The principal power of the Church of England in the House of Lords is vested in the Lords Spiritual - Church of England bishops who are granted seats in the House. Since the Bishopric of Manchester Act 1847, the number of Lords Spiritual has been set at 26. Five of these seats are granted automatically - the Archbishops of Canterbury and York and the Bishops of London, Durham and Winchester - while the others are granted to the most senior bishop when a vacancy arises. This was slightly altered a few years ago when the Lords Spiritual (Women) Act 2015 gave women bishops precedence over male bishops until 2025.
These Lords have exactly the same rights and privileges as Lords Temporal, including voting rights. In addition, one Lord Spiritual is always required to lead prayers at the start of each sitting, so in general, at least one bishop tends to be present during the proceedings of any given day. They are not affiliated with any party, although they do not sit with the crossbenchers either, instead having their own benches within the Chamber - on the government side, closest to the throne.
To put this role in perspective, the 26 Lords Spiritual make up about 3% of the 772 seats in the Lords, and attendance relative to other members is low. A Parliament research briefing published in 2017 provides the following statistics:
The same briefing also examined Bishops' voting attendance:
The average turnout of the Bishops in all divisions since 1999–2000 is
3.91 percent. This figure includes both whipped and un-whipped divisions. Since 2010–12, the Bishops collectively cast, with the
exception of the 2016–17 session, more votes against the Government of
the day than for it. In 2016–17, the Bishops were divided equally,
casting 53 votes for the Government and 53 against over the course of
The Church of England also has the de facto power to control legislation concerning itself. This was granted by the Church of England Assembly (Powers) Act 1919, which allowed the Church Assembly (subsequently transferred to the General Synod in 1969) to propose "Measures" to Parliament. These are scrutinised by the Ecclesiastical Committee, and if passed by both Houses of Parliament, have the same force as primary legislation. I call it a de facto power, as Parliament retains the power to introduce legislation regarding the Church, but as a constitutional convention, does not do so without its consent.
Finally, the Supreme Governor of the Church of England, more commonly known as the Queen, confers peerages, thereby controlling the membership of the House. In practice, this is performed in accordance with the advice of the Prime Minister.
The powers of the Church of England in the Lords, and Parliament as a whole, although fairly slight, are according to Humanists UK "extremely unusual and anti-democratic". In particular, they note:
The only two sovereign states in the world to award clerics of the
established religion votes in their legislatures are the UK and the
Islamic Republic of Iran (a totalitarian theocracy).