Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
224 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C27×C37 | H(3,𝔽3) |
| commutator | C32 | C32×C9×C37 |
| Frattini, cosocle | C3×C9 | C33×C37 |
| socle | C3×C37 | C9×H(3,𝔽3) |
| prime radical | C37×#6908 | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C27×C37 ◁ C3×C27×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C37×#6903 | C3 | semidirect |
| 3. C32×C27×C37 | C3 | semidirect |
| 4. C37×#6873 | C3 | semidirect |
| 5. C37×#6872 | C3 | semidirect |
| 6. C37×#6869 | C3 | semidirect |
| 36. #10624724 | C37 | direct (N × H) |
| 47. #6903 | C3×C37 | semidirect |
| 48. C32×C27 | C3×C37 | semidirect |
| 49. #6873 | C3×C37 | semidirect |
| 50. #6872 | C3×C37 | semidirect |
| 51. #6869 | C3×C37 | semidirect |
| 70. C37 | #10624724 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C37×#6908 | 3 |
| C37×#6903 | 3 |
| C32×C27×C37 | 3 |
| C37×#6873 | 3 |
| C37×#6872 | 3 |
| C37×#6869 | 3 |
| #10624724 | 37 |