Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
456 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 456 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×#6910 | C3 | semidirect |
| 3. C37×#6909 | C3 | semidirect |
| 4. C37×#6909 | C3 | semidirect |
| 5. C3×C37×#732 | C3 | semidirect |
| 6. C3×C37×#731 | C3 | semidirect |
| 7. C37×#6893 | C3 | semidirect |
| 8. C3×3−1+2×C37 | C32 | semidirect |
| 9. C3×3−1+2×C37 | C32 | semidirect |
| 16. C3×H(3,𝔽3)×C37 | C32 | semidirect |
| 17. C32×C9×C37 | C32 | semidirect |
| 50. #10624748 | C37 | direct (N × H) |
| 67. H(5,𝔽3) | C3×C37 | semidirect |
| 68. #6910 | C3×C37 | semidirect |
| 69. #6909 | C3×C37 | semidirect |
| 70. #6909 | C3×C37 | semidirect |
| 71. C3×#732 | C3×C37 | semidirect |
| 72. C3×#731 | C3×C37 | semidirect |
| 73. #6893 | C3×C37 | semidirect |
| 82. C3×3−1+2 | C32×C37 | semidirect |
| 83. C3×3−1+2 | C32×C37 | semidirect |
| 90. C3×H(3,𝔽3) | C32×C37 | semidirect |
| 91. C32×C9 | C32×C37 | semidirect |
| 98. C37 | #10624748 | direct (N × H) |
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×#6910 | 3 |
| C37×#6909 | 3 |
| C3×C37×#732 | 3 |
| C3×C37×#731 | 3 |
| C37×#6893 | 3 |
| #10624748 | 37 |