Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
216 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133192 |
| commutator | C2×C4×C53 | C22×C4 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#339 | C2 |
| socle | C2×C53 | #388 |
| hypercenter | #339 | D53 |
| hypocenter | C53 | #1932 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #339 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16810672 | C2 | semidirect |
| 5. C4×#11719917 | C2 | semidirect |
| 6. #16810393 | C2 | semidirect |
| 7. C53×#339 | C2 | semidirect |
| 16. #13133135 | C4 | semidirect |
| 17. #13133133 | C4 | semidirect |
| 102. #339 | D53 | semidirect |
| 103. C53 | #1932 | semidirect |