Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
950 conjugacy classes of subgroups; only the 482 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 482 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 | C34 |
| commutator, Frattini, cosocle | C32 | C34×C37 |
| prime radical | C32×H(3,𝔽3)×C37 | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C37×C9⋊H(3,𝔽3) | C3 | semidirect |
| 3. C37×#6891 | C3 | semidirect |
| 4. C9×H(3,𝔽3)×C37 | C3 | semidirect |
| 5. C3×C37×C32⋊C9 | C3 | semidirect |
| 7. C3×3−1+2×C37 | C32 | semidirect |
| 11. C32×C9×C37 | C32 | semidirect |
| 14. C37×C32⋊C9 | C32 | semidirect |
| 15. C37×C32⋊C9 | C32 | semidirect |
| 16. C37×C32⋊C9 | C32 | semidirect |
| 26. H(3,𝔽3)×C37 | 3−1+2 | semidirect |
| 30. #10624897 | C37 | direct (N × H) |
| 48. C9⋊H(3,𝔽3) | C3×C37 | semidirect |
| 49. #6891 | C3×C37 | semidirect |
| 50. C9×H(3,𝔽3) | C3×C37 | semidirect |
| 51. C3×C32⋊C9 | C3×C37 | semidirect |
| 60. C3×3−1+2 | C32×C37 | semidirect |
| 64. C32×C9 | C32×C37 | semidirect |
| 67. C32⋊C9 | C32×C37 | semidirect |
| 68. C32⋊C9 | C32×C37 | semidirect |
| 69. C32⋊C9 | C32×C37 | semidirect |
| 70. C37 | #10624897 | direct (N × H) |
| 80. H(3,𝔽3) | 3−1+2×C37 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×H(3,𝔽3)×C37 | 3 |
| C37×C9⋊H(3,𝔽3) | 3 |
| C37×#6891 | 3 |
| C9×H(3,𝔽3)×C37 | 3 |
| C3×C37×C32⋊C9 | 3 |
| #10624897 | 37 |