Finite Group DatabasebetaFinite Group Databasebeta| order | 4320 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C5 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×C32×C5×S4 |
| Fitting, socle | C24×C32×C5 ⋊ D3 |
| derived | A4 ⋊ C23×C32×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32×C5 | S4 |
| commutator, cosocle, hypocenter | A4 | C23×C32×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C32×C5 | D3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C22×C32×C5 | |
1732 conjugacy classes of subgroups; only the 312 normal subgroups are shown.
This group has 1732 conjugacy classes of subgroups — too many to draw. Only its 312 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
900 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×C32×C5×GL(3,𝔽2) |
| automorphism group of | C7×Q8×C31 |
Past the SmallGroups library coverage — no SmallGroups id.