Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
492 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C22⋊D53 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#519 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #519 | D53 |
| hypocenter | C53 | #2949 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #519 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810721 | C2 | semidirect |
| 4. C4×#11719924 | C2 | semidirect |
| 5. #16810659 | C2 | semidirect |
| 6. #16810632 | C2 | semidirect |
| 8. #16810579 | C2 | semidirect |
| 9. C53×#519 | C2 | semidirect |
| 11. C2×#11719924 | C4 | semidirect |
| 20. #13133150 | C22 | semidirect |
| 21. #13133148 | C22 | semidirect |
| 22. #13133141 | C22 | semidirect |
| 24. #13133128 | C22 | semidirect |
| 25. C4×#88322 | C22 | semidirect |
| 28. C4×Q8×C53 | C22 | semidirect |
| 35. C2×Q8×C53 | C2×C4 | semidirect |
| 53. #11719924 | C2×C4 | semidirect |
| 54. #11719924 | C2×C4 | semidirect |
| 59. C2×#88322 | C2×C4 | semidirect |
| 105. #519 | D53 | semidirect |
| 113. C53 | #2949 | semidirect |
| 122. C4×Q8 | C2×D53 | semidirect |
| 139. C2×Q8 | C4×D53 | semidirect |