Finite Group DatabasebetaFinite Group DatabasebetaOrder 21609.
225 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C9×C49 | C7 | 1 | 8 | yes | maximal |
| 2 | C3×C492 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C492 | C9 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 4 | C3×C7×C49 | C3×C7 | 1 | 8 | yes | — |
| 5 | C72×C9 | C72 | 1 | 1 | yes | characteristic |
| 6 | C9×C49 | C49 | 1 | 56 | yes | direct product factor |
| 7 | C7×C49 | C7×C9 | 1 | 8 | yes | — |
| 8 | C3×C72 | C3×C72 | 1 | 1 | yes | characteristic |
| 9 | C3×C49 | C3×C49 | 1 | 56 | yes | — |
| 10 | C7×C9 | C7×C49 | 1 | 8 | yes | — |
| 11 | C72 | C72×C9 | 1 | 1 | yes | characteristic |
| 12 | C49 | C9×C49 | 1 | 56 | yes | direct product factor |
| 13 | C3×C7 | C3×C7×C49 | 1 | 8 | yes | — |
| 14 | C9 | C492 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 15 | C7 | C7×C9×C49 | 1 | 8 | yes | — |
| 16 | C3 | C3×C492 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, socle, prime radical, cosocle | C3×C72 | C3×C72 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C492 | C9 | direct (N × H) |
| 6. C9×C49 | C49 | direct (N × H) |
| 12. C49 | C9×C49 | direct (N × H) |
| 14. C9 | C492 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C9×C49 | 7 |
| C3×C492 | 3 |