Finite Group DatabasebetaFinite Group DatabasebetaOrder 26901.
60 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C72×C61 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C61×C7⋊C9 | — | 7 | 8 | no | maximal |
| 3 | C72×C61 | C9 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×C7×C61 | ΓL(1,𝔽8) | 1 | 8 | yes | — |
| 5 | C9×C61 | — | 49 | 1 | no | Hall |
| 6 | #89204 | C61 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 7 | C7×C61 | C7⋊C9 | 1 | 8 | yes | complemented |
| 8 | C3×C61 | #3861 | 1 | 1 | yes | characteristic |
| 9 | C3×C72 | C3×C61 | 1 | 1 | yes | characteristic |
| 10 | C7⋊C9 | — | 7 | 8 | no | — |
| 11 | C61 | #89204 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 12 | C72 | C9×C61 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C3×C7 | ΓL(1,𝔽8)×C61 | 1 | 8 | yes | — |
| 14 | C9 | — | 49 | 1 | no | Sylow |
| 15 | C7 | C61×C7⋊C9 | 1 | 8 | yes | complemented |
| 16 | C3 | C61×#3861 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C61 | #3861 |
| commutator, hypocenter | C72 | C9×C61 |
| Frattini | C3 | C61×#3861 |
| Fitting, socle, prime radical | C3×C72×C61 | C3 |
| cosocle | C3×C72 | C3×C61 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C3×C61 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C72×C61 | C9 | semidirect |
| 6. #89204 | C61 | direct (N × H) |
| 7. C7×C61 | C7⋊C9 | semidirect |
| 11. C61 | #89204 | direct (N × H) |
| 12. C72 | C9×C61 | semidirect |
| 15. C7 | C61×C7⋊C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C72×C61 | 3 |
| C61×C7⋊C9 | 7 |
| #89204 | 61 |