Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 338 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 | C22×C22⋊D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | Q82×C53 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | Q82 | D53 |
| hypocenter | C53 | #3369 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810975 | C2 | semidirect |
| 5. #16810905 | C2 | semidirect |
| 7. #16810833 | C2 | semidirect |
| 8. #16810754 | C2 | semidirect |
| 9. #16810728 | C2 | semidirect |
| 11. Q82×C53 | C2 | semidirect |
| 117. Q82 | D53 | semidirect |
| 128. C53 | #3369 | semidirect |