verbs - Clauses in Sentences

I understand that a clause contains (in order) a subject, verb and object, like below:

He let his daughter.

"He" is the subject, "let" is the verb and "his daughter" is the object. But what about the sentence below? I need some help identifying the subjects and objects.

He let his daughter listen to the music.

Is this sentence two clauses? He let his daughter and his daughter listen to the music? I enquired about this and I was told that the second cannot be a clause because the verb is not finite. Is this true?

My stance at the moment is:

Part 1

• Subject: He


• Verb: let


• Object: his daughter

Part 2

• Subject: his daughter


• Verb: listen


• Object: the music

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EDIT:
I think I finally have a grasp of the concept after reading the fantastic answers to this question along with some more thinking:

He let his daughter [listen to the music].

In bold I have the subject, italics the verb, bold italics the object and in square brackets I have the phrase.

Could you please confirm if this, and my understanding, is correct?

Answer

Traditionally, a clause is indeed a finite verb and all its dependencies. The subject of the sentence is he, the (direct) object his daughter. The verb let is special in that it often has an object and an infinitive as a tertiary complement (third thingy that strongly depends on it, besides subject and object). You could analyse the infinitive after let as an object complement, because it is very much related to the object, his daughter.

An infinitive is externally much like a noun (it can be governed by a verb); internally it is a verb (it can have arguments that verbs can normally have). It has an argument that depends on it: to the music. One might call the latter an adverbial constituent.

Other linguists use a different definition of clause: they define it as any verb and its dependencies. In that case, the infinitive listen and its argument to the music form a subordinate clause together. It doesn't matter which definition you choose, as long as you are consistent.

But, even according to that definition, I wouldn't call his daughter a subject, because subjects are normally marked as such:

She listened to the music. (she = the subject form of the pronoun)

He let her listen to the music. (her = the non-subject form of the pronoun)

It does fulfil the semantic role that the subject normally has with the verb listen ("experiencer"), whenever there is such a subject; but subject is a syntactic category, not a semantic role, so that is irrelevant. If we called her the subject of listen, then we would have to do the same in this sentence:

She stalked me. (she = subject)

I was stalked by her.

Her expresses the same semantic role as she ("agent"), as is normal in passive sentences with by; but we never call her the subject of the verb in such cases, because semantic role is not what the term subject is all about.