Finite Group DatabasebetaFinite Group DatabasebetaOrder 32428.
84 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C112×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C112×C67 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | Dic11×C67 | — | 11 | 12 | no | maximal |
| 4 | C2×C11×C67 | D11 | 1 | 12 | yes | — |
| 5 | C11×C67 | Dic11 | 1 | 12 | yes | complemented |
| 6 | C11⋊Dic11 | C67 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 7 | C4×C67 | — | 121 | 1 | no | Hall |
| 8 | C2×C112 | C2×C67 | 1 | 1 | yes | characteristic |
| 9 | C2×C67 | C11⋊D11 | 1 | 1 | yes | characteristic |
| 10 | C112 | C4×C67 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C67 | C11⋊Dic11 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 12 | Dic11 | — | 11 | 12 | no | — |
| 13 | C2×C11 | D11×C67 | 1 | 12 | yes | — |
| 14 | C11 | Dic11×C67 | 1 | 12 | yes | complemented |
| 15 | C4 | — | 121 | 1 | no | Sylow |
| 16 | C2 | C67×C11⋊D11 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C67 | C11⋊D11 |
| commutator, hypocenter | C112 | C4×C67 |
| Frattini | C2 | C67×C11⋊D11 |
| Fitting, socle, prime radical | C2×C112×C67 | C2 |
| cosocle | C2×C112 | C2×C67 |
| LCS | G ▷ C112 | |
| UCS | C1 ◁ C2×C67 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C112×C67 | C4 | semidirect |
| 5. C11×C67 | Dic11 | semidirect |
| 6. C11⋊Dic11 | C67 | direct (N × H) |
| 10. C112 | C4×C67 | semidirect |
| 11. C67 | C11⋊Dic11 | direct (N × H) |
| 14. C11 | Dic11×C67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C112×C67 | 2 |
| Dic11×C67 | 11 |
| C11⋊Dic11 | 67 |