Finite Group DatabasebetaFinite Group Databasebeta| order | 8400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | ΓL(1,𝔽8)×#87886 |
| Fitting | C22×C52×C7 ⋊ C22×C3 |
| Hall | C7×#87886 ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ C2×D4 |
| Hall | C52×C7 ⋊ C2×C3×D4 |
| Sylow | C52 ⋊ C2×D4×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×#87886 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D52×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C2×C52×C7 | C23×C3 |
| Fitting | C22×C52×C7 | C22×C3 |
| hypercenter | C22 | D52×ΓL(1,𝔽8) |
| hypocenter | C52×C7 | C2×C3×D4 |
| LCS | G ▷ C2×C52×C7 ▷ C52×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
560 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 140 |
230 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#11717375, AGL(1,𝔽7)×#87886, #72243219 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#11717375, AGL(1,𝔽7)×#87886, #72243219 |
Past the SmallGroups library coverage — no SmallGroups id.