Subgroups — C₄×C₅×Dic₉ #10623587

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×C₅×Dic₉	C₂	1	2	yes	maximal
3	C₄×C₅×Dic₃	—	3	1	no	maximal
4	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
5	C₄×C₅×C₉	C₄	1	2	yes	complemented
6	C₅×Dic₉	C₄	1	4	yes	direct product factor
7	C₄×Dic₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₄×C₅	D₃	1	1	yes	characteristic
9	C₂×C₅×Dic₃	—	3	2	no	—
10	C₂×C₅×C₉	C₂×C₄	1	2	yes	—
11	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₅	—	9	1	no	Hall
13	C₂×C₄×C₉	C₂×C₅	1	1	yes	characteristic
14	C₂×Dic₉	C₂×C₅	1	2	yes	—
15	C₂²×C₃×C₅	C₂×D₃	1	1	yes	characteristic
16	C₃×C₄×C₅	Dic₃	1	2	yes	—
17	C₅×Dic₃	—	3	4	no	—
18	C₄×Dic₃	—	3	1	no	—
19	C₅×C₉	C₄²	1	1	yes	characteristic, complemented, Hall
20	C₂×C₄×C₅	—	9	2	no	—
21	C₂×C₄×C₅	D₉	1	1	yes	characteristic
22	C₂²×C₉	C₂²×C₅	1	1	yes	characteristic
23	C₄×C₉	C₄×C₅	1	2	yes	complemented
24	Dic₉	C₄×C₅	1	4	yes	direct product factor
25	C₂×C₃×C₅	C₄×D₃	1	2	yes	—
26	C₂×C₃×C₅	C₂×Dic₃	1	1	yes	characteristic
27	C₂×C₃×C₄	C₅×D₃	1	1	yes	characteristic
28	C₂×Dic₃	—	3	2	no	—
29	C₂²×C₅	C₂×D₉	1	1	yes	characteristic
30	C₄×C₅	—	9	4	no	—
31	C₄×C₅	Dic₉	1	2	yes	direct product factor
32	C₂×C₉	C₂×C₄×C₅	1	2	yes	—
33	C₂×C₉	C₂×C₄×C₅	1	1	yes	characteristic
34	C₄²	—	9	1	no	Sylow
35	C₃×C₅	C₄×Dic₃	1	1	yes	characteristic
36	C₂²×C₃	C₂×C₅×D₃	1	1	yes	characteristic
37	C₃×C₄	C₅×Dic₃	1	2	yes	—
38	Dic₃	—	3	4	no	—
39	C₂×C₅	C₄×D₉	1	2	yes	—
40	C₂×C₅	C₂×Dic₉	1	1	yes	characteristic
41	C₉	C₄²×C₅	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄	—	9	2	no	—
43	C₂×C₄	C₅×D₉	1	1	yes	characteristic
44	C₂×C₃	C₄×C₅×D₃	1	2	yes	—
45	C₂×C₃	C₂×C₅×Dic₃	1	1	yes	characteristic
46	C₅	C₄×Dic₉	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²	C₂×C₅×D₉	1	1	yes	characteristic
48	C₄	—	9	4	no	—
49	C₄	C₅×Dic₉	1	2	yes	direct product factor
50	C₃	C₄×C₅×Dic₃	1	1	yes	characteristic
51	C₂	C₄×C₅×D₉	1	2	yes	—
52	C₂	C₂×C₅×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅	D₉
commutator, hypocenter	C₉	C₄²×C₅
Frattini	C₂²×C₃	C₂×C₅×D₃
Fitting	C₂×C₄×C₅×C₉	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₃
cosocle	C₂²×C₉	C₂²×C₅
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
5. C₄×C₅×C₉	C₄	semidirect
6. C₅×Dic₉	C₄	direct (N × H)
7. C₄×Dic₉	C₅	direct (N × H)
19. C₅×C₉	C₄²	semidirect
23. C₄×C₉	C₄×C₅	semidirect
24. Dic₉	C₄×C₅	direct (N × H)
31. C₄×C₅	Dic₉	direct (N × H)
41. C₉	C₄²×C₅	semidirect
46. C₅	C₄×Dic₉	direct (N × H)
49. C₄	C₅×Dic₉	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×C₅×Dic₉	2
C₄×C₅×Dic₃	3
C₄×Dic₉	5

Subgroups — C4×C5×Dic9 #10623587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×Dic₉ #10623587