Properties of a matrix that preserves diagonality

Question

Suppose $\Lambda = \text{diag}(\lambda_1,...,\lambda_p)$ is a $p \times p$ full rank real diagonal matrix that has a set of $p'(\leq p)$ distinct eigenvalues that will be denoted by $l_1,...,l_{p'}$, and the $i$th eigenvalue has a $d_i$ multiplicity $$\sum_i d_i=p$$The values on the diagonal are ordered such that $$l_1=\lambda_1=...=\lambda_{d_1}$$ $$...$$ $$l_{p'}=\lambda_{p-d_{p'}+1}=...=\lambda_p$$I would like to characterize the set of matrices $T$ that solve the following equation $$T\Lambda T^H=\Lambda $$It's easy to show that any block diagonal matrix that's composed of blocks of sizes $d_1,...,d_{p'}$ such that each block on the diagonal is unitary, would be a valid $T$. The question is whether that's a necessary condition, or there could exist some matrix of a different structure that satisfy the equation?