Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
576 conjugacy classes of subgroups; only the 285 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 285 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D53 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C22×M16×C53 | C2 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C22×M16 | D53 |
| hypocenter | C53 | C2×#423 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C22×M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810646 | C2 | direct (N × H) |
| 4. C2×#13133146 | C2 | semidirect |
| 15. #13133146 | C22 | semidirect |
| 56. C2×Dic53 | M16 | semidirect |
| 72. Dic53 | C2×M16 | semidirect |
| 75. C2×C53 | #423 | semidirect |
| 83. C53 | C2×#423 | semidirect |
| 152. C2 | #16810646 | direct (N × H) |