Finite Group DatabasebetaFinite Group DatabasebetaOrder 16767.
152 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C9×C23×C27 | C3 | 1 | 4 | yes | maximal |
| 2 | C3×C23×C27 | C9 | 1 | 12 | yes | — |
| 3 | C92×C23 | C32 | 1 | 1 | yes | characteristic |
| 4 | C272 | C23 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 5 | C3×C9×C23 | C3×C9 | 1 | 4 | yes | — |
| 6 | C23×C27 | C27 | 1 | 36 | yes | direct product factor |
| 7 | C9×C27 | C3×C23 | 1 | 4 | yes | — |
| 8 | C32×C23 | C92 | 1 | 1 | yes | characteristic |
| 9 | C9×C23 | C3×C27 | 1 | 12 | yes | — |
| 10 | C3×C27 | C9×C23 | 1 | 12 | yes | — |
| 11 | C92 | C32×C23 | 1 | 1 | yes | characteristic |
| 12 | C3×C23 | C9×C27 | 1 | 4 | yes | — |
| 13 | C3×C9 | C3×C9×C23 | 1 | 4 | yes | — |
| 14 | C27 | C23×C27 | 1 | 36 | yes | direct product factor |
| 15 | C23 | C272 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 16 | C32 | C92×C23 | 1 | 1 | yes | characteristic |
| 17 | C9 | C3×C23×C27 | 1 | 12 | yes | — |
| 18 | C3 | C9×C23×C27 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C92 | C32×C23 |
| socle, prime radical | C32×C23 | C92 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C272 | C23 | direct (N × H) |
| 6. C23×C27 | C27 | direct (N × H) |
| 14. C27 | C23×C27 | direct (N × H) |
| 15. C23 | C272 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×C23×C27 | 3 |
| C272 | 23 |