Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
444 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 151 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, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C7×#125 | C2 |
| prime radical | C22×D21 | C23 |
| hypercenter | #125 | D21 |
| hypocenter | C3×C7 | #533 |
| 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 |
|---|---|---|
| 1. C3×C7×#125 | C2 | semidirect |
| 4. #10621371 | C2 | semidirect |
| 5. #10621370 | C2 | semidirect |
| 7. #10621353 | C2 | semidirect |
| 22. C7×#125 | D3 | semidirect |
| 38. C4×D21 | Q8 | semidirect |
| 53. C3×#125 | D7 | semidirect |
| 98. #125 | D21 | semidirect |
| 122. C3×C7 | #533 | semidirect |
| 158. C7 | #5688 | semidirect |
| 171. C3 | #90379 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C7×#125 | 2 |
| #10621379 | 2 |
| #10621375 | 2 |
| #10621371 | 2 |
| #10621370 | 2 |
| C42×D21 | 2 |
| #10621353 | 2 |
| #90379 | 3 |
| #5688 | 7 |