Finite Group DatabasebetaFinite Group Databasebeta| order | 3360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | AGL(1,𝔽5)×D7⋊A4 |
| Fitting, socle, derived | C22×C5×C7 ⋊ C2×C3×C4 |
| Hall | C22×D7×AGL(1,𝔽5) ⋊ C3 |
| Hall | C5×C7 ⋊ C2×C4×A4 |
| Sylow | C7 ⋊ C2×A4×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C4×D7⋊A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C5×C7 | C2×C3×C4 |
| prime radical | D5×D7⋊A4 | C2 |
| cosocle | C22×C7×D5 | C22×C3 |
| LCS | G ▷ C22×C5×C7 | |
| UCS | C1 | |
264 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 264 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.
60 conjugacy classes — view the full character table.
| maximal subgroup of | A4×AGL(1,𝔽5)×AGL(1,𝔽7) |
| maximal normal subgroup of | A4×AGL(1,𝔽5)×AGL(1,𝔽7) |
| maximal quotient of | AGL(1,𝔽5)×D7⋊SL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id.