Finite Group DatabasebetaFinite Group DatabasebetaOrder 29988.
152 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C17 | #64418⋊C3 |
| commutator, cosocle, hypocenter | #3862 | C22×C3×C17 |
| Frattini | C1 | G |
| Fitting, socle | C2×C72×C17 | C3×D3 |
| LCS | G ▷ #3862 | |
| UCS | C1 ◁ C2×C17 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C17×ΓL(1,𝔽8)2 | C2 | semidirect |
| 2. C17×#64418⋊C3 | C2 | direct (N × H) |
| 3. C2×C17×#64418 | C3 | semidirect |
| 5. C17×ΓL(1,𝔽8)2 | C22 | semidirect |
| 6. C2×C17×#3862 | C2×C3 | semidirect |
| 7. C2×C17×#3861 | D3 | semidirect |
| 9. C17×#64418 | C2×C3 | semidirect |
| 12. C17×#3862 | C22×C3 | semidirect |
| 13. C17×#3861 | C2×D3 | semidirect |
| 16. C2×#64418⋊C3 | C17 | direct (N × H) |
| 17. C2×C72×C17 | C3×D3 | semidirect |
| 22. C2×ΓL(1,𝔽8)2 | C2×C17 | semidirect |
| 23. #64418⋊C3 | C2×C17 | direct (N × H) |
| 24. C72×C17 | C2×C3×D3 | semidirect |
| 35. C2×#64418 | C3×C17 | semidirect |
| 38. ΓL(1,𝔽8)2 | C22×C17 | semidirect |
| 47. C2×#3862 | C2×C3×C17 | semidirect |
| 48. C2×#3861 | D3×C17 | semidirect |
| 51. #64418 | C2×C3×C17 | semidirect |
| 61. #3862 | C22×C3×C17 | semidirect |
| 62. #3861 | C2×D3×C17 | semidirect |
| 73. C2×C72 | C3×D3×C17 | semidirect |
| 82. C72 | C2×C3×D3×C17 | semidirect |
| 93. C2×C17 | #64418⋊C3 | direct (N × H) |
| 104. C17 | C2×#64418⋊C3 | direct (N × H) |
| 126. C2 | C17×#64418⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C17×ΓL(1,𝔽8)2 | 2 |
| C17×#64418⋊C3 | 2 |
| C2×C17×#64418 | 3 |
| C2×C17×#64423 | 3 |
| C2×#64418⋊C3 | 17 |
| C2×C3×D3×C17 | 49 |