Subgroups — C₂×C₅×#611 #10623705

Order 720.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×D₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#611	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#67	—	3	1	no	maximal
6	C₂²×C₅×C₉	—	2	2	no	—
7	C₂²×C₅×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₉	C₂²	1	2	yes	complemented
9	C₂×C₅×D₉	—	2	2	no	—
10	C₂×C₅×D₉	C₂²	1	2	yes	complemented
11	C₅×Dic₉	C₂²	1	2	yes	complemented
12	C₂×#611	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂³×C₃×C₅	D₃	1	1	yes	characteristic
14	C₂²×C₅×D₃	—	3	1	no	—
15	C₅×#67	—	3	4	no	—
16	C₂×C₅×Dic₃	—	3	1	no	—
17	C₂×C₅×C₉	—	2	2	no	—
18	C₂×C₅×C₉	D₄	1	2	yes	complemented
19	C₂×C₅×C₉	C₂³	1	1	yes	characteristic
20	C₅×D₉	—	2	2	no	—
21	C₂×C₅×D₄	—	9	1	no	Hall
22	C₂³×C₉	C₂×C₅	1	1	yes	characteristic, complemented
23	C₂²×D₉	C₂×C₅	1	1	yes	characteristic, complemented
24	#611	C₂×C₅	1	4	yes	direct product factor
25	C₂×Dic₉	C₂×C₅	1	1	yes	characteristic, complemented
26	C₂²×C₃×C₅	—	2	2	no	—
27	C₂²×C₃×C₅	C₂×D₃	1	1	yes	characteristic
28	C₂²×C₃×C₅	C₂×D₃	1	2	yes	—
29	C₂×C₅×D₃	—	3	2	no	—
30	C₂×C₅×D₃	—	6	2	no	—
31	C₅×Dic₃	—	3	2	no	—
32	C₂×#67	—	3	1	no	—
33	C₅×C₉	C₂×D₄	1	1	yes	characteristic, complemented, Hall
34	C₂³×C₅	—	9	1	no	—
35	C₂³×C₅	D₉	1	1	yes	characteristic, complemented
36	C₅×D₄	—	9	4	no	—
37	C₂×C₄×C₅	—	9	1	no	—
38	C₂²×C₉	—	2	2	no	—
39	C₂²×C₉	C₂²×C₅	1	1	yes	characteristic
40	C₂²×C₉	C₂²×C₅	1	2	yes	complemented
41	C₂×D₉	—	2	2	no	—
42	C₂×D₉	C₂²×C₅	1	2	yes	complemented
43	Dic₉	C₂²×C₅	1	2	yes	complemented
44	C₂×C₃×C₅	—	2	2	no	—
45	C₂×C₃×C₅	#67	1	2	yes	—
46	C₂×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
47	C₅×D₃	—	6	2	no	—
48	C₂³×C₃	C₅×D₃	1	1	yes	characteristic
49	C₂²×D₃	—	3	1	no	—
50	#67	—	3	4	no	—
51	C₂×Dic₃	—	3	1	no	—
52	C₂²×C₅	—	2	2	no	—
53	C₂²×C₅	—	9	2	no	—
54	C₂²×C₅	—	18	2	no	—
55	C₂²×C₅	C₂×D₉	1	1	yes	characteristic
56	C₂²×C₅	C₂×D₉	1	2	yes	complemented
57	C₄×C₅	—	9	2	no	—
58	C₂×C₉	—	2	2	no	—
59	C₂×C₉	C₅×D₄	1	2	yes	complemented
60	C₂×C₉	C₂³×C₅	1	1	yes	characteristic
61	D₉	—	2	2	no	—
62	C₂×D₄	—	9	1	no	Sylow
63	C₃×C₅	C₂×#67	1	1	yes	characteristic
64	C₂²×C₃	—	2	2	no	—
65	C₂²×C₃	C₂×C₅×D₃	1	1	yes	characteristic
66	C₂²×C₃	C₂×C₅×D₃	1	2	yes	—
67	C₂×D₃	—	3	2	no	—
68	C₂×D₃	—	6	2	no	—
69	Dic₃	—	3	2	no	—
70	C₂×C₅	—	2	2	no	—
71	C₂×C₅	—	18	2	no	—
72	C₂×C₅	#611	1	2	yes	direct product factor
73	C₂×C₅	C₂²×D₉	1	1	yes	characteristic
74	C₉	C₂×C₅×D₄	1	1	yes	characteristic, complemented, Sylow
75	C₂³	—	9	1	no	—
76	C₂³	C₅×D₉	1	1	yes	characteristic, complemented
77	D₄	—	9	4	no	—
78	C₂×C₄	—	9	1	no	—
79	C₂×C₃	—	2	2	no	—
80	C₂×C₃	C₅×#67	1	2	yes	—
81	C₂×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
82	D₃	—	6	2	no	—
83	C₅	C₂×#611	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²	—	2	2	no	—
85	C₂²	—	9	2	no	—
86	C₂²	—	18	2	no	—
87	C₂²	C₂×C₅×D₉	1	1	yes	characteristic
88	C₂²	C₂×C₅×D₉	1	2	yes	complemented
89	C₄	—	9	2	no	—
90	C₃	C₂×C₅×#67	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	18	2	no	—
93	C₂	C₅×#611	1	2	yes	direct product factor
94	C₂	C₂²×C₅×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₉
commutator, cosocle	C₂×C₉	C₂³×C₅
Frattini	C₂×C₃	C₂²×C₅×D₃
Fitting	C₂³×C₅×C₉	C₂
socle	C₂²×C₃×C₅	C₂×D₃
hypercenter	C₂³×C₅	D₉
hypocenter	C₉	C₂×C₅×D₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅

Complements

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

N	H ≅ G/N	type
1. C₂³×C₅×C₉	C₂	semidirect
2. C₂²×C₅×D₉	C₂	semidirect
3. C₅×#611	C₂	direct (N × H)
4. C₂×C₅×Dic₉	C₂	semidirect
8. C₂²×C₅×C₉	C₂²	semidirect
10. C₂×C₅×D₉	C₂²	semidirect
11. C₅×Dic₉	C₂²	semidirect
12. C₂×#611	C₅	direct (N × H)
18. C₂×C₅×C₉	D₄	semidirect
22. C₂³×C₉	C₂×C₅	semidirect
23. C₂²×D₉	C₂×C₅	semidirect
24. #611	C₂×C₅	direct (N × H)
25. C₂×Dic₉	C₂×C₅	semidirect
33. C₅×C₉	C₂×D₄	semidirect
35. C₂³×C₅	D₉	semidirect
40. C₂²×C₉	C₂²×C₅	semidirect
42. C₂×D₉	C₂²×C₅	semidirect
43. Dic₉	C₂²×C₅	semidirect
56. C₂²×C₅	C₂×D₉	semidirect
59. C₂×C₉	C₅×D₄	semidirect
72. C₂×C₅	#611	direct (N × H)
74. C₉	C₂×C₅×D₄	semidirect
76. C₂³	C₅×D₉	semidirect
83. C₅	C₂×#611	direct (N × H)
88. C₂²	C₂×C₅×D₉	semidirect
93. C₂	C₅×#611	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₉	2
C₂²×C₅×D₉	2
C₅×#611	2
C₂×C₅×Dic₉	2
C₂×C₅×#67	3
C₂×#611	5

Subgroups — C2×C5×#611 #10623705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#611 #10623705