Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | Q8×#11719925 |
| Sylow | C53 ⋊ Q8×Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C22⋊D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | Q82×C53 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | Q82 | D53 |
| hypocenter | C53 | Q8×Q16 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ Q82 | |
336 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
685 conjugacy classes — view the full character table.
| maximal subgroup of | #27374242, #27374245, #27374250, … |
| maximal normal subgroup of | #27374242, #27374245, #27374250, … |
| maximal quotient of | Q8×#13133140, Q8×#13133141, C4⋊C4×#11719925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17392);