Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
292 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 54 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 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C7 |
| Frattini | C2 | C22×C7×S4 |
| Fitting | C7×H(5,𝔽2) | D3 |
| hypercenter | C7×Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C7 ◁ C7×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#994 | C2 | semidirect |
| 2. C7×#986 | C2 | semidirect |
| 3. C7×#985 | C2 | semidirect |
| 6. C7×#41⋊C3 | C22 | semidirect |
| 7. C7×GL(2,𝔽3) | C22 | semidirect |
| 8. C7×#226 | C22 | semidirect |
| 10. C7×H(5,𝔽2) | D3 | semidirect |
| 16. #5925 | C7 | direct (N × H) |
| 35. #994 | C2×C7 | semidirect |
| 36. #986 | C2×C7 | semidirect |
| 37. #985 | C2×C7 | semidirect |
| 57. #41⋊C3 | C22×C7 | semidirect |
| 58. GL(2,𝔽3) | C22×C7 | semidirect |
| 59. #226 | C22×C7 | semidirect |
| 63. H(5,𝔽2) | D3×C7 | semidirect |
| 115. C7 | #5925 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#994 | 2 |
| C7×#986 | 2 |
| C7×#985 | 2 |
| C7×#576 | 3 |
| C7×#239 | 4 |
| #5925 | 7 |