Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1048 conjugacy classes of subgroups; only the 573 normal subgroups are shown.
This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 573 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×C4×Q8×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×C4×Q8 | D53 |
| hypocenter | C53 | C2×#557 |
| 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 |
|---|---|---|
| 1. #16811349 | C2 | direct (N × H) |
| 28. C2×Q8×C53 | Q8 | semidirect |
| 50. Q8×C53 | C2×Q8 | semidirect |
| 78. C2×C53 | #557 | semidirect |
| 90. C53 | C2×#557 | semidirect |
| 112. C2×Q8 | Dic106 | semidirect |
| 133. Q8 | C2×Dic106 | semidirect |
| 159. C2 | #16811349 | direct (N × H) |