Finite Group DatabasebetaFinite Group Databasebeta← ΓL(1,𝔽8)⋊#13095538 #45657432
Order 32634.
112 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 23 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×C37 | C2×C32 |
| LCS | G ▷ C72×C37 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×#11715912 | C2 | semidirect |
| 2. ΓL(1,𝔽8)⋊D259 | C3 | semidirect |
| 3. #24925970 | C3 | semidirect |
| 4. #24925953 | C3 | semidirect |
| 5. #24925948 | C3 | semidirect |
| 6. C37⋊#3862 | C2×C3 | semidirect |
| 7. #19129589 | C2×C3 | semidirect |
| 8. C7×#11715912 | C2×C3 | semidirect |
| 9. C7×ΓL(1,𝔽8)×C37 | C2×C3 | semidirect |
| 12. C7⋊D259 | C32 | semidirect |
| 15. C72×C37 | C2×C32 | semidirect |
| 32. #11715912 | AGL(1,𝔽7) | semidirect |
| 34. ΓL(1,𝔽8)×C37 | AGL(1,𝔽7) | semidirect |
| 46. C7×C37 | C3×AGL(1,𝔽7) | semidirect |
| 47. C7×C37 | C3×AGL(1,𝔽7) | semidirect |
| 55. C7×ΓL(1,𝔽8) | D37⋊C3 | semidirect |
| 66. C72 | C3×D37⋊C3 | semidirect |
| 76. C37 | ΓL(1,𝔽8)⋊AGL(1,𝔽7) | semidirect |
| 85. ΓL(1,𝔽8) | #13095538 | semidirect |
| 92. C7 | ΓL(1,𝔽8)⋊(D37⋊C3) | semidirect |
| 93. C7 | C3×#13095538 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#11715912 | 2 |
| ΓL(1,𝔽8)⋊D259 | 3 |
| #24925970 | 3 |
| #24925953 | 3 |
| #24925948 | 3 |
| C3×#13095538 | 7 |
| ΓL(1,𝔽8)⋊(D37⋊C3) | 7 |
| ΓL(1,𝔽8)⋊AGL(1,𝔽7) | 37 |