Subgroups — C₄²×C₅×C₉ #10623612

Order 720.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₉	C₂	1	3	yes	maximal
2	C₃×C₄²×C₅	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
4	C₄×C₅×C₉	C₄	1	6	yes	direct product factor
5	C₄²×C₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₄×C₅	C₂×C₃	1	3	yes	—
7	C₂×C₅×C₉	C₂×C₄	1	3	yes	—
8	C₄²×C₅	C₉	1	1	yes	characteristic, direct product factor, Hall
9	C₂×C₄×C₉	C₂×C₅	1	3	yes	—
10	C₂²×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
11	C₃×C₄×C₅	C₃×C₄	1	6	yes	—
12	C₃×C₄²	C₃×C₅	1	1	yes	characteristic
13	C₅×C₉	C₄²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₄×C₅	C₂×C₉	1	3	yes	—
15	C₂²×C₉	C₂²×C₅	1	1	yes	characteristic
16	C₄×C₉	C₄×C₅	1	6	yes	direct product factor
17	C₂×C₃×C₅	C₂×C₃×C₄	1	3	yes	—
18	C₂×C₃×C₄	C₂×C₃×C₅	1	3	yes	—
19	C₂²×C₅	C₂²×C₉	1	1	yes	characteristic
20	C₄×C₅	C₄×C₉	1	6	yes	direct product factor
21	C₂×C₉	C₂×C₄×C₅	1	3	yes	—
22	C₄²	C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
23	C₃×C₅	C₃×C₄²	1	1	yes	characteristic
24	C₂²×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
25	C₃×C₄	C₃×C₄×C₅	1	6	yes	—
26	C₂×C₅	C₂×C₄×C₉	1	3	yes	—
27	C₉	C₄²×C₅	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×C₄	C₂×C₅×C₉	1	3	yes	—
29	C₂×C₃	C₂×C₃×C₄×C₅	1	3	yes	—
30	C₅	C₄²×C₉	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²	C₂²×C₅×C₉	1	1	yes	characteristic
32	C₄	C₄×C₅×C₉	1	6	yes	direct product factor
33	C₃	C₃×C₄²×C₅	1	1	yes	characteristic
34	C₂	C₂×C₄×C₅×C₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₃	C₂²×C₃×C₅
socle, prime radical	C₂²×C₃×C₅	C₂²×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₅×C₉	C₄	direct (N × H)
5. C₄²×C₉	C₅	direct (N × H)
8. C₄²×C₅	C₉	direct (N × H)
13. C₅×C₉	C₄²	direct (N × H)
16. C₄×C₉	C₄×C₅	direct (N × H)
20. C₄×C₅	C₄×C₉	direct (N × H)
22. C₄²	C₅×C₉	direct (N × H)
27. C₉	C₄²×C₅	direct (N × H)
30. C₅	C₄²×C₉	direct (N × H)
32. C₄	C₄×C₅×C₉	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×C₄×C₅×C₉	2
C₃×C₄²×C₅	3
C₄²×C₉	5

Subgroups — C42×C5×C9 #10623612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₅×C₉ #10623612