Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C83 · Schur multiplier (H2) = C22 |
| direct product | C4×SD16×C83 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C83 | D4 |
| commutator | C4 | C22×C4×C83 |
| Frattini, cosocle | C2×C4 | C23×C83 |
| socle | C22×C83 | C2×D4 |
| prime radical | C2×D4×C83 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C83 ◁ C42×C83 ◁ G | |
122 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
2324 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C8×SD16×C83, C83×SD16⋊C8, C83×#1582, … |
| maximal normal subgroup of | C8×SD16×C83, C83×SD16⋊C8, C83×#1582, … |
| maximal quotient of | C8×SD16×C83, C83×#1582, C83×#1583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 119);