Finite Group DatabasebetaFinite Group Databasebeta| order | 3360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×AGL(1,𝔽5)×S4 |
| Fitting, socle | C22×C5×C7 ⋊ C4×D3 |
| derived | C5×A4 ⋊ C2×C4×C7 |
| Hall | C5×C7 ⋊ C4×S4 |
| Sylow | C5 ⋊ C4×C7×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | AGL(1,𝔽5)×S4 |
| commutator, hypocenter | C5×A4 | C2×C4×C7 |
| Frattini | C1 | G |
| Fitting, socle | C22×C5×C7 | C4×D3 |
| prime radical | C7×D5×S4 | C2 |
| cosocle | D5×A4 | C22×C7 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 ◁ C7 | |
224 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
175 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×ΓL(1,𝔽8)×S4, D7×AGL(1,𝔽5)×S4 |
| maximal normal subgroup of | AGL(1,𝔽5)×ΓL(1,𝔽8)×S4, D7×AGL(1,𝔽5)×S4 |
| maximal quotient of | C7×AGL(1,𝔽5)×GL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id.