Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 688 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 | C22×C53 | C23×C4 |
| Fitting | C53×#379 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #379 | D53 |
| hypocenter | C53 | #2375 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #379 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16810523 | C2 | semidirect |
| 3. #16810517 | C2 | semidirect |
| 4. #16810511 | C2 | semidirect |
| 5. #16810499 | C2 | semidirect |
| 6. #16810479 | C2 | semidirect |
| 7. C53×#379 | C2 | semidirect |
| 9. C22×C4×D53 | C22 | semidirect |
| 10. #13133186 | C4 | semidirect |
| 11. C2×#11719919 | C22 | semidirect |
| 15. C2×#11719921 | C22 | semidirect |
| 16. C2×C4×Dic53 | C22 | semidirect |
| 17. #13133124 | C22 | semidirect |
| 18. #13133124 | C22 | semidirect |
| 24. C24×C53 | C2×C4 | semidirect |
| 40. C22⋊Dic53 | C2×C4 | semidirect |
| 102. #379 | D53 | semidirect |
| 103. C53 | #2375 | semidirect |
| 121. C24 | C4×D53 | semidirect |