Finite Group DatabasebetaFinite Group Databasebeta← ΓL(1,𝔽8)×#11716328 #30660673
Order 16758.
100 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 25 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. ΓL(1,𝔽8)×#87705 | C2 | semidirect |
| 2. ΓL(1,𝔽8)×D133 | C3 | semidirect |
| 3. #19150301 | C3 | semidirect |
| 4. #19150295 | C3 | semidirect |
| 5. C7×#11716328 | C3 | semidirect |
| 6. C19⋊#3862 | C2×C3 | semidirect |
| 7. #14584116 | C2×C3 | semidirect |
| 8. C7×#87705 | C2×C3 | semidirect |
| 9. C7×C19×ΓL(1,𝔽8) | C2×C3 | semidirect |
| 12. C7×D133 | C32 | semidirect |
| 15. C72×C19 | C2×C32 | semidirect |
| 20. #11716328 | ΓL(1,𝔽8) | direct (N × H) |
| 31. #87705 | C2×ΓL(1,𝔽8) | semidirect |
| 34. C19×ΓL(1,𝔽8) | AGL(1,𝔽7) | semidirect |
| 40. D133 | C3×ΓL(1,𝔽8) | semidirect |
| 46. C7×ΓL(1,𝔽8) | D19⋊C3 | semidirect |
| 48. C7×C19 | C3×AGL(1,𝔽7) | semidirect |
| 49. C7×C19 | C2×C3×ΓL(1,𝔽8) | semidirect |
| 63. C72 | C3×D19⋊C3 | semidirect |
| 81. ΓL(1,𝔽8) | #11716328 | direct (N × H) |
| 82. C19 | ΓL(1,𝔽8)×AGL(1,𝔽7) | semidirect |
| 88. C7 | ΓL(1,𝔽8)×D19⋊C3 | semidirect |
| 89. C7 | C3×#11716328 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#87705 | 2 |
| ΓL(1,𝔽8)×D133 | 3 |
| #19150301 | 3 |
| #19150295 | 3 |
| C7×#11716328 | 3 |
| C3×#11716328 | 7 |
| ΓL(1,𝔽8)×D19⋊C3 | 7 |
| ΓL(1,𝔽8)×AGL(1,𝔽7) | 19 |