Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
816 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 229 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C53×#380 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #380 | D53 |
| hypocenter | C53 | #2315 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810497 | C2 | semidirect |
| 4. #16810494 | C2 | semidirect |
| 5. #16810483 | C2 | semidirect |
| 6. #16810481 | C2 | semidirect |
| 7. C53×#380 | C2 | semidirect |
| 11. C2×C4⋊Dic53 | C22 | semidirect |
| 12. C2×#11719919 | C22 | semidirect |
| 14. C2×#11719921 | C4 | semidirect |
| 19. #13133124 | C22 | semidirect |
| 20. #13133124 | C22 | semidirect |
| 22. C2×C4⋊C4×C53 | C22 | semidirect |
| 24. #95×C53 | C22 | semidirect |
| 33. C22×C4×C53 | C2×C4 | semidirect |
| 38. C22×Dic53 | C2×C4 | semidirect |
| 48. #11719921 | C2×C4 | semidirect |
| 49. #11719921 | C2×C4 | semidirect |
| 64. C22×D53 | C4⋊C4 | semidirect |
| 118. #380 | D53 | semidirect |
| 119. C53 | #2315 | semidirect |
| 126. C2×C4⋊C4 | C2×D53 | semidirect |
| 136. #95 | C2×D53 | semidirect |
| 152. C22×C4 | C4×D53 | semidirect |