Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
190 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×C7×GL(2,𝔽3) | C2 | semidirect |
| 3. C7×#859 | C2 | semidirect |
| 9. C22×C7×Q8 | D3 | semidirect |
| 16. #5421 | C7 | direct (N × H) |
| 21. C7×SL(2,𝔽3) | D4 | semidirect |
| 36. C22×SL(2,𝔽3) | C2×C7 | semidirect |
| 37. C2×GL(2,𝔽3) | C2×C7 | semidirect |
| 38. #859 | C2×C7 | semidirect |
| 53. C7×Q8 | #67 | semidirect |
| 73. C22×Q8 | D3×C7 | semidirect |
| 85. C22×C7 | GL(2,𝔽3) | semidirect |
| 95. SL(2,𝔽3) | C7×D4 | semidirect |
| 129. Q8 | C7×#67 | semidirect |
| 140. C7 | #5421 | direct (N × H) |
| 151. C22 | C7×GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×SL(2,𝔽3) | 2 |
| C2×C7×GL(2,𝔽3) | 2 |
| C7×#859 | 2 |
| C7×#448 | 3 |
| C2×C7×#67 | 4 |
| #5421 | 7 |