On a conformal net $\mathcal{A}$, one can consider collections of unital completely positive maps on each local algebra $\mathcal{A}(I)$, subject to natural compatibility, vacuum preserving and conformal covariance conditions. We call \emph{quantum operations} on $\mathcal{A}$ the subset of extreme such maps. The usual automorphisms of $\mathcal{A}$ (the vacuum preserving invertible unital *-algebra morphisms) are examples of quantum operations, and we show that the fixed point subnet of $\mathcal{A}$ under all quantum operations is the Virasoro net generated by the stress-energy tensor of $\mathcal{A}$. Furthermore, we show that every irreducible conformal subnet $\mathcal{B}\subset\mathcal{A}$ is the fixed points under a subset of quantum operations. When $\mathcal{B}\subset\mathcal{A}$ is discrete (or with finite Jones index), we show that the set of quantum operations on $\mathcal{A}$ that leave $\mathcal{B}$ elementwise fixed has naturally the structure of a compact (or finite) hypergroup, thus extending some results of [Bis17]. Under the same assumptions, we provide a Galois correspondence between intermediate conformal nets and closed subhypergroups. In particular, we show that intermediate conformal nets are in one-to-one correspondence with intermediate subfactors, extending a result of Longo in the finite index/completely rational conformal net setting [Lon03].