Finite Group DatabasebetaFinite Group DatabasebetaOrder 29727.
28 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C9×C367⋊C3 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C367⋊C27 | C3 | 1 | 2 | yes | maximal, complemented |
| 3 | C27×C367 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×C367⋊C3 | C9 | 1 | 1 | yes | characteristic |
| 5 | C9×C367 | C32 | 1 | 1 | yes | characteristic |
| 6 | C367⋊C9 | C9 | 1 | 2 | yes | — |
| 7 | C3×C367 | C3×C9 | 1 | 1 | yes | characteristic |
| 8 | C367⋊C3 | C27 | 1 | 3 | yes | direct product factor |
| 9 | C367 | C3×C27 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C3×C27 | — | 367 | 1 | no | maximal, Sylow |
| 11 | C3×C9 | — | 367 | 1 | no | — |
| 12 | C27 | — | 367 | 2 | no | — |
| 13 | C27 | C367⋊C3 | 1 | 1 | yes | characteristic, direct product factor |
| 14 | C32 | — | 367 | 1 | no | — |
| 15 | C9 | — | 367 | 2 | no | — |
| 16 | C9 | C3×C367⋊C3 | 1 | 1 | yes | characteristic |
| 17 | C3 | — | 367 | 3 | no | — |
| 18 | C3 | C9×C367⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C27 | C367⋊C3 |
| commutator, hypocenter | C367 | C3×C27 |
| Frattini | C9 | C3×C367⋊C3 |
| Fitting | C27×C367 | C3 |
| socle | C3×C367 | C3×C9 |
| prime radical | C3×C367⋊C3 | C9 |
| cosocle | C9×C367 | C32 |
| LCS | G ▷ C367 | |
| UCS | C1 ◁ C27 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C367⋊C27 | C3 | semidirect |
| 3. C27×C367 | C3 | semidirect |
| 8. C367⋊C3 | C27 | direct (N × H) |
| 9. C367 | C3×C27 | semidirect |
| 13. C27 | C367⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×C367⋊C3 | 3 |
| C367⋊C27 | 3 |
| C27×C367 | 3 |
| C3×C27 | 367 |