Shalom Rabbi, a Jew who is not religious and transgresses Shabbat among other Hilchos, can be called to Alyah Torah or to a minyan ?
Thank you for your question.
This is a very sensitive question, and it might depend on the level of how irreligious he is. If he wouldn’t be mechalel Shabbos in from of an important Rov then he wouldn’t be considered mechalel Shabbos in public. However, the general rule is that unfortunately if he is mechalel Shabbos in public then he cannot be counted in to a minyan and cannot get an aliya.
M:B 55-46, 385-4, B’tzel Hachochma 4-5. See Shut Chacham Tzvi 38 that giving him an aliya is even more sensitive because it is an embarrassment to the torah that someone who doesn’t care about it and openly transgresses it, to be given an aliya.