Finite Group DatabasebetaFinite Group DatabasebetaOrder 22932.
168 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 55 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 | C2 | ΓL(1,𝔽8)2×D13 |
| commutator, hypocenter | C72×C13 | C32×C4 |
| Fitting, socle, cosocle | C2×C72×C13 | C2×C32 |
| prime radical | C2×C13×ΓL(1,𝔽8)2 | C2 |
| LCS | G ▷ C72×C13 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Dic13×#3862 | C3 | semidirect |
| 3. Dic13×#3861 | C3 | semidirect |
| 4. C7×ΓL(1,𝔽8)×Dic13 | C3 | semidirect |
| 5. C13×ΓL(1,𝔽8)2 | C4 | semidirect |
| 10. C72×Dic13 | C32 | semidirect |
| 11. C13×#3862 | C3×C4 | semidirect |
| 12. C13×#3861 | C3×C4 | semidirect |
| 13. C7×C13×ΓL(1,𝔽8) | C3×C4 | semidirect |
| 21. ΓL(1,𝔽8)×Dic13 | ΓL(1,𝔽8) | direct (N × H) |
| 24. C72×C13 | C32×C4 | semidirect |
| 34. ΓL(1,𝔽8)2 | Dic13 | direct (N × H) |
| 36. C7×Dic13 | C3×ΓL(1,𝔽8) | semidirect |
| 44. C13×ΓL(1,𝔽8) | C4×ΓL(1,𝔽8) | semidirect |
| 53. #3862 | C3×Dic13 | semidirect |
| 54. #3861 | C3×Dic13 | semidirect |
| 55. C7×ΓL(1,𝔽8) | C3×Dic13 | semidirect |
| 60. C7×C13 | C3×C4×ΓL(1,𝔽8) | semidirect |
| 70. Dic13 | ΓL(1,𝔽8)2 | direct (N × H) |
| 71. C72 | C32×Dic13 | semidirect |
| 88. ΓL(1,𝔽8) | ΓL(1,𝔽8)×Dic13 | direct (N × H) |
| 92. C13 | C4×ΓL(1,𝔽8)2 | semidirect |
| 98. C7 | C3×ΓL(1,𝔽8)×Dic13 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C13×ΓL(1,𝔽8)2 | 2 |
| Dic13×#3862 | 3 |
| Dic13×#3861 | 3 |
| C7×ΓL(1,𝔽8)×Dic13 | 3 |
| C3×ΓL(1,𝔽8)×Dic13 | 7 |
| C4×ΓL(1,𝔽8)2 | 13 |