Subgroups — C₂×C₉×C₂²⋊D₅ #10623695

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×C₉×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₂²⋊D₅	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₉×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×C₂²⋊D₅	C₃	1	1	yes	maximal, characteristic
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×C₉×D₅	—	2	2	no	—
10	C₂×C₉×D₅	C₂²	1	2	yes	complemented
11	C₉×Dic₅	C₂²	1	2	yes	complemented
12	C₂×D₄×C₉	—	5	1	no	maximal, Hall
13	C₂³×C₃×C₅	C₂×C₃	1	1	yes	characteristic
14	C₂²×C₃×D₅	C₂×C₃	1	1	yes	characteristic
15	C₃×C₂²⋊D₅	C₂×C₃	1	4	yes	—
16	C₂×C₃×Dic₅	C₂×C₃	1	1	yes	characteristic
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	C₉×D₅	—	2	2	no	—
21	C₂×C₂²⋊D₅	C₉	1	1	yes	characteristic, direct product factor, Hall
22	C₂³×C₉	—	5	1	no	—
23	C₂³×C₉	D₅	1	1	yes	characteristic, complemented
24	D₄×C₉	—	5	4	no	—
25	C₂×C₄×C₉	—	5	1	no	—
26	C₂²×C₃×C₅	—	2	2	no	—
27	C₂²×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
28	C₂²×C₃×C₅	C₂²×C₃	1	2	yes	—
29	C₂×C₃×D₅	—	2	2	no	—
30	C₂×C₃×D₅	C₂²×C₃	1	2	yes	—
31	C₃×Dic₅	C₂²×C₃	1	2	yes	—
32	C₂×C₃×D₄	—	5	1	no	—
33	C₅×C₉	C₂×D₄	1	1	yes	characteristic, complemented, Hall
34	C₂³×C₅	C₂×C₉	1	1	yes	characteristic, complemented
35	C₂²×D₅	C₂×C₉	1	1	yes	characteristic, complemented
36	C₂²⋊D₅	C₂×C₉	1	4	yes	direct product factor
37	C₂×Dic₅	C₂×C₉	1	1	yes	characteristic, complemented
38	C₂²×C₉	—	2	2	no	—
39	C₂²×C₉	—	5	2	no	—
40	C₂²×C₉	—	10	2	no	—
41	C₂²×C₉	C₂×D₅	1	1	yes	characteristic
42	C₂²×C₉	C₂×D₅	1	2	yes	complemented
43	C₄×C₉	—	5	2	no	—
44	C₂×C₃×C₅	—	2	2	no	—
45	C₂×C₃×C₅	C₃×D₄	1	2	yes	—
46	C₂×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
47	C₃×D₅	—	2	2	no	—
48	C₂³×C₃	—	5	1	no	—
49	C₂³×C₃	C₃×D₅	1	1	yes	characteristic
50	C₃×D₄	—	5	4	no	—
51	C₂×C₃×C₄	—	5	1	no	—
52	C₂²×C₅	—	2	2	no	—
53	C₂²×C₅	C₂²×C₉	1	1	yes	characteristic
54	C₂²×C₅	C₂²×C₉	1	2	yes	complemented
55	C₂×D₅	—	2	2	no	—
56	C₂×D₅	C₂²×C₉	1	2	yes	complemented
57	Dic₅	C₂²×C₉	1	2	yes	complemented
58	C₂×C₉	—	2	2	no	—
59	C₂×C₉	—	10	2	no	—
60	C₂×C₉	C₂²⋊D₅	1	2	yes	direct product factor
61	C₂×C₉	C₂²×D₅	1	1	yes	characteristic
62	C₂×D₄	—	5	1	no	Sylow
63	C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
64	C₂²×C₃	—	2	2	no	—
65	C₂²×C₃	—	5	2	no	—
66	C₂²×C₃	—	10	2	no	—
67	C₂²×C₃	C₂×C₃×D₅	1	1	yes	characteristic
68	C₂²×C₃	C₂×C₃×D₅	1	2	yes	—
69	C₃×C₄	—	5	2	no	—
70	C₂×C₅	—	2	2	no	—
71	C₂×C₅	D₄×C₉	1	2	yes	complemented
72	C₂×C₅	C₂³×C₉	1	1	yes	characteristic
73	D₅	—	2	2	no	—
74	C₉	C₂×C₂²⋊D₅	1	1	yes	characteristic, direct product factor, Sylow
75	C₂³	—	5	1	no	—
76	C₂³	C₉×D₅	1	1	yes	characteristic, complemented
77	D₄	—	5	4	no	—
78	C₂×C₄	—	5	1	no	—
79	C₂×C₃	—	2	2	no	—
80	C₂×C₃	—	10	2	no	—
81	C₂×C₃	C₃×C₂²⋊D₅	1	2	yes	—
82	C₂×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
83	C₅	C₂×D₄×C₉	1	1	yes	characteristic, complemented, Sylow
84	C₂²	—	2	2	no	—
85	C₂²	—	5	2	no	—
86	C₂²	—	10	2	no	—
87	C₂²	C₂×C₉×D₅	1	1	yes	characteristic
88	C₂²	C₂×C₉×D₅	1	2	yes	complemented
89	C₄	—	5	2	no	—
90	C₃	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	10	2	no	—
93	C₂	C₉×C₂²⋊D₅	1	2	yes	direct product factor
94	C₂	C₂²×C₉×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₅
commutator	C₂×C₅	C₂³×C₉
Frattini	C₂×C₃	C₂²×C₃×D₅
Fitting	C₂³×C₅×C₉	C₂
socle	C₂²×C₃×C₅	C₂²×C₃
prime radical	C₂×C₃×C₂²⋊D₅	C₃
cosocle	C₂×C₃×C₅	C₂³×C₃
hypercenter	C₂³×C₉	D₅
hypocenter	C₅	C₂×D₄×C₉
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₉ ◁ C₂³×C₉

Complements

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

N	H ≅ G/N	type
1. C₂³×C₅×C₉	C₂	semidirect
2. C₂²×C₉×D₅	C₂	semidirect
3. C₉×C₂²⋊D₅	C₂	direct (N × H)
4. C₂×C₉×Dic₅	C₂	semidirect
8. C₂²×C₅×C₉	C₂²	semidirect
10. C₂×C₉×D₅	C₂²	semidirect
11. C₉×Dic₅	C₂²	semidirect
18. C₂×C₅×C₉	D₄	semidirect
21. C₂×C₂²⋊D₅	C₉	direct (N × H)
23. C₂³×C₉	D₅	semidirect
33. C₅×C₉	C₂×D₄	semidirect
34. C₂³×C₅	C₂×C₉	semidirect
35. C₂²×D₅	C₂×C₉	semidirect
36. C₂²⋊D₅	C₂×C₉	direct (N × H)
37. C₂×Dic₅	C₂×C₉	semidirect
42. C₂²×C₉	C₂×D₅	semidirect
54. C₂²×C₅	C₂²×C₉	semidirect
56. C₂×D₅	C₂²×C₉	semidirect
57. Dic₅	C₂²×C₉	semidirect
60. C₂×C₉	C₂²⋊D₅	direct (N × H)
71. C₂×C₅	D₄×C₉	semidirect
74. C₉	C₂×C₂²⋊D₅	direct (N × H)
76. C₂³	C₉×D₅	semidirect
83. C₅	C₂×D₄×C₉	semidirect
88. C₂²	C₂×C₉×D₅	semidirect
93. C₂	C₉×C₂²⋊D₅	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×C₉×D₅	2
C₉×C₂²⋊D₅	2
C₂×C₉×Dic₅	2
C₂×C₃×C₂²⋊D₅	3
C₂×D₄×C₉	5

Subgroups — C2×C9×C22⋊D5 #10623695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×C₂²⋊D₅ #10623695