Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C83 · Schur multiplier (H2) = C26 |
| direct product | C2×C4×D4×C83 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C83 | C22 |
| commutator | C2 | C23×C4×C83 |
| Frattini, cosocle | C22 | C24×C83 |
| socle | C23×C83 | C23 |
| prime radical | C22×D4×C83 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C83 ◁ G | |
426 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 274 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
3320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C83×#1459, C83×#1460, C83×#1465, … |
| maximal normal subgroup of | C83×#1459, C83×#1460, C83×#1465, … |
| maximal quotient of | C2×C4×C22⋊C4×C83, C2×C4×C4⋊C4×C83, C42×D4×C83, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 196);