53 | #2372 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16810530 | C2 | semidirect |
| 3. #16810515 | C2 | semidirect |
| 4. #16810513 | C2 | semidirect |
| 6. C4×#11719921 | C2 | semidirect |
| 7. #16810470 | C2 | semidirect |
| 8. #16810468 | C2 | semidirect |
| 9. C53×#391 | C2 | semidirect |
| 12. #13133206 | C4 | semidirect |
| 35. C2×Q8×C53 | C2×C4 | semidirect |
| 55. #11719921 | C2×C4 | semidirect |
| 56. #11719921 | C2×C4 | semidirect |
| 61. C4×Dic53 | C2×C4 | semidirect |
| 126. #391 | D53 | semidirect |
| 133. C53 | #2372 | semidirect |
| 159. C2×Q8 | C4×D53 | semidirect |