Subgroups — C₉×#3960 #13083789

Order 1440.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×D₄₀	C₂	1	1	yes	maximal, characteristic
3	C₉×#672	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#3960	C₃	1	1	yes	maximal, characteristic
5	C₅×Q₈×C₉	—	2	1	no	—
6	C₉×D₂₀	—	2	1	no	—
7	C₅×C₈×C₉	C₂²	1	1	yes	characteristic
8	C₉×SD₃₂	—	5	1	no	maximal, Hall
9	C₃×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
10	C₃×D₄₀	C₂×C₃	1	1	yes	characteristic
11	C₃×#672	C₂×C₃	1	1	yes	characteristic
12	C₂×C₉×D₅	—	4	1	no	—
13	C₄×C₅×C₉	—	4	1	no	—
14	C₄×C₅×C₉	D₄	1	1	yes	characteristic
15	#3960	C₉	1	1	yes	characteristic, direct product factor, Hall
16	C₉×Q₁₆	D₅	1	1	yes	characteristic, complemented
17	C₉×D₈	—	5	1	no	—
18	C₉×C₁₆	—	5	1	no	—
19	C₃×C₅×Q₈	—	2	1	no	—
20	C₃×D₂₀	—	2	1	no	—
21	C₃×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
22	C₃×SD₃₂	—	5	1	no	—
23	C₂×C₅×C₉	D₈	1	1	yes	characteristic
24	C₉×D₅	—	8	1	no	—
25	C₅×Q₁₆	C₂×C₉	1	1	yes	characteristic, complemented
26	D₄₀	C₂×C₉	1	1	yes	characteristic
27	#672	C₂×C₉	1	1	yes	characteristic, complemented
28	Q₈×C₉	—	2	1	no	—
29	D₄×C₉	—	10	1	no	—
30	C₈×C₉	C₂×D₅	1	1	yes	characteristic
31	C₂×C₃×D₅	—	4	1	no	—
32	C₃×C₄×C₅	—	4	1	no	—
33	C₃×C₄×C₅	C₃×D₄	1	1	yes	characteristic
34	C₃×Q₁₆	C₃×D₅	1	1	yes	characteristic
35	C₃×D₈	—	5	1	no	—
36	C₃×C₁₆	—	5	1	no	—
37	C₅×C₉	SD₃₂	1	1	yes	characteristic, complemented, Hall
38	C₅×Q₈	—	2	1	no	—
39	D₂₀	—	2	1	no	—
40	C₅×C₈	C₂²×C₉	1	1	yes	characteristic
41	C₂²×C₉	—	20	1	no	—
42	C₄×C₉	—	4	1	no	—
43	C₄×C₉	C₂²⋊D₅	1	1	yes	characteristic
44	SD₃₂	—	5	1	no	Sylow
45	C₂×C₃×C₅	C₃×D₈	1	1	yes	characteristic
46	C₃×D₅	—	8	1	no	—
47	C₃×Q₈	—	2	1	no	—
48	C₃×D₄	—	10	1	no	—
49	C₃×C₈	C₂×C₃×D₅	1	1	yes	characteristic
50	C₂×D₅	—	4	1	no	—
51	C₄×C₅	—	4	1	no	—
52	C₄×C₅	D₄×C₉	1	1	yes	characteristic
53	C₂×C₉	—	40	1	no	—
54	C₂×C₉	#686	1	1	yes	characteristic
55	Q₁₆	C₉×D₅	1	1	yes	characteristic, complemented
56	D₈	—	5	1	no	—
57	C₁₆	—	5	1	no	—
58	C₃×C₅	C₃×SD₃₂	1	1	yes	characteristic
59	C₂²×C₃	—	20	1	no	—
60	C₃×C₄	—	4	1	no	—
61	C₃×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
62	C₂×C₅	C₉×D₈	1	1	yes	characteristic
63	D₅	—	8	1	no	—
64	C₉	#3960	1	1	yes	characteristic, direct product factor, Sylow
65	Q₈	—	2	1	no	—
66	D₄	—	10	1	no	—
67	C₈	C₂×C₉×D₅	1	1	yes	characteristic
68	C₂×C₃	—	40	1	no	—
69	C₂×C₃	C₃×#686	1	1	yes	characteristic
70	C₅	C₉×SD₃₂	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	20	1	no	—
72	C₄	—	4	1	no	—
73	C₄	C₉×C₂²⋊D₅	1	1	yes	characteristic
74	C₃	C₃×#3960	1	1	yes	characteristic
75	C₂	—	40	1	no	—
76	C₂	C₉×#686	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#686
commutator	C₅×C₈	C₂²×C₉
Frattini	C₃×C₈	C₂×C₃×D₅
Fitting	C₅×C₉×Q₁₆	C₂
socle	C₂×C₃×C₅	C₃×D₈
prime radical	C₃×D₄₀	C₂×C₃
cosocle	C₃×C₅×C₈	C₂²×C₃
hypercenter	C₉×Q₁₆	D₅
hypocenter	C₅	C₉×SD₃₂
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ C₈×C₉ ◁ C₉×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₅×C₉×Q₁₆	C₂	semidirect
3. C₉×#672	C₂	semidirect
15. #3960	C₉	direct (N × H)
16. C₉×Q₁₆	D₅	semidirect
25. C₅×Q₁₆	C₂×C₉	semidirect
27. #672	C₂×C₉	semidirect
37. C₅×C₉	SD₃₂	semidirect
55. Q₁₆	C₉×D₅	semidirect
64. C₉	#3960	direct (N × H)
70. C₅	C₉×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₉×Q₁₆	2
C₉×D₄₀	2
C₉×#672	2
C₃×#3960	3
C₉×SD₃₂	5

Subgroups — C9×#3960 #13083789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3960 #13083789