Subgroups — C₉×#685 #10623574

Order 720.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#685	C₃	1	1	yes	maximal, characteristic
5	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
6	C₂×C₉×D₅	—	2	1	no	—
7	C₂×C₉×D₅	—	2	1	no	—
8	C₂×C₉×D₅	C₄	1	2	yes	complemented
9	C₄×C₅×C₉	—	2	1	no	—
10	C₉×Dic₅	—	2	1	no	—
11	C₉×C₂²⋊C₄	—	5	1	no	maximal, Hall
12	C₂²×C₃×D₅	C₂×C₃	1	1	yes	characteristic
13	C₂×C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×Dic₅	C₂×C₃	1	1	yes	characteristic
15	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₅×C₉	D₄	1	1	yes	characteristic
17	C₂×C₅×C₉	D₄	1	1	yes	characteristic
18	C₉×D₅	—	2	2	no	—
19	#685	C₉	1	1	yes	characteristic, direct product factor, Hall
20	C₂³×C₉	—	5	1	no	—
21	C₂×C₄×C₉	—	5	1	no	—
22	C₂×C₄×C₉	D₅	1	1	yes	characteristic, complemented
23	C₂²×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
24	C₂×C₃×D₅	—	2	1	no	—
25	C₂×C₃×D₅	—	2	1	no	—
26	C₂×C₃×D₅	C₃×C₄	1	2	yes	—
27	C₃×C₄×C₅	—	2	1	no	—
28	C₃×Dic₅	—	2	1	no	—
29	C₃×C₂²⋊C₄	—	5	1	no	—
30	C₅×C₉	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
31	C₂²×D₅	C₂×C₉	1	1	yes	characteristic
32	C₂×C₄×C₅	C₂×C₉	1	1	yes	characteristic, complemented
33	C₂×Dic₅	C₂×C₉	1	1	yes	characteristic, complemented
34	C₂²×C₉	—	5	2	no	—
35	C₂²×C₉	—	10	1	no	—
36	C₂²×C₉	—	10	1	no	—
37	C₂²×C₉	C₂×D₅	1	1	yes	characteristic
38	C₄×C₉	—	2	1	no	—
39	C₄×C₉	—	10	1	no	—
40	C₂×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
41	C₂×C₃×C₅	C₃×D₄	1	1	yes	characteristic
42	C₂×C₃×C₅	C₃×D₄	1	1	yes	characteristic
43	C₃×D₅	—	2	2	no	—
44	C₂³×C₃	—	5	1	no	—
45	C₂×C₃×C₄	—	5	1	no	—
46	C₂×C₃×C₄	C₃×D₅	1	1	yes	characteristic
47	C₂²×C₅	C₂²×C₉	1	1	yes	characteristic
48	C₂×D₅	—	2	1	no	—
49	C₂×D₅	—	2	1	no	—
50	C₂×D₅	C₄×C₉	1	2	yes	complemented
51	C₄×C₅	—	2	1	no	—
52	Dic₅	—	2	1	no	—
53	C₂×C₉	—	10	2	no	—
54	C₂×C₉	C₄×D₅	1	1	yes	characteristic
55	C₂×C₉	D₂₀	1	1	yes	characteristic
56	C₂×C₉	C₂²⋊D₅	1	1	yes	characteristic
57	C₂²⋊C₄	—	5	1	no	Sylow
58	C₃×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₃	—	5	2	no	—
60	C₂²×C₃	—	10	1	no	—
61	C₂²×C₃	—	10	1	no	—
62	C₂²×C₃	C₂×C₃×D₅	1	1	yes	characteristic
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	—	10	1	no	—
65	C₂×C₅	C₂×C₄×C₉	1	1	yes	characteristic
66	C₂×C₅	D₄×C₉	1	1	yes	characteristic
67	C₂×C₅	D₄×C₉	1	1	yes	characteristic
68	D₅	—	2	2	no	—
69	C₉	#685	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³	—	5	1	no	—
71	C₂×C₄	—	5	1	no	—
72	C₂×C₄	C₉×D₅	1	1	yes	characteristic, complemented
73	C₂×C₃	—	10	2	no	—
74	C₂×C₃	C₃×C₄×D₅	1	1	yes	characteristic
75	C₂×C₃	C₃×D₂₀	1	1	yes	characteristic
76	C₂×C₃	C₃×C₂²⋊D₅	1	1	yes	characteristic
77	C₅	C₉×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	5	2	no	—
79	C₂²	—	10	1	no	—
80	C₂²	—	10	1	no	—
81	C₂²	C₂×C₉×D₅	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	10	1	no	—
84	C₃	C₃×#685	1	1	yes	characteristic
85	C₂	—	10	2	no	—
86	C₂	C₄×C₉×D₅	1	1	yes	characteristic
87	C₂	C₉×D₂₀	1	1	yes	characteristic
88	C₂	C₉×C₂²⋊D₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×C₉	C₂	semidirect
3. C₂×C₉×Dic₅	C₂	semidirect
8. C₂×C₉×D₅	C₄	semidirect
19. #685	C₉	direct (N × H)
22. C₂×C₄×C₉	D₅	semidirect
30. C₅×C₉	C₂²⋊C₄	semidirect
32. C₂×C₄×C₅	C₂×C₉	semidirect
33. C₂×Dic₅	C₂×C₉	semidirect
50. C₂×D₅	C₄×C₉	semidirect
69. C₉	#685	direct (N × H)
72. C₂×C₄	C₉×D₅	semidirect
77. C₅	C₉×C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×D₅	2
C₂×C₄×C₅×C₉	2
C₂×C₉×Dic₅	2
C₃×#685	3
C₉×C₂²⋊C₄	5

Subgroups — C9×#685 #10623574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#685 #10623574