Finite Group DatabasebetaFinite Group Databasebeta| order | 3360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | S4×C7⋊AGL(1,𝔽5) |
| Fitting, socle | C22×C5×C7 ⋊ C4×D3 |
| derived | C5×C7×A4 ⋊ C2×C4 |
| Hall | C5×C7 ⋊ C4×S4 |
| Sylow | C7 ⋊ AGL(1,𝔽5)×S4 |
| Sylow | C5 ⋊ S4×Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C5×C7×A4 | C2×C4 |
| Fitting, socle | C22×C5×C7 | C4×D3 |
| prime radical | C7×D5×S4 | C2 |
| cosocle | C7×D5×A4 | C22 |
| LCS | G ▷ C5×C7×A4 | |
| UCS | C1 | |
224 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
85 conjugacy classes — view the full character table.
| maximal subgroup of | S4×ΓL(1,𝔽8)⋊AGL(1,𝔽5), D7×AGL(1,𝔽5)×S4 |
| maximal normal subgroup of | S4×ΓL(1,𝔽8)⋊AGL(1,𝔽5), D7×AGL(1,𝔽5)×S4 |
| maximal quotient of | GL(2,𝔽3)×C7⋊AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id.