Finite Group DatabasebetaFinite Group Databasebeta| order | 4320 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#11724050 |
| derived | C22⋊(C22⋊C9) ⋊ C2×C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | #11724050 |
| commutator, cosocle, hypocenter | C22⋊(C22⋊C9) | C2×C3×C5 |
| Frattini | C3 | C3×C5×#1020 |
| Fitting | C24×C32×C5 | D3 |
| socle | C24×C3×C5 | C3×D3 |
| LCS | G ▷ C22⋊(C22⋊C9) | |
| UCS | C1 ◁ C5 | |
458 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
230 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#72213207, D5×#11724050 |
| maximal normal subgroup of | D5×#11724050 |
Past the SmallGroups library coverage — no SmallGroups id.