The buttons are inside the LoggedInTemplate of the LoginView. The controls inside the template are not accessible directly.
Button Button1 = (Button)LoginView1.FindControl("Button1");
The buttons will only be available when a user is logged in, otherwise the AnonymousTemplate is used. So it's best to check if a button was retrieved by the previous code before doing anything to it to avoid a null reference error.
if(Button1 != null)
{
//Do your stuff
}