Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
540 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C4×C4⋊C4×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C4×C4⋊C4 | D53 |
| hypocenter | C53 | #2319 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810519 | C2 | semidirect |
| 4. #16810514 | C2 | semidirect |
| 5. #16810485 | C2 | semidirect |
| 6. #16810482 | C2 | semidirect |
| 7. #16810469 | C2 | semidirect |
| 8. C4×C4⋊C4×C53 | C2 | semidirect |
| 11. C4×D212 | C4 | semidirect |
| 103. C4×C4⋊C4 | D53 | semidirect |
| 105. C53 | #2319 | semidirect |