Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1000 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 157 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×#390 | C2 |
| hypercenter | #390 | D53 |
| hypocenter | C53 | #2679 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133187 | C2 | semidirect |
| 2. C2×#13133185 | C2 | semidirect |
| 3. #16810540 | C2 | semidirect |
| 4. #16810539 | C2 | semidirect |
| 5. #16810525 | C2 | semidirect |
| 6. C53×#390 | C2 | semidirect |
| 7. C24×D53 | C22 | semidirect |
| 13. C2×C4⋊Dic53 | C22 | semidirect |
| 14. C2×C22⋊Dic53 | C22 | semidirect |
| 15. C2×#11719921 | C22 | semidirect |
| 16. C2×#11719921 | C22 | semidirect |
| 17. C2×#11719921 | C22 | semidirect |
| 18. #13133124 | C22 | semidirect |
| 20. C2×C4⋊C4×C53 | C22 | semidirect |
| 21. C2×C22⋊C4×C53 | C22 | semidirect |
| 22. C2×C42×C53 | C22 | semidirect |
| 32. C2×D4×C53 | D4 | semidirect |
| 44. C2×D212 | D4 | semidirect |
| 121. #390 | D53 | semidirect |
| 122. C53 | #2679 | semidirect |
| 131. C2×C4⋊C4 | C2×D53 | semidirect |
| 136. C2×C22⋊C4 | C2×D53 | semidirect |
| 137. C2×C42 | C2×D53 | semidirect |
| 156. C2×D4 | D212 | semidirect |