This paper proposes a quasi-maximum likelihood (QML) estimator of the break point for large-dimensional factor models with multiple structural breaks in the factor loading matrix. We show that the QML estimator is consistent for the true break point when the covariance matrix of the pre- or post-break factor loading (or both) is singular. Consistency here means that the deviation of the estimated break date from the true break date $k_0$ converges to zero as the sample size grows. This is a much stronger result than the break fraction $\hat k/T$ being $T$-consistent (super-consistent) for $k_0/T$. Also, we consider the likelihood ratio (LR) test for the estimated factors. Simulation results confirm the theoretical properties of our estimator, and it significantly outperforms existing estimators for change points in factor models.