Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
372 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 92 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×C37 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33×C37 |
| prime radical | C3×C37×AΣL(1,𝔽27) | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ C32×C9×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×3−1+2×C37 | C3 | semidirect |
| 3. C37×#6894 | C3 | semidirect |
| 4. C9×H(3,𝔽3)×C37 | C3 | semidirect |
| 5. C37×3−1+2⋊C9 | C3 | semidirect |
| 6. C37×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 7. C37×#6866 | C3 | semidirect |
| 35. H(3,𝔽3)×C37 | 3−1+2 | semidirect |
| 45. #10624794 | C37 | direct (N × H) |
| 64. C9×C37 | AΣL(1,𝔽27) | semidirect |
| 65. C32×3−1+2 | C3×C37 | semidirect |
| 67. #6894 | C3×C37 | semidirect |
| 68. C9×H(3,𝔽3) | C3×C37 | semidirect |
| 69. 3−1+2⋊C9 | C3×C37 | semidirect |
| 70. H(3,𝔽3)⋊C9 | C3×C37 | semidirect |
| 71. #6866 | C3×C37 | semidirect |
| 94. C37 | #10624794 | direct (N × H) |
| 107. H(3,𝔽3) | 3−1+2×C37 | semidirect |
| 135. C9 | C37×AΣL(1,𝔽27) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×3−1+2×C37 | 3 |
| C3×C37×AΣL(1,𝔽27) | 3 |
| C37×#6894 | 3 |
| C9×H(3,𝔽3)×C37 | 3 |
| C37×3−1+2⋊C9 | 3 |
| C37×H(3,𝔽3)⋊C9 | 3 |
| C37×#6866 | 3 |
| #10624794 | 37 |