Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
948 conjugacy classes of subgroups; only the 470 normal subgroups are shown.
This group has 948 conjugacy classes of subgroups — too many to draw. Only its 470 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 | C32 | C34×C37 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C37 ◁ C3×3−1+2×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C37×3−1+4 | C3 | semidirect |
| 2. C3×C37×#737 | C3 | semidirect |
| 3. C37×#6913 | C3 | semidirect |
| 4. C37×#6912 | C3 | semidirect |
| 5. C37×#6911 | C3 | semidirect |
| 6. C37×#6909 | C3 | semidirect |
| 7. C37×#6908 | C3 | semidirect |
| 8. C3×C37×#731 | C3 | semidirect |
| 11. C37×#737 | C32 | semidirect |
| 16. C3×3−1+2×C37 | C32 | semidirect |
| 20. C32×C9×C37 | C32 | semidirect |
| 21. C37×#733 | C32 | semidirect |
| 23. C37×#731 | C32 | semidirect |
| 24. C37×#731 | C32 | semidirect |
| 25. C37×AΣL(1,𝔽27) | C32 | semidirect |
| 47. #10624935 | C37 | direct (N × H) |
| 60. 3−1+4 | C3×C37 | semidirect |
| 61. C3×#737 | C3×C37 | semidirect |
| 62. #6913 | C3×C37 | semidirect |
| 63. #6912 | C3×C37 | semidirect |
| 64. #6911 | C3×C37 | semidirect |
| 65. #6909 | C3×C37 | semidirect |
| 66. #6908 | C3×C37 | semidirect |
| 67. C3×#731 | C3×C37 | semidirect |
| 75. #737 | C32×C37 | semidirect |
| 80. C3×3−1+2 | C32×C37 | semidirect |
| 84. C32×C9 | C32×C37 | semidirect |
| 85. #733 | C32×C37 | semidirect |
| 87. #731 | C32×C37 | semidirect |
| 88. #731 | C32×C37 | semidirect |
| 89. AΣL(1,𝔽27) | C32×C37 | semidirect |
| 90. C37 | #10624935 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C37×3−1+4 | 3 |
| C3×C37×#737 | 3 |
| C37×#6913 | 3 |
| C37×#6912 | 3 |
| C37×#6911 | 3 |
| C37×#6909 | 3 |
| C37×#6908 | 3 |
| C3×C37×#731 | 3 |
| #10624935 | 37 |