Finite Group DatabasebetaFinite Group Databasebeta← C32×C37×AΣL(1,𝔽27) #41004708
Order 26973.
2672 conjugacy classes of subgroups; only the 512 normal subgroups are shown.
This group has 2672 conjugacy classes of subgroups — too many to draw. Only its 512 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C37 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34×C37 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33×C37 ◁ C34×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C35×C37 | C3 | semidirect |
| 2. C32×3−1+2×C37 | C3 | semidirect |
| 3. C32×H(3,𝔽3)×C37 | C3 | semidirect |
| 4. C3×C37×AΣL(1,𝔽27) | C3 | direct (N × H) |
| 10. C34×C37 | C32 | semidirect |
| 12. C3×3−1+2×C37 | C32 | semidirect |
| 14. C3×H(3,𝔽3)×C37 | C32 | semidirect |
| 16. C37×AΣL(1,𝔽27) | C32 | direct (N × H) |
| 28. C33×C37 | C33 | semidirect |
| 30. 3−1+2×C37 | C33 | semidirect |
| 32. H(3,𝔽3)×C37 | C33 | semidirect |
| 34. C32×AΣL(1,𝔽27) | C37 | direct (N × H) |
| 43. C32×C37 | AΣL(1,𝔽27) | direct (N × H) |
| 48. C35 | C3×C37 | semidirect |
| 49. C32×3−1+2 | C3×C37 | semidirect |
| 50. C32×H(3,𝔽3) | C3×C37 | semidirect |
| 51. C3×AΣL(1,𝔽27) | C3×C37 | direct (N × H) |
| 56. C3×C37 | C3×AΣL(1,𝔽27) | direct (N × H) |
| 63. C34 | C32×C37 | semidirect |
| 65. C3×3−1+2 | C32×C37 | semidirect |
| 67. C3×H(3,𝔽3) | C32×C37 | semidirect |
| 69. AΣL(1,𝔽27) | C32×C37 | direct (N × H) |
| 70. C37 | C32×AΣL(1,𝔽27) | direct (N × H) |
| 82. C33 | C33×C37 | semidirect |
| 84. 3−1+2 | C33×C37 | semidirect |
| 86. H(3,𝔽3) | C33×C37 | semidirect |
| 96. C32 | C37×AΣL(1,𝔽27) | direct (N × H) |
| 105. C3 | C3×C37×AΣL(1,𝔽27) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C35×C37 | 3 |
| C32×3−1+2×C37 | 3 |
| C32×H(3,𝔽3)×C37 | 3 |
| C3×C37×AΣL(1,𝔽27) | 3 |
| C32×AΣL(1,𝔽27) | 37 |