Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
1718 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 1718 conjugacy classes of subgroups — too many to draw. Only its 274 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C33×C37⋊C3 |
| commutator, socle, cosocle | C32×C37 | C34 |
| Fitting | C37×C32⋊H(3,𝔽3) | C3 |
| hypercenter | C32⋊H(3,𝔽3) | C37⋊C3 |
| hypocenter | C37 | #10624880 |
| LCS | G ▷ C32×C37 ▷ C37 | |
| UCS | C1 ◁ C32 ◁ C32⋊H(3,𝔽3) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×C32⋊(C37⋊C3) | C3 | semidirect |
| 2. C3×H(3,𝔽3)×C37⋊C3 | C3 | semidirect |
| 3. #22582550 | C3 | semidirect |
| 4. (C3×C37⋊C3)⋊H(3,𝔽3) | C3 | semidirect |
| 5. (C3×C37⋊C3)⋊H(3,𝔽3) | C3 | semidirect |
| 6. C37×C32⋊H(3,𝔽3) | C3 | semidirect |
| 8. C33×C37⋊C3 | C32 | semidirect |
| 9. C33×C37⋊C3 | C32 | semidirect |
| 11. C3×C32⋊(C37⋊C3) | C32 | semidirect |
| 12. C3×C32⋊(C37⋊C3) | C32 | semidirect |
| 13. C3×C32⋊(C37⋊C3) | C32 | semidirect |
| 14. C3×C32⋊(C37⋊C3) | C32 | semidirect |
| 15. C3×C32⋊(C37⋊C3) | C32 | semidirect |
| 17. C34×C37 | C32 | semidirect |
| 18. C3×H(3,𝔽3)×C37 | C32 | semidirect |
| 19. C3×H(3,𝔽3)×C37 | C32 | semidirect |
| 30. C32×C37⋊C3 | H(3,𝔽3) | semidirect |
| 65. C32⋊H(3,𝔽3) | C37⋊C3 | semidirect |
| 77. C34 | C3×C37⋊C3 | semidirect |
| 85. C3×H(3,𝔽3) | C3×C37⋊C3 | semidirect |
| 86. C3×H(3,𝔽3) | C3×C37⋊C3 | semidirect |
| 87. C37 | #10624880 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C32⋊(C37⋊C3) | 3 |
| C3×H(3,𝔽3)×C37⋊C3 | 3 |
| #22582550 | 3 |
| (C3×C37⋊C3)⋊H(3,𝔽3) | 3 |
| C37×C32⋊H(3,𝔽3) | 3 |
| #10624880 | 37 |