Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
338 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 338 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 | C22 | C2×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 |
| hypercenter | C2×C4 | C7⋊S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ 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. #10621504 | C2 | direct (N × H) |
| 3. C2×C7⋊GL(2,𝔽3) | C2 | semidirect |
| 6. C7×#41⋊C3 | C22 | semidirect |
| 7. C2×C7×SL(2,𝔽3) | C22 | semidirect |
| 8. C7⋊GL(2,𝔽3) | C22 | semidirect |
| 9. C2×C7×#41 | D3 | semidirect |
| 20. C7×SL(2,𝔽3) | C23 | semidirect |
| 23. C7×#41 | C2×D3 | semidirect |
| 36. C2×#41⋊C3 | D7 | semidirect |
| 61. #41⋊C3 | C2×D7 | semidirect |
| 62. C2×SL(2,𝔽3) | C2×D7 | semidirect |
| 67. C2×#41 | D21 | semidirect |
| 86. SL(2,𝔽3) | C22×D7 | semidirect |
| 90. #41 | C2×D21 | semidirect |
| 104. C2×C7 | #986 | semidirect |
| 127. C7 | C2×#986 | semidirect |
| 143. C2 | #10621504 | 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 |
| #10621504 | 2 |
| C2×C7⋊GL(2,𝔽3) | 2 |
| C2×#6512 | 3 |
| C2×D84 | 4 |
| C2×#986 | 7 |