Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1134 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 165 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 | C2×#123×C53 | C2 |
| hypercenter | C2×#123 | D53 |
| hypocenter | C53 | #2613 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D212 | C2 | semidirect |
| 2. C2×#13133185 | C2 | semidirect |
| 3. #16810540 | C2 | semidirect |
| 4. #16810532 | C2 | semidirect |
| 5. #16810486 | C2 | semidirect |
| 6. #16810478 | C2 | semidirect |
| 7. C2×#123×C53 | C2 | semidirect |
| 12. C22×C22⋊D53 | C22 | semidirect |
| 15. C2×#11719921 | C22 | semidirect |
| 16. C2×#11719921 | C22 | semidirect |
| 17. #13133124 | C22 | semidirect |
| 21. C2×C4⋊C4×C53 | C22 | semidirect |
| 23. C2×C22⋊C4×C53 | C22 | semidirect |
| 29. C23×D53 | D4 | semidirect |
| 34. C22×C4×C53 | D4 | semidirect |
| 105. C2×#123 | D53 | semidirect |
| 111. C53 | #2613 | semidirect |
| 121. C2×C4⋊C4 | C2×D53 | semidirect |
| 126. C2×C22⋊C4 | C2×D53 | semidirect |
| 144. C22×C4 | C22⋊D53 | semidirect |