Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.
This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 469 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×#560 |
| 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. #16811360 | C2 | direct (N × H) |
| 2. C2×#13133211 | C2 | semidirect |
| 3. C2×#13133198 | C2 | semidirect |
| 4. C2×#13133189 | C2 | semidirect |
| 5. C2×#13133185 | C2 | semidirect |
| 6. C2×#127×C53 | C2 | semidirect |
| 9. C23×Dic53 | C22 | semidirect |
| 10. #13133211 | C22 | semidirect |
| 11. #13133198 | C22 | semidirect |
| 12. #13133189 | C22 | semidirect |
| 13. #13133185 | C22 | semidirect |
| 14. C2×#11719919 | C22 | semidirect |
| 16. C2×C22⋊Dic53 | C22 | semidirect |
| 19. C22×D4×C53 | C22 | semidirect |
| 20. #127×C53 | C22 | semidirect |
| 21. C2×C42×C53 | C22 | semidirect |
| 30. C2×D4×C53 | C23 | semidirect |
| 36. C22×Dic53 | C23 | semidirect |
| 37. C42×C53 | C23 | semidirect |
| 39. C22⋊Dic53 | C23 | semidirect |
| 41. #11719919 | C23 | semidirect |
| 75. C2×C53 | #560 | semidirect |
| 79. C2×#127 | D53 | semidirect |
| 84. C53 | C2×#560 | semidirect |
| 88. C22×D4 | C2×D53 | semidirect |
| 90. #127 | C2×D53 | semidirect |
| 99. C2×C42 | C2×D53 | semidirect |
| 110. C2×D4 | C22×D53 | semidirect |
| 119. C42 | C22×D53 | semidirect |
| 153. C2 | #16811360 | direct (N × H) |