Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
344 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D21 |
| commutator, socle, prime radical, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C7×#125 | C2 |
| hypercenter | #125 | D21 |
| hypocenter | C3×C7 | #564 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 86. C3×C7 | #564 | semidirect |
| 111. C7 | C3⋊#564 | semidirect |
| 120. C3 | C7⋊#564 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C7×#125 | 2 |
| #10621373 | 2 |
| #10621372 | 2 |
| #10621353 | 2 |
| C7⋊#564 | 3 |
| C3⋊#564 | 7 |