Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
408 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×A4×D7 |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Fitting | C7×2−1+4 | C2×C3 |
| socle | C2×C7 | C23×A4 |
| hypercenter | D4 | A4×D7 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×2−1+4⋊C3 | C2 | semidirect |
| 2. #10621530 | C2 | semidirect |
| 3. D7×#41⋊C3 | C2 | semidirect |
| 4. #10621527 | C2 | semidirect |
| 5. C2×D7×SL(2,𝔽3) | C2 | semidirect |
| 6. D7×2−1+4 | C3 | semidirect |
| 8. C7×#41⋊C3 | C22 | semidirect |
| 9. C2×C7×SL(2,𝔽3) | C22 | semidirect |
| 11. D7×SL(2,𝔽3) | C22 | semidirect |
| 12. #66662 | C22 | semidirect |
| 13. C7×2−1+4 | C2×C3 | semidirect |
| 16. #6554 | C2×C3 | semidirect |
| 19. D7×#41 | C2×C3 | semidirect |
| 21. #6551 | C2×C3 | semidirect |
| 23. C2×Q8×D7 | C2×C3 | semidirect |
| 33. C7×#41 | C22×C3 | semidirect |
| 35. C2×C7×Q8 | C22×C3 | semidirect |
| 38. #1161 | C22×C3 | semidirect |
| 45. Q8×D7 | C22×C3 | semidirect |
| 67. 2−1+4⋊C3 | D7 | direct (N × H) |
| 113. #41⋊C3 | C2×D7 | semidirect |
| 116. C2×SL(2,𝔽3) | C2×D7 | semidirect |
| 126. 2−1+4 | C3×D7 | semidirect |
| 167. #41 | C2×C3×D7 | semidirect |
| 179. C2×Q8 | C2×C3×D7 | semidirect |
| 191. D7 | 2−1+4⋊C3 | direct (N × H) |
| 232. C7 | C2×2−1+4⋊C3 | semidirect |
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 |
| #10621530 | 2 |
| D7×#41⋊C3 | 2 |
| #10621527 | 2 |
| C2×D7×SL(2,𝔽3) | 2 |
| D7×2−1+4 | 3 |
| C3×D4×D7 | 4 |
| C2×2−1+4⋊C3 | 7 |