Subgroups — C₃²×C₄×C₅×C₆₁ #24926721

Order 10980.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×C₆₁	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×C₆₁	C₄	1	1	yes	characteristic, direct product factor, Hall
4	C₃²×C₄×C₆₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×C₆₁	C₂×C₃	1	4	yes	—
6	C₄×C₅×C₆₁	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₃²×C₆₁	C₂×C₅	1	1	yes	characteristic
8	C₃×C₅×C₆₁	C₃×C₄	1	4	yes	direct product factor
9	C₃×C₄×C₆₁	C₃×C₅	1	4	yes	direct product factor
10	C₂×C₅×C₆₁	C₂×C₃²	1	1	yes	characteristic
11	C₃²×C₆₁	C₄×C₅	1	1	yes	characteristic, direct product factor, Hall
12	C₂×C₃×C₆₁	C₂×C₃×C₅	1	4	yes	—
13	C₅×C₆₁	C₃²×C₄	1	1	yes	characteristic, direct product factor, Hall
14	C₄×C₆₁	C₃²×C₅	1	1	yes	characteristic, direct product factor, Hall
15	C₃×C₆₁	C₃×C₄×C₅	1	4	yes	direct product factor
16	C₃²×C₄×C₅	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₆₁	C₂×C₃²×C₅	1	1	yes	characteristic
18	C₂×C₃²×C₅	C₂×C₆₁	1	1	yes	characteristic
19	C₆₁	C₃²×C₄×C₅	1	1	yes	characteristic, direct product factor, Sylow
20	C₃×C₄×C₅	C₃×C₆₁	1	4	yes	direct product factor
21	C₃²×C₅	C₄×C₆₁	1	1	yes	characteristic, direct product factor, Hall
22	C₃²×C₄	C₅×C₆₁	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₃×C₅	C₂×C₃×C₆₁	1	4	yes	—
24	C₄×C₅	C₃²×C₆₁	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₃²	C₂×C₅×C₆₁	1	1	yes	characteristic
26	C₃×C₅	C₃×C₄×C₆₁	1	4	yes	direct product factor
27	C₃×C₄	C₃×C₅×C₆₁	1	4	yes	direct product factor
28	C₂×C₅	C₂×C₃²×C₆₁	1	1	yes	characteristic
29	C₃²	C₄×C₅×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₃	C₂×C₃×C₅×C₆₁	1	4	yes	—
31	C₅	C₃²×C₄×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
32	C₄	C₃²×C₅×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	C₃×C₄×C₅×C₆₁	1	4	yes	direct product factor
34	C₂	C₂×C₃²×C₅×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃²×C₅×C₆₁
socle, prime radical	C₂×C₃²×C₅×C₆₁	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×C₆₁	C₃	direct (N × H)
3. C₃²×C₅×C₆₁	C₄	direct (N × H)
4. C₃²×C₄×C₆₁	C₅	direct (N × H)
6. C₄×C₅×C₆₁	C₃²	direct (N × H)
8. C₃×C₅×C₆₁	C₃×C₄	direct (N × H)
9. C₃×C₄×C₆₁	C₃×C₅	direct (N × H)
11. C₃²×C₆₁	C₄×C₅	direct (N × H)
13. C₅×C₆₁	C₃²×C₄	direct (N × H)
14. C₄×C₆₁	C₃²×C₅	direct (N × H)
15. C₃×C₆₁	C₃×C₄×C₅	direct (N × H)
16. C₃²×C₄×C₅	C₆₁	direct (N × H)
19. C₆₁	C₃²×C₄×C₅	direct (N × H)
20. C₃×C₄×C₅	C₃×C₆₁	direct (N × H)
21. C₃²×C₅	C₄×C₆₁	direct (N × H)
22. C₃²×C₄	C₅×C₆₁	direct (N × H)
24. C₄×C₅	C₃²×C₆₁	direct (N × H)
26. C₃×C₅	C₃×C₄×C₆₁	direct (N × H)
27. C₃×C₄	C₃×C₅×C₆₁	direct (N × H)
29. C₃²	C₄×C₅×C₆₁	direct (N × H)
31. C₅	C₃²×C₄×C₆₁	direct (N × H)
32. C₄	C₃²×C₅×C₆₁	direct (N × H)
33. C₃	C₃×C₄×C₅×C₆₁	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×C₃²×C₅×C₆₁	2
C₃×C₄×C₅×C₆₁	3
C₃²×C₄×C₆₁	5
C₃²×C₄×C₅	61

Subgroups — C32×C4×C5×C61 #24926721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₅×C₆₁ #24926721