Why is Dodge usually left out of the Ford and Chevy rivalry?

Sometimes it seems like Chrysler is rather irrelevant and overshadowed by Ford and GM

It isn't in the Pick Up Truck market.

But decades the Ford Vs Chevy Rivalry goes back decades.

They do not care. They are selling Jeeps like hotcakes right now. Dodge hit it big in the SUV era.

Funny. All I see are Ram trucks in my hood.

Maybe because it is overshowed in the advertising stakes
the dodge brand is not terribly reliable and that could be one of the reasons

Back in the 60's Dodges and Plymouths were among the hottest and best made cars out there. My buddy's '68 Road Runner left my '57 Chevy in the dust.

Today's Mopars don't enjoy the same respect, as their quality has been poor. Those Hellcats and Demons will still leave most cars in the dust… But they just aren't reliable.

Mopar has not kept up since the 60s. Had Daimler not bought Mopar they would be gone today. New offerings from Mopar since the 60s have been substandard. Daimler put modern engines in nearly every Mopar product bringing the company to the modern age of automobiles. I'm certain that if we could compare the big 3's investment in engineers and designers we would find mopar coming up way short.

Now their trucks have finally improved so they rebranded them as not being Dodge. Weird choice.

Been driving-owning Chrysler products of varying forms since 1974, and will keep doing so. Worked at Chevy for years and have had to work on Fords too. None of them things are superior.

Years ago the Chrysler family of cars abandoned the performance group of cars. That happened back in the 70's during the gas crunch. Chrysler sunk a lot of money into the "K" car and the "Minivan" market. They also cut back on research and engineering for the drive train group. Engines and transmission development was all but surrendered to Mitsubishi engineering. Mitsubishi might be a good name for some things, but what they provided Chrysler was inferior in all respects.
This left Dodge short in performance.
Then the Chrysler corporation bought out Jeep. Here again they turned to Mitsubishi for engines and transmissions. Not that Mitsubishi makes bad transmissions (Ford used the five speed manuals by them in the Bronco II's) but they just didn't measure up to Jeep four wheel drive units.
Because Chrysler was counting on the Jeep name for utility vehicles, the truck division suffered. If for no other reason then the Dodge staff that should have been working on Dodge products was not depending on Jeep. Jeep had not invested in new engineering for some time. They had outsourced many parts to GM and Chrysler.
So Dodge gets left out because Lee Iacocca did away with the engineering staff the corporation needed to be self sufficient.

Hi so ask yourself why simple answer gives a lot of bother.

Dodge didn't lay off tens of thousands of American workers right before Christmas and is still laying off today. It doesn't seem fair nor right that Americans invested into GM but GM is not willing to do the same for Americans. Now GM says that they are too big to fail. I say let's prove them wrong and stop buying GM products. America also invested in Chrysler but Chrysler is investing back into America by moving plants from Mexico to the USA and opening new plants in the USA, creating jobs. I would boycott GM until Mary T. Barra resigns and GM starts doing right by America or next time there will be no more bailouts.