Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C41 · Schur multiplier (H2) = C2 |
| direct product | C41×#1405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C41 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C4×C41 |
| Frattini, cosocle | C4⋊Q8 | C22×C41 |
| prime radical | 2−1+4×C41 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C41 ◁ C22×C41 ◁ C2×C4×C41 ◁ C4⋊Q8×C41 ◁ G | |
106 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
697 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C41×#1405, C41×#13572, C41×#13573, … |
| maximal normal subgroup of | C2×C41×#1405, C41×#13572, C41×#13573, … |
| maximal quotient of | C41×#7345, C41×#7376, C41×#7409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 137);