Finite Group DatabasebetaFinite Group DatabasebetaOrder 34263.
472 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C47 | H(3,𝔽3) |
| commutator | C32 | C32×C9×C47 |
| Frattini, cosocle | C33 | C33×C47 |
| socle | C32×C47 | C3×H(3,𝔽3) |
| prime radical | C3×C47×AΣL(1,𝔽27) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C47 ◁ C32×C9×C47 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×C9×C47 | C3 | semidirect |
| 3. C9×3−1+2×C47 | C3 | semidirect |
| 4. C9×H(3,𝔽3)×C47 | C3 | semidirect |
| 5. C47×3−1+2⋊C9 | C3 | semidirect |
| 6. C47×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 7. C47×#6866 | C3 | semidirect |
| 17. C47×AΣL(1,𝔽27) | C9 | direct (N × H) |
| 28. C33×C47 | C3×C9 | semidirect |
| 31. 3−1+2×C47 | C3×C9 | semidirect |
| 33. H(3,𝔽3)×C47 | C3×C9 | semidirect |
| 45. C9×AΣL(1,𝔽27) | C47 | direct (N × H) |
| 64. C9×C47 | AΣL(1,𝔽27) | direct (N × H) |
| 65. C33×C9 | C3×C47 | semidirect |
| 67. C9×3−1+2 | C3×C47 | semidirect |
| 68. C9×H(3,𝔽3) | C3×C47 | semidirect |
| 69. 3−1+2⋊C9 | C3×C47 | semidirect |
| 70. H(3,𝔽3)⋊C9 | C3×C47 | semidirect |
| 71. #6866 | C3×C47 | semidirect |
| 88. AΣL(1,𝔽27) | C9×C47 | direct (N × H) |
| 94. C47 | C9×AΣL(1,𝔽27) | direct (N × H) |
| 100. C33 | C3×C9×C47 | semidirect |
| 103. 3−1+2 | C3×C9×C47 | semidirect |
| 105. H(3,𝔽3) | C3×C9×C47 | semidirect |
| 135. C9 | C47×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 |
|---|---|
| C33×C9×C47 | 3 |
| C3×C47×AΣL(1,𝔽27) | 3 |
| C9×3−1+2×C47 | 3 |
| C9×H(3,𝔽3)×C47 | 3 |
| C47×3−1+2⋊C9 | 3 |
| C47×H(3,𝔽3)⋊C9 | 3 |
| C47×#6866 | 3 |
| C9×AΣL(1,𝔽27) | 47 |