Finite Group DatabasebetaFinite Group Databasebeta| order | 2352 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×D7×S4 |
| Fitting, socle | C22×C72 ⋊ C2×D3 |
| derived | C7×A4 ⋊ C22×C7 |
| Sylow | C72 ⋊ C2×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | D7×S4 |
| commutator, cosocle, hypocenter | C7×A4 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C22×C72 | C2×D3 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 ◁ C7 | |
165 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 26 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 | C7×#1020⋊D7, S4×#64423, D72×S4, … |
| maximal normal subgroup of | S4×#64423, D72×S4, C7×S4×AGL(1,𝔽7) |
| maximal quotient of | C7×#1020⋊D7, S4×D49⋊C7 |
Past the SmallGroups library coverage — no SmallGroups id.