Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133187 |
| commutator | C42×C53 | C23 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×D4⋊C8 | C2 |
| socle | C22×C53 | #120 |
| hypercenter | D4⋊C8 | D53 |
| hypocenter | C53 | #1650 |
| LCS | G ▷ C42×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ D4⋊C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811028 | C2 | semidirect |
| 3. #16810909 | C2 | semidirect |
| 5. #16810368 | C2 | semidirect |
| 6. #16810355 | C2 | semidirect |
| 96. C53 | #1650 | semidirect |