Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
152 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | A4×D7 |
| commutator, hypocenter | C7×Q8 | C2×C3×C4 |
| Frattini | C4 | C2×A4×D7 |
| Fitting | C7×#131 | C2×C3 |
| socle | C2×C7 | C2×C4×A4 |
| prime radical | D7×#41⋊C3 | C2 |
| cosocle | C7×#41 | C22×C3 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#131⋊C3 | C2 | semidirect |
| 3. #10620795 | C2 | semidirect |
| 4. D7×#131 | C3 | semidirect |
| 10. C7×#131 | C2×C3 | semidirect |
| 11. #6511 | C2×C3 | semidirect |
| 39. #131⋊C3 | D7 | direct (N × H) |
| 70. #131 | C3×D7 | semidirect |
| 105. D7 | #131⋊C3 | direct (N × H) |
| 125. C7 | C2×#131⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D7×#41⋊C3 | 2 |
| C7×#131⋊C3 | 2 |
| #10620795 | 2 |
| D7×#131 | 3 |
| C3×C8×D7 | 4 |
| C2×#131⋊C3 | 7 |