Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
440 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 72 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 | D7⋊A4 |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Frattini | C2 | C22×D7⋊A4 |
| Fitting | C2×C7×#41 | C2×C3 |
| socle | C22×C7 | C22×A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. D7⋊(#41⋊C3) | C2 | direct (N × H) |
| 3. C2×D7⋊SL(2,𝔽3) | C2 | semidirect |
| 4. C2×#66665 | C2 | semidirect |
| 5. C2×D7×#41 | C3 | semidirect |
| 6. #41⋊ΓL(1,𝔽8) | C22 | semidirect |
| 7. C2×C7⋊SL(2,𝔽3) | C22 | semidirect |
| 8. D7⋊SL(2,𝔽3) | C22 | semidirect |
| 9. #66665 | C22 | semidirect |
| 11. C2×C7×#41 | C2×C3 | semidirect |
| 13. D7×#41 | C2×C3 | semidirect |
| 14. C2×#1161 | C2×C3 | semidirect |
| 22. C7⋊SL(2,𝔽3) | C23 | semidirect |
| 28. C7×#41 | C22×C3 | semidirect |
| 35. #1161 | C22×C3 | semidirect |
| 102. C2×#41 | AGL(1,𝔽7) | semidirect |
| 112. C2×D7 | #41⋊C3 | semidirect |
| 133. #41 | C2×AGL(1,𝔽7) | semidirect |
| 152. C2×C7 | C2×#41⋊C3 | semidirect |
| 155. D7 | C2×#41⋊C3 | semidirect |
| 188. C7 | C22×#41⋊C3 | semidirect |
| 209. C2 | D7⋊(#41⋊C3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41⋊ΓL(1,𝔽8) | 2 |
| D7⋊(#41⋊C3) | 2 |
| C2×D7⋊SL(2,𝔽3) | 2 |
| C2×#66665 | 2 |
| C2×D7×#41 | 3 |
| C2×C4×AGL(1,𝔽7) | 4 |
| C22×#41⋊C3 | 7 |