Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 548 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, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C53×#396 | C2 |
| hypercenter | #396 | D53 |
| hypocenter | C53 | #2842 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #396 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16810560 | C2 | semidirect |
| 2. #16810558 | C2 | semidirect |
| 3. #16810547 | C2 | semidirect |
| 4. #16810546 | C2 | semidirect |
| 5. #16810545 | C2 | semidirect |
| 7. C53×#396 | C2 | semidirect |
| 9. C2×C4⋊Dic53 | C22 | semidirect |
| 10. C2×#11719919 | C22 | semidirect |
| 11. C2×#11719919 | C22 | semidirect |
| 17. #13133124 | C22 | semidirect |
| 18. C2×C4⋊C4×C53 | C22 | semidirect |
| 20. #95×C53 | C22 | semidirect |
| 95. #396 | D53 | semidirect |
| 97. C53 | #2842 | semidirect |
| 102. C2×C4⋊C4 | C2×D53 | semidirect |
| 110. #95 | C2×D53 | semidirect |