I don't think all radicals should be suppressed, because the word radical can be used in so many contexts, but to save a semantic argument, I will say yes. If there are radicals, especially violent ones, they should not be allowed to operate and should be suppressed immediately. It's dangerous.
Yes, the government should suppress radical ideologies that could threaten the American people. There should be plenty of evidence to support such a conclusion prior to suppressing the ideologies. Radical anything is bad. Most of it tends to sort itself out within given communities, but sometimes a more powerful hand need to step in and take over for the safety of the people.
No, radical ideologies should not be suppressed by the government. Under the first amendment, we have the right to free speech, so we can say whatever we want, and also believe whatever we want. If someone believes in a radical idea, then there is nothing wrong with it at all.
This would be the same as supressing thing like White Supremacy, or on the other hand, Black Power. It's not part of our democracy to do that. They are supressed enough as it is because when we send delegates to the house of representatives, they will not have many delegates.
No, the government should not suppress radical ideologies because it will cause those ideologies to spread and multiply. The more a government tries to stop radical ideas, the more interesting those radical ideas become. Suppressing ideologies causes movements to grow and gain momentum, as is the case with Pussy Riot. The intervention of the government caused Pussy Riot's fame to soar.