COVID-19 is a contagious infection that has severe effects on the global economy and our daily life. Accurate diagnosis of COVID-19 is of importance for consultants, patients, and radiologists. In this study, we use the deep learning network AlexNet as the backbone, and enhance it with the following two aspects: 1) adding batch normalization to help accelerate the training, reducing the internal covariance shift; 2) replacing the fully connected layer in AlexNet with three classifiers: SNN, ELM, and RVFL. Therefore, we have three novel models from the deep COVID network (DC-Net) framework, which are named DC-Net-S, DC-Net-E, and DC-Net-R, respectively. After comparison, we find the proposed DC-Net-R achieves an average accuracy of 90.91% on a private dataset (available upon email request) comprising of 296 images while the specificity reaches 96.13%, and has the best performance among all three proposed classifiers. In addition, we show that our DC-Net-R also performs much better than other existing algorithms in the literature. Supplementary Information The online version contains supplementary material available at 10.1007/s11390-020-0679-8.