Excellent link just given , not just for firms but to find individuals.
A solicitor does not have letters after the surname. They may write formally e.g "James Smith, solicitor "which is enough because it is a criminal offence to falsely pass oneself off as a solicitor. Apart from that, there's a subtlety within firms of solicitors, reflected in the Law Society lists in the link. Some are partners. These are usually the most senior members of the firm, they are the 'bosses' who run it. Associates are solicitors who are salaried employees of the firm. They may be quite senior, particularly in very big firms e.g there's a big firm in Cambridge,with over three hundred employees, where there are associate solicitors who have 25 years in practice .I think their most junior partner is 33 years in practice ( and charges accordingly!)