Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
488 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#459 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #459 | D53 |
| hypocenter | C53 | #3109 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811171 | C2 | semidirect |
| 2. C4×D4×D53 | C2 | semidirect |
| 3. #16810940 | C2 | semidirect |
| 4. #16810914 | C2 | semidirect |
| 5. #16810913 | C2 | semidirect |
| 6. #16810815 | C2 | semidirect |
| 7. #16810755 | C2 | semidirect |
| 8. #16810735 | C2 | semidirect |
| 9. #16810732 | C2 | semidirect |
| 10. #16810679 | C2 | semidirect |
| 11. C53×#459 | C2 | semidirect |
| 16. #13133217 | C22 | semidirect |
| 23. D4×Dic53 | C22 | semidirect |
| 30. #13133149 | C22 | semidirect |
| 31. #13133148 | C22 | semidirect |
| 32. C4⋊#88322 | C22 | semidirect |
| 35. #13133138 | C22 | semidirect |
| 37. #13133127 | C22 | semidirect |
| 39. #127×C53 | C22 | semidirect |
| 41. C4⋊C8×C53 | C22 | semidirect |
| 42. #102×C53 | C22 | semidirect |
| 96. D4×C53 | #41 | semidirect |
| 112. D212 | #41 | semidirect |
| 161. #459 | D53 | semidirect |
| 166. C53 | #3109 | semidirect |
| 173. #127 | C2×D53 | semidirect |
| 191. C4⋊C8 | C2×D53 | semidirect |
| 195. #102 | C2×D53 | semidirect |
| 256. D4 | #11719947 | semidirect |