Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
448 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 448 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 | C22×C22⋊D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | D42×C53 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | D42 | D53 |
| hypocenter | C53 | #3324 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ D42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811343 | C2 | semidirect |
| 2. #16810969 | C2 | semidirect |
| 3. #16810944 | C2 | semidirect |
| 4. #16810903 | C2 | semidirect |
| 5. #16810867 | C2 | semidirect |
| 6. #16810820 | C2 | semidirect |
| 7. #16810774 | C2 | semidirect |
| 8. C4×#11719923 | C2 | semidirect |
| 9. #16810701 | C2 | semidirect |
| 10. #16810628 | C2 | semidirect |
| 13. C2×#11719923 | C22 | semidirect |
| 14. C2×#11719917 | C22 | semidirect |
| 15. #13133217 | C22 | semidirect |
| 18. #13133196 | C22 | semidirect |
| 19. C4×Dic106 | C22 | semidirect |
| 21. #13133176 | C22 | semidirect |
| 22. #13133150 | C22 | semidirect |
| 23. #13133148 | C22 | semidirect |
| 24. C4⋊#88322 | C22 | semidirect |
| 25. #13133140 | C22 | semidirect |
| 28. #13133138 | C22 | semidirect |
| 29. #13133138 | C22 | semidirect |
| 30. #13133138 | C22 | semidirect |
| 31. C22⋊#88322 | C22 | semidirect |
| 32. C4×#88322 | C22 | semidirect |
| 147. C53 | #3324 | semidirect |