Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
330 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C2×C3×C7 |
| Frattini | C2 | C2×C7×C22⋊A4 |
| Fitting | C7×#585 | C3 |
| socle | C2×C7 | C2×C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C4×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#997 | C2 | semidirect |
| 2. C7×#585 | C3 | semidirect |
| 10. #5950 | C7 | direct (N × H) |
| 17. C7×#41 | A4 | semidirect |
| 26. C22×C4×C7 | A4 | semidirect |
| 27. #997 | C2×C7 | semidirect |
| 30. #585 | C3×C7 | semidirect |
| 68. #41 | C7×A4 | semidirect |
| 77. C22×C4 | C7×A4 | semidirect |
| 97. C7 | #5950 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#997 | 2 |
| C7×#585 | 3 |
| C7×#41⋊C3 | 4 |
| C4×C7×A4 | 4 |
| #5950 | 7 |