Finite Group DatabasebetaFinite Group DatabasebetaOrder 32193.
60 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C72×C73 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | #17987159 | — | 7 | 8 | no | maximal |
| 3 | C72×C73 | C9 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×C7×C73 | ΓL(1,𝔽8) | 1 | 8 | yes | — |
| 5 | #10620321 | — | 49 | 1 | no | Hall |
| 6 | C7×C73 | C7⋊C9 | 1 | 8 | yes | complemented |
| 7 | #89204 | — | 73 | 1 | no | maximal, Hall |
| 8 | C3×C73 | #3861 | 1 | 1 | yes | characteristic |
| 9 | C3×C72 | C73⋊C3 | 1 | 1 | yes | characteristic |
| 10 | C73 | #89204 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C7⋊C9 | — | 511 | 8 | no | — |
| 12 | C72 | #10620321 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C3×C7 | #13095275 | 1 | 8 | yes | — |
| 14 | C9 | — | 3577 | 1 | no | Sylow |
| 15 | C7 | #17987159 | 1 | 8 | yes | complemented |
| 16 | C3 | #24924363 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | #24924363 |
| commutator, hypocenter | C72×C73 | C9 |
| Fitting, socle, prime radical, cosocle | C3×C72×C73 | C3 |
| LCS | G ▷ C72×C73 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C72×C73 | C9 | semidirect |
| 6. C7×C73 | C7⋊C9 | semidirect |
| 10. C73 | #89204 | semidirect |
| 12. C72 | #10620321 | semidirect |
| 15. C7 | #17987159 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C72×C73 | 3 |
| #17987159 | 7 |
| #89204 | 73 |