Subgroups — C₄×C₉×Dic₅ #10623571

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	1	yes	maximal, characteristic
2	C₂×C₉×Dic₅	C₂	1	2	yes	maximal
3	C₃×C₄×Dic₅	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
5	C₄×C₅×C₉	C₄	1	2	yes	complemented
6	C₉×Dic₅	C₄	1	4	yes	direct product factor
7	C₄²×C₉	—	5	1	no	maximal, Hall
8	C₂×C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic
9	C₂×C₃×Dic₅	C₂×C₃	1	2	yes	—
10	C₂×C₅×C₉	C₂×C₄	1	2	yes	—
11	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
12	C₄×Dic₅	C₉	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₄×C₉	—	5	2	no	—
14	C₂×C₄×C₉	D₅	1	1	yes	characteristic
15	C₂²×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
16	C₃×C₄×C₅	C₃×C₄	1	2	yes	—
17	C₃×Dic₅	C₃×C₄	1	4	yes	—
18	C₃×C₄²	—	5	1	no	—
19	C₅×C₉	C₄²	1	1	yes	characteristic, complemented, Hall
20	C₂×C₄×C₅	C₂×C₉	1	1	yes	characteristic
21	C₂×Dic₅	C₂×C₉	1	2	yes	—
22	C₂²×C₉	C₂×D₅	1	1	yes	characteristic
23	C₄×C₉	—	5	4	no	—
24	C₄×C₉	Dic₅	1	2	yes	direct product factor
25	C₂×C₃×C₅	C₂×C₃×C₄	1	2	yes	—
26	C₂×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄	—	5	2	no	—
28	C₂×C₃×C₄	C₃×D₅	1	1	yes	characteristic
29	C₂²×C₅	C₂²×C₉	1	1	yes	characteristic
30	C₄×C₅	C₄×C₉	1	2	yes	complemented
31	Dic₅	C₄×C₉	1	4	yes	direct product factor
32	C₂×C₉	C₄×D₅	1	2	yes	—
33	C₂×C₉	C₂×Dic₅	1	1	yes	characteristic
34	C₄²	—	5	1	no	Sylow
35	C₃×C₅	C₃×C₄²	1	1	yes	characteristic
36	C₂²×C₃	C₂×C₃×D₅	1	1	yes	characteristic
37	C₃×C₄	—	5	4	no	—
38	C₃×C₄	C₃×Dic₅	1	2	yes	—
39	C₂×C₅	C₂×C₄×C₉	1	2	yes	—
40	C₂×C₅	C₂×C₄×C₉	1	1	yes	characteristic
41	C₉	C₄×Dic₅	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₄	—	5	2	no	—
43	C₂×C₄	C₉×D₅	1	1	yes	characteristic
44	C₂×C₃	C₃×C₄×D₅	1	2	yes	—
45	C₂×C₃	C₂×C₃×Dic₅	1	1	yes	characteristic
46	C₅	C₄²×C₉	1	1	yes	characteristic, complemented, Sylow
47	C₂²	C₂×C₉×D₅	1	1	yes	characteristic
48	C₄	—	5	4	no	—
49	C₄	C₉×Dic₅	1	2	yes	direct product factor
50	C₃	C₃×C₄×Dic₅	1	1	yes	characteristic
51	C₂	C₄×C₉×D₅	1	2	yes	—
52	C₂	C₂×C₉×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₉	D₅
commutator, hypocenter	C₅	C₄²×C₉
Frattini	C₂²×C₃	C₂×C₃×D₅
Fitting	C₂×C₄×C₅×C₉	C₂
socle, prime radical, cosocle	C₂²×C₃×C₅	C₂²×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₉

Complements

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

N	H ≅ G/N	type
5. C₄×C₅×C₉	C₄	semidirect
6. C₉×Dic₅	C₄	direct (N × H)
12. C₄×Dic₅	C₉	direct (N × H)
19. C₅×C₉	C₄²	semidirect
24. C₄×C₉	Dic₅	direct (N × H)
30. C₄×C₅	C₄×C₉	semidirect
31. Dic₅	C₄×C₉	direct (N × H)
41. C₉	C₄×Dic₅	direct (N × H)
46. C₅	C₄²×C₉	semidirect
49. C₄	C₉×Dic₅	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
C₂×C₉×Dic₅	2
C₃×C₄×Dic₅	3
C₄²×C₉	5

Subgroups — C4×C9×Dic5 #10623571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₉×Dic₅ #10623571