Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×D4×D167 |
| Fitting | C22×D4×C167 ⋊ C2 |
| Sylow | C167 ⋊ C23×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D167 |
| commutator, cosocle | C2×C167 | C25 |
| Frattini | C2 | C24×D167 |
| Fitting | C22×D4×C167 | C2 |
| socle | C23×C167 | C23 |
| hypercenter | C22×D4 | D167 |
| hypocenter | C167 | C23×D4 |
| LCS | G ▷ C2×C167 ▷ C167 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
1362 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 503 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1700 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #36324051, #36324057, #36324067, … |
| maximal normal subgroup of | #36324051, #36324057, #36324067, … |
| maximal quotient of | C22×C22⋊C4×D167, C22×C4⋊C4×D167, C2×C4×D4×D167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 1369);