Finite Group DatabasebetaFinite Group Databasebeta← ΓL(1,𝔽8)×AΣL(1,𝔽16) #13051079
Order 1344.
188 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 39 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 |
|---|---|---|
| 2. ΓL(1,𝔽8)×#100 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#99 | C2 | semidirect |
| 4. C7×AΣL(1,𝔽16) | C3 | semidirect |
| 5. C24×ΓL(1,𝔽8) | C4 | semidirect |
| 11. C22⋊C4×ΓL(1,𝔽8) | C4 | semidirect |
| 13. C7×#100 | C2×C3 | semidirect |
| 14. C7×#99 | C2×C3 | semidirect |
| 27. C24×C7 | C3×C4 | semidirect |
| 33. C7×C22⋊C4 | C3×C4 | semidirect |
| 50. AΣL(1,𝔽16) | ΓL(1,𝔽8) | direct (N × H) |
| 75. #100 | C2×ΓL(1,𝔽8) | semidirect |
| 76. #99 | C2×ΓL(1,𝔽8) | semidirect |
| 101. ΓL(1,𝔽8) | AΣL(1,𝔽16) | direct (N × H) |
| 102. C24 | C4×ΓL(1,𝔽8) | semidirect |
| 108. C22⋊C4 | C4×ΓL(1,𝔽8) | semidirect |
| 138. C7 | C3×AΣL(1,𝔽16) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#120 | 2 |
| ΓL(1,𝔽8)×#100 | 2 |
| ΓL(1,𝔽8)×#99 | 2 |
| C7×AΣL(1,𝔽16) | 3 |
| C3×AΣL(1,𝔽16) | 7 |