Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
560 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 560 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 | C3×C37 | C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33×C37 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C37 ◁ C33×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C37×H(5,𝔽3) | C3 | semidirect |
| 2. C37×#6911 | C3 | semidirect |
| 3. C37×#6909 | C3 | semidirect |
| 4. C3×C37×AΣL(1,𝔽27) | C3 | semidirect |
| 5. C37×C32⋊H(3,𝔽3) | C3 | semidirect |
| 6. C34×C37 | C32 | semidirect |
| 7. C3×3−1+2×C37 | C32 | semidirect |
| 13. C3×H(3,𝔽3)×C37 | C32 | semidirect |
| 23. C33×C37 | H(3,𝔽3) | semidirect |
| 34. H(3,𝔽3)×C37 | H(3,𝔽3) | semidirect |
| 36. UT(4,𝔽3) | C37 | direct (N × H) |
| 52. H(5,𝔽3) | C3×C37 | semidirect |
| 53. #6911 | C3×C37 | semidirect |
| 54. #6909 | C3×C37 | semidirect |
| 55. C3×AΣL(1,𝔽27) | C3×C37 | semidirect |
| 56. C32⋊H(3,𝔽3) | C3×C37 | semidirect |
| 64. C34 | C32×C37 | semidirect |
| 65. C3×3−1+2 | C32×C37 | semidirect |
| 71. C3×H(3,𝔽3) | C32×C37 | semidirect |
| 75. C37 | UT(4,𝔽3) | direct (N × H) |
| 82. C33 | H(3,𝔽3)×C37 | semidirect |
| 93. H(3,𝔽3) | H(3,𝔽3)×C37 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C37×H(5,𝔽3) | 3 |
| C37×#6911 | 3 |
| C37×#6909 | 3 |
| C3×C37×AΣL(1,𝔽27) | 3 |
| C37×C32⋊H(3,𝔽3) | 3 |
| UT(4,𝔽3) | 37 |