Finite Group DatabasebetaFinite Group Databasebeta| order | 3000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×D5×#1043 |
| Fitting, socle, Hall | C3×C53 ⋊ C2×C4 |
| derived, Sylow | C53 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D5×#1043 |
| commutator, hypocenter | C53 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C3×C53 | C2×C4 |
| prime radical | C3×D5×C5⋊D5 | C2 |
| cosocle | C5×C5⋊D5 | C22×C3 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C3 | |
304 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 300 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AGL(1,𝔽5)×#1043, D3×D5×#1043, ΓL(1,𝔽16)×#1043, … |
| maximal normal subgroup of | C3×AGL(1,𝔽5)×#1043, D3×D5×#1043, ΓL(1,𝔽16)×#1043, … |
Past the SmallGroups library coverage — no SmallGroups id.