Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
672 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 153 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 | C4×C22⋊C4×C53 | C2 |
| hypercenter | C4×C22⋊C4 | D53 |
| hypocenter | C53 | #2561 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133196 | C2 | semidirect |
| 3. #16810550 | C2 | semidirect |
| 4. #16810536 | C2 | semidirect |
| 5. #16810514 | C2 | semidirect |
| 6. #16810484 | C2 | semidirect |
| 8. C4×C22⋊C4×C53 | C2 | semidirect |
| 10. C22×Dic106 | C22 | semidirect |
| 13. C2×C4⋊Dic53 | C22 | semidirect |
| 14. C2×#11719919 | C22 | semidirect |
| 21. #13133124 | C22 | semidirect |
| 22. #13133124 | C22 | semidirect |
| 25. C2×C42×C53 | C22 | semidirect |
| 26. #95×C53 | C22 | semidirect |
| 118. C4×C22⋊C4 | D53 | semidirect |
| 119. C53 | #2561 | semidirect |
| 134. C2×C42 | C2×D53 | semidirect |
| 137. #95 | C2×D53 | semidirect |