Finite Group DatabasebetaFinite Group Databasebeta| order | 8400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C22×ΓL(1,𝔽8)⋊#1043 |
| Fitting, socle | C22×C52×C7 ⋊ C3×C4 |
| derived, Hall | C52×C7 ⋊ C22×C3×C4 |
| Hall | C22×C7⋊#1043 ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ C22×C4 |
| Sylow | C52 ⋊ C22×ΓL(1,𝔽8)⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ C22×#1043 |
| Sylow | C7 ⋊ C22×C3×#1043 |
864 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 700 |
176 conjugacy classes — view the full character table.
| maximal subgroup of | C22×AGL(1,𝔽7)×#1043, #72243219, #72243227 |
| maximal normal subgroup of | C22×AGL(1,𝔽7)×#1043, #72243219, #72243227 |
Past the SmallGroups library coverage — no SmallGroups id.