Finite Group DatabasebetaFinite Group DatabasebetaOrder 31347.
304 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C43 | C3×H(3,𝔽3) |
| commutator | C32 | C32×C9×C43 |
| Frattini, cosocle | C33 | C33×C43 |
| prime radical | C3×C43×AΣL(1,𝔽27) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C43 ◁ C32×C9×C43 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C9×3−1+2×C43 | C3 | semidirect |
| 3. C9×H(3,𝔽3)×C43 | C3 | semidirect |
| 4. C3×C43×C32⋊C9 | C3 | semidirect |
| 5. C43×3−1+2⋊C9 | C3 | semidirect |
| 6. C43×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 7. C43×#6866 | C3 | semidirect |
| 14. C43×AΣL(1,𝔽27) | C9 | semidirect |
| 28. C33×C43 | C3×C9 | semidirect |
| 31. 3−1+2×C43 | C3×C9 | semidirect |
| 43. AΣL(1,𝔽27)⋊C9 | C43 | direct (N × H) |
| 64. C9×3−1+2 | C3×C43 | semidirect |
| 65. C9×H(3,𝔽3) | C3×C43 | semidirect |
| 66. C3×C32⋊C9 | C3×C43 | semidirect |
| 67. 3−1+2⋊C9 | C3×C43 | semidirect |
| 68. H(3,𝔽3)⋊C9 | C3×C43 | semidirect |
| 69. #6866 | C3×C43 | semidirect |
| 83. AΣL(1,𝔽27) | C9×C43 | semidirect |
| 92. C43 | AΣL(1,𝔽27)⋊C9 | direct (N × H) |
| 98. C33 | C3×C9×C43 | semidirect |
| 101. 3−1+2 | C3×C9×C43 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C43×AΣL(1,𝔽27) | 3 |
| C9×3−1+2×C43 | 3 |
| C9×H(3,𝔽3)×C43 | 3 |
| C3×C43×C32⋊C9 | 3 |
| C43×3−1+2⋊C9 | 3 |
| C43×H(3,𝔽3)⋊C9 | 3 |
| C43×#6866 | 3 |
| AΣL(1,𝔽27)⋊C9 | 43 |