Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
532 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 125 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×#493 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #493 | D53 |
| hypocenter | C53 | #3205 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811171 | C2 | semidirect |
| 2. #127×D53 | C2 | semidirect |
| 3. #16810999 | C2 | semidirect |
| 4. #16810970 | C2 | semidirect |
| 5. #16810969 | C2 | semidirect |
| 6. #16810816 | C2 | semidirect |
| 7. #16810755 | C2 | semidirect |
| 8. #16810674 | C2 | semidirect |
| 9. C53×#493 | C2 | semidirect |
| 14. C2×#11719922 | C22 | semidirect |
| 15. #13133217 | C22 | semidirect |
| 24. D4×Dic53 | C22 | semidirect |
| 26. #13133146 | C22 | semidirect |
| 27. #13133139 | C22 | semidirect |
| 28. #13133138 | C22 | semidirect |
| 30. C4×#88322 | C22 | semidirect |
| 31. C2×D8×C53 | C22 | semidirect |
| 33. #127×C53 | C22 | semidirect |
| 34. C4×C8×C53 | C22 | semidirect |
| 100. C4×D53 | D8 | semidirect |
| 136. #493 | D53 | semidirect |
| 142. C53 | #3205 | semidirect |
| 148. C2×D8 | C2×D53 | semidirect |
| 155. #127 | C2×D53 | semidirect |
| 167. C4×C8 | C2×D53 | semidirect |