Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 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, socle | C23×C53 | C24 |
| Fitting | C22×C22⋊C4×C53 | C2 |
| hypercenter | C22×C22⋊C4 | D53 |
| hypocenter | C53 | #2404 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C22×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#11719945 | C2 | semidirect |
| 2. C2×#13133186 | C2 | semidirect |
| 3. #16810535 | C2 | semidirect |
| 4. #16810532 | C2 | semidirect |
| 5. #16810502 | C2 | semidirect |
| 6. #16810499 | C2 | semidirect |
| 7. #16810486 | C2 | semidirect |
| 8. C22×C22⋊C4×C53 | C2 | semidirect |
| 11. C22×Dic106 | C22 | semidirect |
| 12. C22×D212 | C22 | semidirect |
| 15. C22×C4×D53 | C22 | semidirect |
| 19. C2×#11719921 | C22 | semidirect |
| 20. #13133124 | C22 | semidirect |
| 21. #13133124 | C22 | semidirect |
| 22. #13133124 | C22 | semidirect |
| 39. C22×C4×C53 | D4 | semidirect |
| 48. C2×C22⋊D53 | D4 | semidirect |
| 76. C23×C53 | #41 | semidirect |
| 136. C22×C22⋊C4 | D53 | semidirect |
| 142. C53 | #2404 | semidirect |
| 189. C22×C4 | C22⋊D53 | semidirect |
| 213. C23 | #11719945 | semidirect |