Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
388 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C4×Q8×C53 | C4 |
| hypercenter | C4×Q8 | #6157 |
| hypocenter | C53 | Q8×M16 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811673 | C2 | semidirect |
| 2. Q8×#88324 | C2 | semidirect |
| 28. #11719936 | Q8 | direct (N × H) |
| 37. Q8×C53 | M16 | semidirect |
| 50. #88324 | C2×Q8 | semidirect |
| 72. C53 | Q8×M16 | semidirect |
| 105. Q8 | #11719936 | direct (N × H) |