When you go to certain churches, you might notice that there are no leadership opportunities for women; some churches, usually the more "liberal" and "progressive" kinds, will install female pastors. Some churches allow for some middle ground and will allow women to teach Sunday schools, small groups, or to lead in worship; sometimes, they'll be deaconesses. Still, some denominations will allow for female prophets and apostles. The Seventh Day Adventists are otherwise conservative Christians who are famous for their so-called prophetess Ellen G. White.
Commentary on this I’ve found is summarized best in the following statement. Later, I’ll also pull out one of the footnotes (#6) to reiterate what I'm saying.
The very phrase which Paul uses is found in a number of proximate Jewish writings, and its meaning is not in doubt. It is a clear reference to a principle drawn from the Biblical text (not a direct quote), either to the Pentateuch(1) or some other part of the Old Testament.(2)
Several commentators note that this appeal to ‘the Law’ is a standard form of argument in Paul’s writings.(3) (4) (5) (6)
Of twelve standard modern Bible commentaries,(7) almost all of them understand this is as a reference to the Law of Moses or a general principle from Genesis or the Old Testament; of these commentaries only one egalitarian commentary disagrees.(8)
What this excerpt shows is that there simply is no direct command; it’s a “principle” or tradition based on inferences found throughout the scriptures. That’s eisegesis, not exegesis.
In John Gill's commentary on verse 34 of 1 Corinthians, we find that even he refers back to the Genesis fall, saying the following:
In Acts, evangelist Philip had daughters, and they were said to be prophetesses. And it’s obvious that in the meeting in Acts 2, men and women were prophesying in languages by miracle in order to witness.
 ‘Fourth, “as the law says” does not refer to secular law restricting women’s actions in the public arena but to the OT law.34 Paul’s presumed impatience with the law is exaggerated. He appeals to it in the context in 14:21 and also in 7:19 and 9:8–10 (cf. Rom. 3:19; 7:7). The problem is that he does not cite a text from the law, and no OT passage instructs women to be silent. Perhaps he refers to a general assumption that the law calls for the wife’s submission to her husband.’, Garland (egalitarian), ‘1 Corinthians’, Baker Exegetical Commentary on the New Testament, p. 672 (2003).
It's very obvious that Paul isn’t citing a specific law from Torah but is rather building a case based on traditional inferences based on a few verses, namely the Genesis order of creation (male first, female second). It’s my view, based on other instances where women are in positions of leadership and, or, public speaking, that Paul has a rather narrow and unnecessary restriction on the role of women. It's obvious to me that Paul is piecing together a standard that could be argued contradicts Jesus' clear intention of making a Kingdom of equals.
This is an on going study and is subject to revisions according to further studies.