This depends on what you mean by 'army'. Do you mean the total size of the armed land forces, including militia and garrisons, or do you mean how large a field army for a single battle could be?

If the latter, we have many examples from battles of the era - note that this is a number more limited by logistics and infrastructure rather than the ability of the nation to raise and equip troops. There were usually other fronts, garrisons, militias and reserves that could not be brought to the front due to problems with transport and logistics. This is especially true for the Crimean War, where both sides were at the very end of an extremely long supply line. As railroads were developed, this rapidly changed.,

The Austrian army sent to crush the Hungarian revolt 1848 numbered some 70 000.

The joint British-French-Ottoman force at Alma 1854 numbered some 61 000 and faced about 37 000 Russians.

At the Battle of Solferino 1859, about 129 000 Austrians faced a Sardinian and French force of about 138 000.

At Königgrätz 1866, 215 000 Austrians faced 221 000 Prussians. At the same time, the Austrians had about 130 000 men facing the Italians in northern Italy.

I think you might want to narrow down which part of the 1800s you're referring to, considering the number of major European and North American conflicts through the century and the huge societal and military changes.