Finite Group DatabasebetaFinite Group Databasebeta| order | 7840 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C7 |
| direct product | C22×AGL(1,𝔽5)×C7⋊D7 |
| Fitting, socle | C22×C5×C72 ⋊ C2×C4 |
| derived, Hall | C5×C72 ⋊ C23×C4 |
| Sylow | C72 ⋊ C23×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C22×C4×C7⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | AGL(1,𝔽5)×C7⋊D7 |
| commutator, hypocenter | C5×C72 | C23×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C5×C72 | C2×C4 |
| prime radical | C22×D5×C7⋊D7 | C2 |
| cosocle | C72×D5 | C24 |
| LCS | G ▷ C5×C72 | |
| UCS | C1 ◁ C22 | |
2360 conjugacy classes of subgroups; only the 422 normal subgroups are shown.
This group has 2360 conjugacy classes of subgroups — too many to draw. Only its 422 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 980 |
520 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×AGL(1,𝔽5)×#64424, AGL(1,𝔽5)×#11716115 |
| maximal normal subgroup of | C22×AGL(1,𝔽5)×#64424, AGL(1,𝔽5)×#11716115 |
Past the SmallGroups library coverage — no SmallGroups id.