Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
472 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 472 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 | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#462 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #462 | D53 |
| hypocenter | C53 | #3276 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #462 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133249 | C2 | semidirect |
| 2. Q8×D212 | C2 | semidirect |
| 4. #16811002 | C2 | semidirect |
| 5. #16810880 | C2 | semidirect |
| 6. #16810829 | C2 | semidirect |
| 7. #16810788 | C2 | semidirect |
| 8. #16810777 | C2 | semidirect |
| 9. C4×#11719924 | C2 | semidirect |
| 10. #16810700 | C2 | semidirect |
| 11. C53×#462 | C2 | semidirect |
| 67. #11719924 | D4 | semidirect |
| 123. C4×C53 | #137 | semidirect |
| 140. #462 | D53 | semidirect |
| 146. C53 | #3276 | semidirect |
| 261. C4 | #13133249 | semidirect |