Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
14 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C37×C243 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C37×C81 | C9 | 1 | 1 | yes | characteristic |
| 3 | C27×C37 | C27 | 1 | 1 | yes | characteristic |
| 4 | C729 | — | 37 | 1 | no | maximal, Sylow |
| 5 | C9×C37 | C81 | 1 | 1 | yes | characteristic |
| 6 | C243 | C37⋊C3 | 1 | 1 | yes | characteristic |
| 7 | C3×C37 | C243 | 1 | 1 | yes | characteristic |
| 8 | C81 | #66524 | 1 | 1 | yes | characteristic |
| 9 | C37 | C729 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C27 | #11758601 | 1 | 1 | yes | characteristic |
| 11 | C9 | #15688500 | 1 | 1 | yes | characteristic |
| 12 | C3 | #22582440 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C243 | C37⋊C3 |
| commutator, hypocenter | C37 | C729 |
| Fitting, cosocle | C37×C243 | C3 |
| socle, prime radical | C3×C37 | C243 |
| LCS | G ▷ C37 | |
| UCS | C1 ◁ C243 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. C37 | C729 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C37×C243 | 3 |
| C729 | 37 |