Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D32×D28 |
| Fitting | C32×C4×C7 ⋊ C23 |
| Hall | C32×C7 ⋊ C22×D4 |
| Sylow | C32 ⋊ C22×D28 |
| Sylow | C7 ⋊ D32×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32×D7 |
| commutator, socle, cosocle | C2×C32×C7 | C24 |
| Fitting | C32×C4×C7 | C23 |
| hypercenter | C4 | D32×D7 |
| hypocenter | C32×C7 | C22×D4 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
998 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 998 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
153 conjugacy classes — view the full character table.
| maximal subgroup of | D32×D4×D7, #72246690, #72246700, … |
| maximal normal subgroup of | D32×D4×D7, #72246690, #72246700, … |
| maximal quotient of | #72246874 |
Past the SmallGroups library coverage — no SmallGroups id.