Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
168 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×Dic53 |
| commutator | C4×C53 | C2×C42 |
| Frattini | C2×C8 | C22×D53 |
| Fitting | C2×M32×C53 | C2 |
| socle | C22×C53 | #97 |
| hypercenter | C2×M32 | D53 |
| hypocenter | C53 | C2×#347 |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C2×M32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810400 | C2 | direct (N × H) |
| 7. #13133157 | C4 | semidirect |
| 9. C2×#11719908 | C4 | semidirect |
| 13. M32×C53 | C4 | semidirect |
| 15. C2×C16×C53 | C4 | semidirect |
| 29. C16×C53 | C2×C4 | semidirect |
| 30. #11719908 | C2×C4 | semidirect |
| 57. C2×C53 | #347 | semidirect |
| 63. C53 | C2×#347 | semidirect |
| 69. M32 | Dic53 | semidirect |
| 72. C2×C16 | Dic53 | semidirect |
| 89. C16 | C2×Dic53 | semidirect |
| 115. C2 | #16810400 | direct (N × H) |