Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | #137×D83 |
| Fitting | #137×C83 ⋊ C2 |
| Sylow | C83 ⋊ C2×#137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D83 |
| commutator | C4×C83 | C24 |
| Frattini | C4 | C23×D83 |
| Fitting | #137×C83 | C2 |
| socle | C2×C83 | C22×D4 |
| hypercenter | #137 | D83 |
| hypocenter | C83 | C2×#137 |
| LCS | G ▷ C4×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #137 | |
258 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
473 conjugacy classes — view the full character table.
| maximal subgroup of | #455×D83, #456×D83, #24916838, … |
| maximal normal subgroup of | #455×D83, #456×D83, #24916838, … |
| maximal quotient of | #419×D83, #417×D83, #428×D83, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 1339);