Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#87547 |
| Fitting, Hall | C7×H(7,𝔽2) ⋊ C3 |
| Sylow | H(7,𝔽2) ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3×C7 |
| Frattini | C2 | C22×C7×C22⋊A4 |
| Fitting | C7×H(7,𝔽2) | C3 |
| hypercenter | C7×D4 | C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C7 ◁ C7×D4 | |
2154 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 2154 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
245 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#87547 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#87547 |
Past the SmallGroups library coverage — no SmallGroups id.