Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#41×D167 |
| Fitting | C2×#41×C167 ⋊ C2 |
| Sylow | C167 ⋊ C22×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D167 |
| commutator, cosocle | C2×C167 | C25 |
| Frattini | C2 | C24×D167 |
| Fitting | C2×#41×C167 | C2 |
| socle | C22×C167 | C24 |
| hypercenter | C2×#41 | D167 |
| hypocenter | C167 | C22×#41 |
| LCS | G ▷ C2×C167 ▷ C167 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
890 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 455 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 | #36324080, #408×D167, #36332580, … |
| maximal normal subgroup of | #36324080, #408×D167, #36332580, … |
| maximal quotient of | C2×#117×D167, C2×C4×D4×D167, C2×C4×Q8×D167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 1374);