Finite Group DatabasebetaFinite Group DatabasebetaOrder 19683.
129214865 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×H(7,𝔽3) | C3 | 1 | 3280 | yes | maximal, complemented |
| 2 | H(7,𝔽3) | C32 | 1 | 597780 | yes | complemented |
| 3 | C32×H(5,𝔽3) | C32 | 1 | 298480 | yes | complemented |
| 4 | C3×H(5,𝔽3) | C33 | 1 | 24176880 | yes | complemented |
| 5 | C33×H(3,𝔽3) | C33 | 1 | 918400 | yes | complemented |
| 6 | C35 | C34 | 1 | 91840 | yes | complemented |
| 7 | H(5,𝔽3) | C34 | 1 | 48958182 | yes | complemented |
| 8 | C32×H(3,𝔽3) | C34 | 1 | 26863200 | yes | complemented |
| 9 | C34 | — | 81 | 91840 | no | — |
| 10 | C34 | C35 | 1 | 918400 | yes | — |
| 11 | C3×H(3,𝔽3) | C35 | 1 | 24176880 | yes | — |
| 12 | C33 | — | 27 | 918400 | no | — |
| 13 | C33 | C36 | 1 | 298480 | yes | — |
| 14 | H(3,𝔽3) | C36 | 1 | 597780 | yes | — |
| 15 | C32 | — | 9 | 298480 | no | — |
| 16 | C32 | C37 | 1 | 3280 | yes | — |
| 17 | C3 | — | 3 | 3280 | no | — |
| 18 | C3 | C38 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C3 | C38 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×H(7,𝔽3) | C3 | semidirect |
| 2. H(7,𝔽3) | C32 | semidirect |
| 3. C32×H(5,𝔽3) | C32 | semidirect |
| 4. C3×H(5,𝔽3) | C33 | semidirect |
| 5. C33×H(3,𝔽3) | C33 | semidirect |
| 6. C35 | C34 | semidirect |
| 7. H(5,𝔽3) | C34 | semidirect |
| 8. C32×H(3,𝔽3) | C34 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×H(7,𝔽3) | 3 |