We explore unsupervised machine learning for galaxy morphology analyses using a combination of feature extraction with a vector-quantized variational autoencoder (VQ-VAE) and hierarchical clustering (HC). We propose a new methodology that includes: (1) consideration of the clustering performance simultaneously when learning features from images; (2) allowing for various distance thresholds within the HC algorithm; (3) using the galaxy orientation to determine the number of clusters. This set-up provides 27 clusters created with this unsupervised learning that we show are well separated based on galaxy shape and structure (e.g. Sérsic index, concentration, asymmetry, Gini coefficient). These resulting clusters also correlate well with physical properties such as the colour-magnitude diagram, and span the range of scaling relations such as mass versus size amongst the different machine-defined clusters. When we merge these multiple clusters into two large preliminary clusters to provide a binary classification, an accuracy of $\sim 87{{\ \rm per\ cent}}$ is reached using an imbalanced data set, matching real galaxy distributions, which includes 22.7 per cent early-type galaxies and 77.3 per cent late-type galaxies. Comparing the given clusters with classic Hubble types (ellipticals, lenticulars, early spirals, late spirals, and irregulars), we show that there is an intrinsic vagueness in visual classification systems, in particular galaxies with transitional features such as lenticulars and early spirals. Based on this, the main result in this work is not how well our unsupervised method matches visual classifications and physical properties, but that the method provides an independent classification that may be more physically meaningful than any visually based ones.