Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#16863298 |
| Fitting | C22×C42×C67 ⋊ C2 |
| Sylow | C67 ⋊ C2×#390 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D67 |
| commutator | C22×C67 | C23×C4 |
| Frattini | C23 | C23×D67 |
| Fitting | C22×C42×C67 | C2 |
| socle | C24×C67 | C23 |
| hypercenter | C22×C42 | D67 |
| hypocenter | C67 | C2×#390 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C24 ◁ C22×C42 | |
1360 conjugacy classes of subgroups; only the 381 normal subgroups are shown.
This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 381 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
2168 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#22558029, #30743815, #30743817, … |
| maximal normal subgroup of | C2×#22558029, #30743815, #30743817, … |
| maximal quotient of | C2×#22558439, C2×#22558447, C4×#16863298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 6966);