Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
568 conjugacy classes of subgroups; only the 333 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 333 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 | C2×C4×Q8×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×C4×Q8 | D53 |
| hypocenter | C53 | Q8×C4⋊C4 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. C4×Q8×C53 | C4 | semidirect |
| 30. C4⋊Dic53 | Q8 | direct (N × H) |
| 33. Q8×C53 | C4⋊C4 | semidirect |
| 58. C4×C53 | C4×Q8 | semidirect |
| 74. C53 | Q8×C4⋊C4 | semidirect |
| 77. C4×Q8 | Dic53 | semidirect |
| 102. Q8 | C4⋊Dic53 | direct (N × H) |
| 129. C4 | Q8×Dic53 | semidirect |