Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
1064 conjugacy classes of subgroups; only the 476 normal subgroups are shown.
This group has 1064 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 | C3×C9×C37 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34×C37 |
| socle | C32×C37 | C3×H(3,𝔽3) |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C37 ◁ C32×C9×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C37×#737 | C3 | semidirect |
| 2. C33×C9×C37 | C3 | semidirect |
| 3. C37×#6908 | C3 | direct (N × H) |
| 4. C3×C37×#733 | C3 | semidirect |
| 5. C3×C37×#732 | C3 | semidirect |
| 6. C3×C37×#731 | C3 | semidirect |
| 7. C3×C37×AΣL(1,𝔽27) | C3 | semidirect |
| 10. C37×#737 | C32 | semidirect |
| 21. C32×C9×C37 | C32 | semidirect |
| 22. C37×#733 | C32 | semidirect |
| 23. C37×#732 | C32 | semidirect |
| 24. C37×#731 | C32 | semidirect |
| 25. C37×AΣL(1,𝔽27) | C32 | semidirect |
| 51. C3×#6908 | C37 | direct (N × H) |
| 67. C3×#737 | C3×C37 | semidirect |
| 68. C33×C9 | C3×C37 | semidirect |
| 69. #6908 | C3×C37 | direct (N × H) |
| 70. C3×#733 | C3×C37 | semidirect |
| 71. C3×#732 | C3×C37 | semidirect |
| 72. C3×#731 | C3×C37 | semidirect |
| 73. C3×AΣL(1,𝔽27) | C3×C37 | semidirect |
| 78. C3×C37 | #6908 | direct (N × H) |
| 82. #737 | C32×C37 | semidirect |
| 93. C32×C9 | C32×C37 | semidirect |
| 94. #733 | C32×C37 | semidirect |
| 95. #732 | C32×C37 | semidirect |
| 96. #731 | C32×C37 | semidirect |
| 97. AΣL(1,𝔽27) | C32×C37 | semidirect |
| 98. C37 | C3×#6908 | direct (N × H) |
| 143. C3 | C37×#6908 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C37×#737 | 3 |
| C33×C9×C37 | 3 |
| C37×#6908 | 3 |
| C3×C37×#733 | 3 |
| C3×C37×#732 | 3 |
| C3×C37×#731 | 3 |
| C3×C37×AΣL(1,𝔽27) | 3 |
| C3×#6908 | 37 |