Finite Group DatabasebetaFinite Group Databasebeta← ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊D19 #30660686
Order 16758.
100 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C19 | C2×C32 |
| LCS | G ▷ C72×C19 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C19×ΓL(1,𝔽8)2 | C2 | semidirect |
| 2. ΓL(1,𝔽8)×D133 | C3 | semidirect |
| 3. #19150304 | C3 | semidirect |
| 4. #19150298 | C3 | semidirect |
| 5. C7×ΓL(1,𝔽8)⋊D19 | C3 | semidirect |
| 6. C19×#3862 | C2×C3 | semidirect |
| 7. C19×#3861 | C2×C3 | semidirect |
| 8. C7×C19×ΓL(1,𝔽8) | C2×C3 | semidirect |
| 9. C7×C19×ΓL(1,𝔽8) | C2×C3 | semidirect |
| 12. C7×D133 | C32 | semidirect |
| 15. C72×C19 | C2×C32 | semidirect |
| 21. ΓL(1,𝔽8)⋊D19 | ΓL(1,𝔽8) | direct (N × H) |
| 25. ΓL(1,𝔽8)2 | D19 | semidirect |
| 33. C19×ΓL(1,𝔽8) | AGL(1,𝔽7) | semidirect |
| 34. C19×ΓL(1,𝔽8) | C2×ΓL(1,𝔽8) | semidirect |
| 40. D133 | C3×ΓL(1,𝔽8) | semidirect |
| 43. #3862 | C3×D19 | semidirect |
| 44. #3861 | C3×D19 | semidirect |
| 45. C7×ΓL(1,𝔽8) | C3×D19 | semidirect |
| 46. C7×ΓL(1,𝔽8) | C3×D19 | semidirect |
| 48. C7×C19 | C3×AGL(1,𝔽7) | semidirect |
| 49. C7×C19 | C2×C3×ΓL(1,𝔽8) | semidirect |
| 63. C72 | C32×D19 | semidirect |
| 80. ΓL(1,𝔽8) | ΓL(1,𝔽8)×D19 | semidirect |
| 81. ΓL(1,𝔽8) | ΓL(1,𝔽8)⋊D19 | direct (N × H) |
| 82. C19 | ΓL(1,𝔽8)×AGL(1,𝔽7) | semidirect |
| 88. C7 | C3×ΓL(1,𝔽8)⋊D19 | semidirect |
| 89. C7 | C3×ΓL(1,𝔽8)×D19 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C19×ΓL(1,𝔽8)2 | 2 |
| ΓL(1,𝔽8)×D133 | 3 |
| #19150304 | 3 |
| #19150298 | 3 |
| C7×ΓL(1,𝔽8)⋊D19 | 3 |
| C3×ΓL(1,𝔽8)×D19 | 7 |
| C3×ΓL(1,𝔽8)⋊D19 | 7 |
| ΓL(1,𝔽8)×AGL(1,𝔽7) | 19 |