Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
1304 conjugacy classes of subgroups; only the 476 normal subgroups are shown.
This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 476 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 | C32 | C34×C37 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ C34×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 |
| 2. C32×H(3,𝔽3)×C37 | C3 | semidirect |
| 3. C37×#6911 | C3 | direct (N × H) |
| 4. C3×C37×#732 | C3 | semidirect |
| 9. C3×3−1+2×C37 | C32 | semidirect |
| 12. C3×H(3,𝔽3)×C37 | C32 | semidirect |
| 13. C3×H(3,𝔽3)×C37 | C32 | semidirect |
| 14. C32×C9×C37 | C32 | semidirect |
| 15. C37×#732 | C32 | semidirect |
| 27. H(3,𝔽3)×C37 | C33 | semidirect |
| 30. C3×C9×C37 | C33 | semidirect |
| 31. C3×#6911 | C37 | direct (N × H) |
| 44. C32×3−1+2 | C3×C37 | semidirect |
| 45. C32×H(3,𝔽3) | C3×C37 | semidirect |
| 46. #6911 | C3×C37 | direct (N × H) |
| 47. C3×#732 | C3×C37 | semidirect |
| 52. C3×C37 | #6911 | direct (N × H) |
| 58. C3×3−1+2 | C32×C37 | semidirect |
| 61. C3×H(3,𝔽3) | C32×C37 | semidirect |
| 62. C3×H(3,𝔽3) | C32×C37 | semidirect |
| 63. C32×C9 | C32×C37 | semidirect |
| 64. #732 | C32×C37 | semidirect |
| 65. C37 | C3×#6911 | direct (N × H) |
| 77. H(3,𝔽3) | C33×C37 | semidirect |
| 80. C3×C9 | C33×C37 | semidirect |
| 97. C3 | C37×#6911 | direct (N × H) |
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 |
| C32×H(3,𝔽3)×C37 | 3 |
| C37×#6911 | 3 |
| C3×C37×#732 | 3 |
| C3×#6911 | 37 |