The Greek word that is translated "groaned" expresses that Jesus was deeply moved, but not necessarily with sorrow. It was more a groan of anger at Satan who had caused all the grief Jesus was seeing around Him. Jesus came to "destroy him that had the power of death, that is, the devil" (Hebrews 2:14), and He was grieved to see the pain that His enemy had inflicted on those He loved.

This is the type of groaning that the Holy Spirit does for us. It is not just the Holy Spirit sympathizing with us, but it is the Holy Spirit doing battle for us when we don't know how to pray. In this case, it was the Holy Spirit doing battle through Jesus against death and the doubt of those present, that would have kept Lazarus in the grave.

Everyone who has the indwelling of the Holy Spirit has or will have this happen to them. This groaning of the Holy Spirit is not just grief but a groan of anger and resistance against Satan's devices in our lives. 

Many times Christians don't discern this because they think it is just them grieved with their situation. But this is the Holy Spirit desiring to get into intercession with us against our problems.

Although the groaning is unutterable, we can discern it, and many times people react with audible groans or other outward acts. This has led to religious doctrines and traditions that are unscriptural and offensive to many people. There is nothing wrong with us reacting to the inner working of the Holy Spirit as long as we don't confuse our reactions with the Holy Spirit's actions. This intercession cannot be uttered. Any counterfeits that religion may have produced only shows that there has to be a genuine. 

The genuine groaning in the Spirit is priceless.