Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
| direct product | C2×#16863173 |
| Sylow | C67 ⋊ C2×#347 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×Dic67 |
| commutator | C4×C67 | C2×C42 |
| Frattini | C2×C8 | C22×D67 |
| Fitting | C2×M32×C67 | C2 |
| socle | C22×C67 | #97 |
| hypercenter | C2×M32 | D67 |
| hypocenter | C67 | C2×#347 |
| LCS | G ▷ C4×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C4 ◁ C2×M32 | |
168 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #347×Dic67, C4×#16863173, #30770660, … |
| maximal normal subgroup of | #347×Dic67, C4×#16863173, #30770660, … |
| maximal quotient of | C2×#22558129, C2×#22558131, #30764831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 6111);