Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C4×C22⋊D83 |
| Fitting | C23×C4×C83 ⋊ C2 |
| Sylow | C83 ⋊ C2×C4×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D83 |
| commutator | C2×C83 | C23×C4 |
| Frattini | C22 | C23×D83 |
| Fitting | C23×C4×C83 | C2 |
| socle | C23×C83 | C23 |
| hypercenter | C23×C4 | D83 |
| hypocenter | C83 | C2×C4×D4 |
| LCS | G ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 | |
426 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1352 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4×#14581706, #24907448, #24907752, … |
| maximal normal subgroup of | C4×#14581706, #24907448, #24907752, … |
| maximal quotient of | C2×C4×C22⋊Dic83, C2×C4×(C2×D83)⋊C4, C2×C4×Dic83⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 1008);