Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | #126×Dic67 |
| Sylow | C67 ⋊ C4×#126 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D67 |
| commutator | C22×C67 | C23×C4 |
| Fitting | C2×#126×C67 | C2 |
| socle | C23×C67 | C24 |
| hypercenter | C2×#126 | D67 |
| hypocenter | C67 | C4×#126 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C23 ◁ C2×#126 | |
516 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 229 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
980 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#126×D67, C2×#126×Dic67, C4×#16863929, … |
| maximal normal subgroup of | C4×#126×D67, C2×#126×Dic67, C4×#16863929, … |
| maximal quotient of | #388×Dic67, #387×Dic67, #383×Dic67, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 7062);