Finite Group DatabasebetaFinite Group Databasebeta| order | 12224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#20259293 |
| Fitting | C22×D4×C191 ⋊ C2 |
| Sylow | C191 ⋊ C2×#121 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D191 |
| commutator, cosocle | C22×C191 | C24 |
| Frattini | C22 | C23×D191 |
| Fitting | C22×D4×C191 | C2 |
| socle | C23×C191 | C23 |
| hypercenter | C22×D4 | D191 |
| hypocenter | C191 | C2×#121 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
426 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1928 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #37572834, #37572839, #37572945, … |
| maximal normal subgroup of | #37572834, #37572839, #37572945, … |
| maximal quotient of | C2×#27132520, C2×#27132539, C2×#27132551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12224, 1038);