Interpretable computer vision models can produce transparent predictions, where the features of an image are compared with prototypes from a training dataset and the similarity between them forms a basis for classification. Nevertheless these methods are computationally expensive to train, introduce additional complexity and may require domain knowledge to adapt hyper-parameters to a new dataset. Inspired by developments in object detection, segmentation and large-scale self-supervised foundation vision models, we introduce Component Features (ComFe), a novel explainable-by-design image classification approach using a transformer-decoder head and hierarchical mixture-modelling. With only global image labels and no segmentation or part annotations, ComFe can identify consistent image components, such as the head, body, wings and tail of a bird, and the image background, and determine which of these features are informative in making a prediction. We demonstrate that ComFe obtains higher accuracy compared to previous interpretable models across a range of fine-grained vision benchmarks, without the need to individually tune hyper-parameters for each dataset. We also show that ComFe outperforms a non-interpretable linear head across a range of datasets, including ImageNet, and improves performance on generalisation and robustness benchmarks.