Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
354 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 117 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×#500 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #500 | D53 |
| hypocenter | C53 | #3357 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #500 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811334 | C2 | semidirect |
| 3. #16810966 | C2 | semidirect |
| 4. #16810938 | C2 | semidirect |
| 5. #16810915 | C2 | semidirect |
| 6. #16810910 | C2 | semidirect |
| 7. #16810870 | C2 | semidirect |
| 8. C4×#11719913 | C2 | semidirect |
| 9. #16810702 | C2 | semidirect |
| 10. #16810664 | C2 | semidirect |
| 11. C53×#500 | C2 | semidirect |
| 124. #500 | D53 | semidirect |
| 133. C53 | #3357 | semidirect |