Finite Group DatabasebetaFinite Group Databasebeta| order | 8400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×ΓL(1,𝔽8)×D5⋊AGL(1,𝔽5) |
| Fitting, socle | C2×C52×C7 ⋊ C2×C3×C4 |
| derived, Hall | C52×C7 ⋊ C22×C3×C4 |
| Hall | C2×C7×D5⋊AGL(1,𝔽5) ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ C22×C4 |
| Sylow | C52 ⋊ C22×C4×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C2×C3×D5⋊AGL(1,𝔽5) |
496 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 140 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽7)×D5⋊AGL(1,𝔽5), ΓL(1,𝔽8)×#11717375, #72243208 |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×D5⋊AGL(1,𝔽5), ΓL(1,𝔽8)×#11717375, #72243208 |
| maximal quotient of | ΓL(1,𝔽8)×#11717375 |
Past the SmallGroups library coverage — no SmallGroups id.