Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1260 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 485 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×#127×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#127 | D53 |
| hypocenter | C53 | C2×#540 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811227 | C2 | direct (N × H) |
| 2. C2×#13133209 | C2 | semidirect |
| 3. C2×#13133198 | C2 | semidirect |
| 4. C2×#13133189 | C2 | semidirect |
| 5. C2×#13133188 | C2 | semidirect |
| 6. C2×D4×Dic53 | C2 | semidirect |
| 7. C2×#13133174 | C2 | semidirect |
| 8. C2×#127×C53 | C2 | semidirect |
| 12. #13133209 | C22 | semidirect |
| 13. #13133198 | C22 | semidirect |
| 14. #13133189 | C22 | semidirect |
| 15. #13133188 | C22 | semidirect |
| 16. D4×Dic53 | C22 | semidirect |
| 17. #13133174 | C22 | semidirect |
| 27. #127×C53 | C22 | semidirect |
| 94. C2×C53 | #540 | semidirect |
| 99. C2×#127 | D53 | semidirect |
| 106. C53 | C2×#540 | semidirect |
| 112. #127 | C2×D53 | semidirect |
| 192. C2 | #16811227 | direct (N × H) |