Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
306 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C7⋊S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C23 |
| Frattini | C2 | C22×C7⋊S4 |
| Fitting | C2×C7×#41 | D3 |
| socle | C22×C7 | C2×S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#41⋊C3 | C2 | semidirect |
| 2. #10621503 | C2 | direct (N × H) |
| 3. C2×C7⋊GL(2,𝔽3) | C2 | semidirect |
| 4. C2×C7⋊#226 | C2 | semidirect |
| 7. C7×#41⋊C3 | C22 | semidirect |
| 9. C7⋊GL(2,𝔽3) | C22 | semidirect |
| 10. C7⋊#226 | C22 | semidirect |
| 11. C2×C7×#41 | D3 | semidirect |
| 27. C7×#41 | C2×D3 | semidirect |
| 46. C2×#41⋊C3 | D7 | semidirect |
| 76. #41⋊C3 | C2×D7 | semidirect |
| 84. C2×#41 | D21 | semidirect |
| 110. #41 | C2×D21 | semidirect |
| 128. C2×C7 | #985 | semidirect |
| 155. C7 | C2×#985 | semidirect |
| 171. C2 | #10621503 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#41⋊C3 | 2 |
| #10621503 | 2 |
| C2×C7⋊GL(2,𝔽3) | 2 |
| C2×C7⋊#226 | 2 |
| C2×#6513 | 3 |
| C2×C4×D21 | 4 |
| C2×#985 | 7 |