Consider the group $SU(4)$. We can apply the group action to a vector space. We also find that any non-trivial stabilizer (isotropy group) is conjugate to any of the following groups: $S(U(1)\times U(3))$, $S(U(2)^2)$, $S(U(1)^2 \times U(2))$, and $S(U(1)^4)$. My question is, how can you find these distinct groups, the groups that the stabilizer is conjugate to?
This example is taken from a paper by S. Sen and PJ Housten called "Symmetry breaking patterns and extended Morse theory."(https://iopscience.iop.org/article/10.1088/0305-4470/17/6/012/pdf) They provide a citation, which is "Symmetry defects and broken symmetry. Configurations Hidden Symmetry" by Louis Michel (https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.52.617 (sorry, no direct access to this paper)). As far as I can tell, the latter paper does not provide a step-by-step how-to on finding the groups that the stabilizer is conjugate to.