Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
530 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 530 conjugacy classes of subgroups — too many to draw. Only its 98 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 | C33 | C33×C37 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ C32×C9×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×C9×C37 | C3 | semidirect |
| 2. C3×C37×#733 | C3 | semidirect |
| 3. C3×C37×#732 | C3 | semidirect |
| 4. C3×C37×#731 | C3 | semidirect |
| 5. C3×C37×AΣL(1,𝔽27) | C3 | semidirect |
| 6. C37×#6893 | C3 | semidirect |
| 49. #10624734 | C37 | direct (N × H) |
| 66. C33×C9 | C3×C37 | semidirect |
| 67. C3×#733 | C3×C37 | semidirect |
| 68. C3×#732 | C3×C37 | semidirect |
| 69. C3×#731 | C3×C37 | semidirect |
| 70. C3×AΣL(1,𝔽27) | C3×C37 | semidirect |
| 71. #6893 | C3×C37 | semidirect |
| 94. C37 | #10624734 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×C9×C37 | 3 |
| C3×C37×#733 | 3 |
| C3×C37×#732 | 3 |
| C3×C37×#731 | 3 |
| C3×C37×AΣL(1,𝔽27) | 3 |
| C37×#6893 | 3 |
| #10624734 | 37 |