  We classify connected \'etale algebras $A$'s in multiplicity-free modular
fusion categories (MFCs) $\mathcal{B}$'s at rank six, namely
$\text{rank}(\mathcal{B})=6$. There are eight MFCs in total and the result
indicates that only $so(5)_2$ has nontrivial connected \'etale algebra. We
briefly mention anyon condensation as it is used to determine the category of
right $A$-modules in $so(5)_2$. Finally, we discuss physical applications,
specifically proving spontaneous $\mathcal{B}$-symmetry breaking (SSB) of these
MFCs. The discussion also includes predicting ground state degeneracies and SSB
in massive renormalization group flows from two non-unitary minimal models.
