Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1296 conjugacy classes of subgroups; only the 517 normal subgroups are shown.
This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 517 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D53 |
| commutator, socle | C22×C53 | C25 |
| Fitting | C53×#543 | C2 |
| hypercenter | #543 | D53 |
| hypocenter | C53 | C2×#543 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #543 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811249 | C2 | semidirect |
| 2. #16811248 | C2 | semidirect |
| 3. C22⋊Q8×D53 | C2 | semidirect |
| 5. C53×#543 | C2 | semidirect |
| 7. C2×Q8×D53 | C22 | semidirect |
| 9. #13133197 | C22 | semidirect |
| 10. #13133196 | C22 | semidirect |
| 11. #13133195 | C22 | semidirect |
| 12. #13133194 | C22 | semidirect |
| 13. #13133193 | C22 | semidirect |
| 14. C4⋊C4×D53 | C22 | semidirect |
| 19. C22⋊Q8×C53 | C22 | semidirect |
| 23. C23×D53 | Q8 | semidirect |
| 26. C2×Q8×C53 | C23 | semidirect |
| 33. C2×Dic106 | C23 | semidirect |
| 34. C4⋊C4×C53 | C23 | semidirect |
| 38. C4⋊Dic53 | C23 | semidirect |
| 39. #11719919 | C23 | semidirect |
| 41. C23×C53 | C2×Q8 | semidirect |
| 47. C22×D53 | C2×Q8 | semidirect |
| 75. D53 | #543 | direct (N × H) |
| 78. #543 | D53 | direct (N × H) |
| 81. C53 | C2×#543 | semidirect |
| 90. C22⋊Q8 | C2×D53 | semidirect |
| 104. C2×Q8 | C22×D53 | semidirect |
| 112. C4⋊C4 | C22×D53 | semidirect |
| 124. C23 | Q8×D53 | semidirect |