Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#11716109 |
| Sylow | C72 ⋊ C2×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D72 |
| commutator, cosocle | C2×C72 | C24 |
| Frattini | C2 | C22×D72 |
| Fitting | C23×C72 | C22 |
| socle | C22×C72 | C23 |
| hypercenter | C23 | D72 |
| hypocenter | C72 | C2×#41 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C23 | |
409 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13096688, C2×#13096712, D3×#11716109, … |
| maximal normal subgroup of | C2×#13096688, C2×#13096712, C2×#72214099 |
| maximal quotient of | D3×#11716109 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 871);