Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16810502 |
| commutator | C2×C4×C53 | C22×C4 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#568 | C2 |
| socle | C2×C53 | #386 |
| hypercenter | #568 | D53 |
| hypocenter | C53 | #1903 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #568 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16810640 | C2 | semidirect |
| 3. #16810629 | C2 | semidirect |
| 4. C2×#13133133 | C2 | semidirect |
| 5. #16810395 | C2 | semidirect |
| 6. #16810381 | C2 | semidirect |
| 7. C53×#568 | C2 | semidirect |
| 11. C2×#11719924 | C4 | semidirect |
| 12. C2×#11719922 | C4 | semidirect |
| 114. #568 | D53 | semidirect |
| 120. C53 | #1903 | semidirect |