Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
146 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 38 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 | D4×A4 |
| commutator, cosocle | #41 | C22×C3×C7 |
| Frattini | C4 | C22×C7×A4 |
| Fitting | C7×#577 | C3 |
| hypercenter | C7×SD16 | A4 |
| hypocenter | Q8 | C3×C7×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 ◁ C7×SD16 | |
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×#994 | C2 | semidirect |
| 3. C7×#131⋊C3 | C2 | semidirect |
| 4. C7×#577 | C3 | semidirect |
| 10. C7×H(5,𝔽2) | C2×C3 | semidirect |
| 15. C7×#131 | C2×C3 | semidirect |
| 16. #577⋊C3 | C7 | direct (N × H) |
| 37. 2−1+4⋊C3 | C2×C7 | semidirect |
| 38. #994 | C2×C7 | semidirect |
| 39. #131⋊C3 | C2×C7 | semidirect |
| 43. #577 | C3×C7 | semidirect |
| 68. H(5,𝔽2) | C2×C3×C7 | semidirect |
| 73. #131 | C2×C3×C7 | semidirect |
| 127. C7 | #577⋊C3 | 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×#994 | 2 |
| C7×#131⋊C3 | 2 |
| C7×#577 | 3 |
| C3×C7×SD16 | 4 |
| #577⋊C3 | 7 |