Lutheranism has no characteristic polity, and that is itself informative. Luther had no theory of church government that required a particular shape, so the shape was supplied by whoever held power where the reform succeeded.
In Germany that meant the territorial prince. The peace of Augsburg in 1555 settled the wars of religion on the principle that the ruler determined the religion of his territory, and Lutheran churches accordingly became state churches with the sovereign as their effective head and a consistory of clergy and lawyers running them. In Scandinavia the reform was carried out by the crown and the existing episcopal structure was simply retained, which is why Sweden and Finland have Lutheran bishops who claim historic succession while German Lutherans generally do not and do not think it matters.
The consequence is that Lutheran churches today are governed by synods combining clergy and lay delegates, with bishops or presidents or superintendents at the head depending on the country, and that Lutherans regard the question as one on which the gospel does not legislate. What they do treat as binding is doctrine: a Lutheran body is defined by which of the Book of Concord documents it subscribes and how strictly.
That is where the family divides. The Lutheran World Federation gathers the large majority of Lutherans worldwide. Outside it sit confessional bodies, the Lutheran Church-Missouri Synod and the Wisconsin Synod in the United States, and their partners abroad, which hold to a stricter subscription, do not ordain women, and practise closed communion. The disagreement between them is not about polity at all. It is about how much of the confessions is still load-bearing.