"As per Hubble constant" galaxies don't even have to move.
What the constant tells us is the rate at which space is expanding. In between each bit of space, more space is being added. All the time. Everywhere. Including inside things (even atoms).
Locally, the rate is very small (over a human lifetime, a 1-metre space would grow by less than the size of an atom).
Since this happens everywhere, the effect does add up over long distances. The rate is given for a distance of a million parsecs (Mpc) which is equal to approximately 3,261,600 light-years (the size of the Local Group of galaxies, roughly). Every second, over that distance, space expands by 70 km.
Hubble rate = 70 km/ s*Mpc (the real rate is slightly different, but 70 is a good approximation for rough calculations -- all calculations here are approximations)
So even without anything moving, if you were to take the distance between two objects located at 10 million parsecs from each other, then you take the distance between the same two objects 1 second later, the distance would have increased by 700 km.
They would appear to "move away from each other" at the speed of 700 km/s, even though neither of them is moving relative to its own local space.
Over a distance of 4250 Mpc (13.8 billion light-years) it just so happens that the rate of "new space" being added over that distance is equal to the speed of light. If a photon from that distance was trying to come our way (so that we could "see" it), it would never get here: the distance between that photon and your eyes would remain the same. Every second, the photon would move this way by 300,000 km, while every second, 300,000 km of new space would be added.
Because light travels at... the speed of light (one year to cover one light-year), the furthest back we can ever observe anything from the universe is 13.8 billion years ago. That is why we say the Observable Universe is 13.8 billion years old. The whole universe? We don't know (other than it must be at least 13.8 billion years old).
We already know (from other observations) that the whole universe is bigger than whatever we can observe. Therefore, there are things that are located beyond our 13.8 billion light-year horizon. The amount of new space being added between us and any such object must be greater than the speed of light (therefore, we will NEVER see those things). You may have the impression that these things are moving away from us faster than the speed of light, but that would be a wrong interpretation. These things might very well be stopped (relative to their own local space); it is only the amount of new space that adds us to make it look like a speed.
Thus, no object needs to go at or above the speed of light. The "speed of cosmological recession" is an illusion cause by the expansion of space between the two objects.