Subgroups — C₃⁴×C₄×C₅₃ #72195055

Order 17172.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⁴×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄×C₅₃	C₃	1	40	yes	maximal, direct product factor
3	C₂×C₃³×C₅₃	C₂×C₃	1	40	yes	—
4	C₃⁴×C₅₃	C₄	1	1	yes	characteristic, direct product factor, Hall
5	C₃²×C₄×C₅₃	C₃²	1	130	yes	direct product factor
6	C₃³×C₅₃	C₃×C₄	1	40	yes	direct product factor
7	C₂×C₃²×C₅₃	C₂×C₃²	1	130	yes	—
8	C₃×C₄×C₅₃	C₃³	1	40	yes	direct product factor
9	C₃²×C₅₃	C₃²×C₄	1	130	yes	direct product factor
10	C₃⁴×C₄	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅₃	C₂×C₃³	1	40	yes	—
12	C₄×C₅₃	C₃⁴	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₃⁴	C₂×C₅₃	1	1	yes	characteristic
14	C₃×C₅₃	C₃³×C₄	1	40	yes	direct product factor
15	C₃³×C₄	C₃×C₅₃	1	40	yes	direct product factor
16	C₂×C₅₃	C₂×C₃⁴	1	1	yes	characteristic
17	C₃⁴	C₄×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
18	C₂×C₃³	C₂×C₃×C₅₃	1	40	yes	—
19	C₅₃	C₃⁴×C₄	1	1	yes	characteristic, direct product factor, Sylow
20	C₃²×C₄	C₃²×C₅₃	1	130	yes	direct product factor
21	C₃³	C₃×C₄×C₅₃	1	40	yes	direct product factor
22	C₂×C₃²	C₂×C₃²×C₅₃	1	130	yes	—
23	C₃×C₄	C₃³×C₅₃	1	40	yes	direct product factor
24	C₃²	C₃²×C₄×C₅₃	1	130	yes	direct product factor
25	C₂×C₃	C₂×C₃³×C₅₃	1	40	yes	—
26	C₄	C₃⁴×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
27	C₃	C₃³×C₄×C₅₃	1	40	yes	direct product factor
28	C₂	C₂×C₃⁴×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃⁴×C₅₃
socle, prime radical	C₂×C₃⁴×C₅₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃³×C₄×C₅₃	C₃	direct (N × H)
4. C₃⁴×C₅₃	C₄	direct (N × H)
5. C₃²×C₄×C₅₃	C₃²	direct (N × H)
6. C₃³×C₅₃	C₃×C₄	direct (N × H)
8. C₃×C₄×C₅₃	C₃³	direct (N × H)
9. C₃²×C₅₃	C₃²×C₄	direct (N × H)
10. C₃⁴×C₄	C₅₃	direct (N × H)
12. C₄×C₅₃	C₃⁴	direct (N × H)
14. C₃×C₅₃	C₃³×C₄	direct (N × H)
15. C₃³×C₄	C₃×C₅₃	direct (N × H)
17. C₃⁴	C₄×C₅₃	direct (N × H)
19. C₅₃	C₃⁴×C₄	direct (N × H)
20. C₃²×C₄	C₃²×C₅₃	direct (N × H)
21. C₃³	C₃×C₄×C₅₃	direct (N × H)
23. C₃×C₄	C₃³×C₅₃	direct (N × H)
24. C₃²	C₃²×C₄×C₅₃	direct (N × H)
26. C₄	C₃⁴×C₅₃	direct (N × H)
27. C₃	C₃³×C₄×C₅₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃⁴×C₅₃	2
C₃³×C₄×C₅₃	3
C₃⁴×C₄	53

Subgroups — C34×C4×C53 #72195055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⁴×C₄×C₅₃ #72195055