Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
304 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 304 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 | C7×2−1+4 | D3 |
| hypercenter | C7×D4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C7 ◁ C7×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. C7×#986 | C2 | semidirect |
| 3. C7×#985 | C2 | semidirect |
| 4. C7×#983 | C2 | semidirect |
| 5. C2×C7×GL(2,𝔽3) | C2 | semidirect |
| 8. C7×#41⋊C3 | C22 | semidirect |
| 9. C2×C7×SL(2,𝔽3) | C22 | semidirect |
| 10. C7×GL(2,𝔽3) | C22 | semidirect |
| 11. C7×GL(2,𝔽3) | C22 | semidirect |
| 12. C7×#226 | C22 | semidirect |
| 13. C7×2−1+4 | D3 | semidirect |
| 24. #5926 | C7 | direct (N × H) |
| 36. C7×#41 | C2×D3 | semidirect |
| 38. C2×C7×Q8 | C2×D3 | semidirect |
| 57. 2−1+4⋊C3 | C2×C7 | semidirect |
| 58. #986 | C2×C7 | semidirect |
| 59. #985 | C2×C7 | semidirect |
| 60. #983 | C2×C7 | semidirect |
| 61. C2×GL(2,𝔽3) | C2×C7 | semidirect |
| 93. #41⋊C3 | C22×C7 | semidirect |
| 94. C2×SL(2,𝔽3) | C22×C7 | semidirect |
| 95. GL(2,𝔽3) | C22×C7 | semidirect |
| 96. GL(2,𝔽3) | C22×C7 | semidirect |
| 97. #226 | C22×C7 | semidirect |
| 102. 2−1+4 | D3×C7 | semidirect |
| 135. #41 | C2×D3×C7 | semidirect |
| 137. C2×Q8 | C2×D3×C7 | semidirect |
| 190. C7 | #5926 | direct (N × H) |
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 |
| C7×#986 | 2 |
| C7×#985 | 2 |
| C7×#983 | 2 |
| C2×C7×GL(2,𝔽3) | 2 |
| C7×#577 | 3 |
| D3×C7×D4 | 4 |
| #5926 | 7 |