Finite Group DatabasebetaFinite Group Databasebeta| order | 5040 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | D7×C3⋊D3×AGL(1,𝔽5) |
| Fitting, socle, derived, Hall | C32×C5×C7 ⋊ C22×C4 |
| Hall | C32×C7 ⋊ C22×AGL(1,𝔽5) |
| Hall | C32×C5 ⋊ C2×C4×D7 |
| Hall | C5×C7 ⋊ C2×C4×C3⋊D3 |
| Sylow | C32 ⋊ C2×D7×AGL(1,𝔽5) |
| Sylow | C7 ⋊ C2×C3⋊D3×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C4×D7×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C5×C7 | C22×C4 |
| prime radical | D5×D7×C3⋊D3 | C2 |
| cosocle | C32×C7×D5 | C23 |
| LCS | G ▷ C32×C5×C7 | |
| UCS | C1 | |
648 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
150 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×AGL(1,𝔽5)×AGL(1,𝔽7), D7×AGL(1,𝔽5)×#157, D32×D7×AGL(1,𝔽5) |
| maximal normal subgroup of | C3⋊D3×AGL(1,𝔽5)×AGL(1,𝔽7), D7×AGL(1,𝔽5)×#157, D32×D7×AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id.