Finite Group DatabasebetaFinite Group Databasebeta| order | 4200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×(C5⋊AGL(1,𝔽7))⋊D5 |
| Fitting, socle | C2×C52×C7 ⋊ C22×C3 |
| derived, Hall | C52×C7 ⋊ C23×C3 |
| Hall | C2×(C5⋊D5)⋊D7 ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ C23 |
| Sylow | C52 ⋊ C22×AGL(1,𝔽7) |
| Hall | ΓL(1,𝔽8) ⋊ C2×D52 |
| Sylow | C7 ⋊ C2×C3×D52 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C5⋊AGL(1,𝔽7))⋊D5 |
| commutator, cosocle, hypocenter | C52×C7 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C52×C7 | C22×C3 |
| LCS | G ▷ C52×C7 | |
| UCS | C1 ◁ C2 | |
296 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 140 |
124 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72218917, C2×D52×AGL(1,𝔽7), C2×#72218916, … |
| maximal normal subgroup of | C2×#72218917, C2×D52×AGL(1,𝔽7), C2×#72218916, … |
| maximal quotient of | #72243197 |
Past the SmallGroups library coverage — no SmallGroups id.