Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
144 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C22×C7 |
| Frattini | C4 | C2×C7×S4 |
| Fitting | C7×2−1+4 | D3 |
| hypercenter | C7×D4 | S4 |
| hypocenter | SL(2,𝔽3) | C7×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 ◁ C7×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×2−1+4⋊C3 | C2 | semidirect |
| 2. C7×#986 | C2 | semidirect |
| 3. C7×GU(2,𝔽3) | C2 | semidirect |
| 9. C7×2−1+4 | D3 | semidirect |
| 16. #5431 | C7 | direct (N × H) |
| 35. 2−1+4⋊C3 | C2×C7 | semidirect |
| 36. #986 | C2×C7 | semidirect |
| 37. GU(2,𝔽3) | C2×C7 | semidirect |
| 64. 2−1+4 | D3×C7 | semidirect |
| 120. C7 | #5431 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×2−1+4⋊C3 | 2 |
| C7×#986 | 2 |
| C7×GU(2,𝔽3) | 2 |
| C7×#454 | 3 |
| C7×#213 | 4 |
| #5431 | 7 |