Alternatives to Logistic Regression

Question

I have age, gender, height, weight and some other similar parameters of 15000 subjects. I also have one column showing if they had a medical condition (present in about 20% subjects). I now want to analyze the factors which are most contributing to the medical condition. The usual test for this is Logistic regression analysis. However, that assumes that the relationships are linear. What other machine learning tests can I apply for this purpose with this kind of data? Thanks for your insight.