Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×D4×D83 |
| Fitting | C4×D4×C83 ⋊ C2 |
| Sylow | C83 ⋊ C2×C4×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D83 |
| commutator | C2×C83 | C23×C4 |
| Frattini | C22 | C23×D83 |
| Fitting | C4×D4×C83 | C2 |
| socle | C22×C83 | C24 |
| hypercenter | C4×D4 | D83 |
| hypocenter | C83 | C2×C4×D4 |
| LCS | G ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
426 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
860 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D4×D83, #435×D83, #436×D83, … |
| maximal normal subgroup of | C8×D4×D83, #435×D83, #436×D83, … |
| maximal quotient of | C4×C22⋊C4×D83, C4×C4⋊C4×D83, C4×(C22⋊D83)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 1006);