Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
336 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C22⋊D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | Q82×C53 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | Q82 | D53 |
| hypocenter | C53 | Q8×Q16 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 35. #11719925 | Q8 | direct (N × H) |
| 44. Q8×C53 | Q16 | semidirect |
| 78. C53 | Q8×Q16 | semidirect |
| 118. Q8 | #11719925 | direct (N × H) |