Subgroups — C₅×(C₂²⋊A₄)⋊C₉ #72247366

Order 2160.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₂²⋊A₄	C₃	1	1	yes	maximal, characteristic
2	C₅×C₂²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×C₂²⋊C₉	C₃	1	2	yes	maximal, complemented
4	C₅×C₉×A₄	—	4	2	no	maximal
5	(C₂²⋊A₄)⋊C₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₅×C₂²⋊C₉	—	3	2	no	—
7	C₂⁴×C₃×C₅	C₃²	1	1	yes	characteristic
8	C₅×C₂²⋊A₄	C₉	1	3	yes	complemented
9	C₃×C₅×A₄	—	4	2	no	—
10	C₃×C₅×A₄	—	12	1	no	—
11	C₂²×C₅×C₉	—	4	2	no	—
12	C₅×C₂²⋊C₉	—	4	2	no	—
13	C₅×C₂²⋊C₉	—	12	1	no	—
14	C₅×C₂²⋊C₉	A₄	1	2	yes	complemented
15	C₃×C₂²⋊A₄	C₃×C₅	1	1	yes	characteristic
16	C₂²⋊(C₂²⋊C₉)	C₃×C₅	1	1	yes	characteristic, complemented
17	C₂²×C₂²⋊C₉	C₃×C₅	1	2	yes	complemented
18	C₃×C₅×C₉	—	16	1	no	Hall
19	C₂³×C₃×C₅	—	3	2	no	—
20	C₂³×C₃×C₅	—	9	1	no	—
21	C₉×A₄	—	4	2	no	—
22	C₂×C₅×C₉	—	12	2	no	—
23	C₂⁴×C₅	C₃×C₉	1	1	yes	characteristic, complemented, Hall
24	C₂×C₂²⋊C₉	—	3	2	no	—
25	C₂²×C₃×C₅	—	3	1	no	—
26	C₂²×C₃×C₅	—	3	1	no	—
27	C₂²×C₃×C₅	—	9	1	no	—
28	C₂²×C₃×C₅	—	9	2	no	—
29	C₂²×C₃×C₅	C₃×A₄	1	2	yes	—
30	C₅×A₄	—	4	6	no	—
31	C₅×A₄	—	12	3	no	—
32	C₂⁴×C₃	C₃²×C₅	1	1	yes	characteristic
33	C₂²⋊A₄	C₅×C₉	1	3	yes	complemented
34	C₃²×C₅	—	16	1	no	—
35	C₅×C₉	—	4	2	no	—
36	C₅×C₉	—	16	1	no	—
37	C₂³×C₅	—	3	2	no	—
38	C₂³×C₅	—	9	1	no	—
39	C₃×A₄	—	4	2	no	—
40	C₃×A₄	—	12	1	no	—
41	C₂²×C₉	—	4	2	no	—
42	C₂²⋊C₉	—	4	2	no	—
43	C₂²⋊C₉	—	12	1	no	—
44	C₂²⋊C₉	C₅×A₄	1	2	yes	complemented
45	C₂×C₃×C₅	—	3	2	no	—
46	C₂×C₃×C₅	—	9	1	no	—
47	C₃×C₉	—	16	1	no	Sylow
48	C₂³×C₃	—	3	2	no	—
49	C₂³×C₃	—	9	1	no	—
50	C₂²×C₅	—	3	1	no	—
51	C₂²×C₅	—	3	1	no	—
52	C₂²×C₅	—	9	1	no	—
53	C₂²×C₅	—	9	2	no	—
54	C₂²×C₅	C₉×A₄	1	2	yes	complemented
55	C₂×C₉	—	12	2	no	—
56	C₂⁴	C₃×C₅×C₉	1	1	yes	characteristic, complemented, Sylow
57	C₃×C₅	—	16	3	no	—
58	C₃×C₅	A₄²	1	1	yes	characteristic
59	C₂²×C₃	—	3	1	no	—
60	C₂²×C₃	—	3	1	no	—
61	C₂²×C₃	—	9	1	no	—
62	C₂²×C₃	—	9	2	no	—
63	C₂²×C₃	C₃×C₅×A₄	1	2	yes	—
64	A₄	—	4	6	no	—
65	A₄	—	12	3	no	—
66	C₂×C₅	—	3	2	no	—
67	C₂×C₅	—	9	1	no	—
68	C₃²	—	16	1	no	—
69	C₉	—	4	2	no	—
70	C₉	—	16	1	no	—
71	C₂³	—	3	2	no	—
72	C₂³	—	9	1	no	—
73	C₂×C₃	—	3	2	no	—
74	C₂×C₃	—	9	1	no	—
75	C₅	(C₂²⋊A₄)⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²	—	3	1	no	—
77	C₂²	—	3	1	no	—
78	C₂²	—	9	1	no	—
79	C₂²	—	9	2	no	—
80	C₂²	C₅×C₉×A₄	1	2	yes	complemented
81	C₃	—	16	3	no	—
82	C₃	C₅×A₄²	1	1	yes	characteristic
83	C₂	—	3	2	no	—
84	C₂	—	9	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅	A₄²
commutator, hypocenter	C₂⁴	C₃×C₅×C₉
Frattini	C₃	C₅×A₄²
Fitting, socle	C₂⁴×C₃×C₅	C₃²
prime radical	C₃×C₅×C₂²⋊A₄	C₃
cosocle	C₂⁴×C₃	C₃²×C₅
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×C₂²⋊(C₂²⋊C₉)	C₃	semidirect
3. C₂²×C₅×C₂²⋊C₉	C₃	semidirect
5. (C₂²⋊A₄)⋊C₉	C₅	direct (N × H)
8. C₅×C₂²⋊A₄	C₉	semidirect
14. C₅×C₂²⋊C₉	A₄	semidirect
16. C₂²⋊(C₂²⋊C₉)	C₃×C₅	semidirect
17. C₂²×C₂²⋊C₉	C₃×C₅	semidirect
23. C₂⁴×C₅	C₃×C₉	semidirect
33. C₂²⋊A₄	C₅×C₉	semidirect
44. C₂²⋊C₉	C₅×A₄	semidirect
54. C₂²×C₅	C₉×A₄	semidirect
56. C₂⁴	C₃×C₅×C₉	semidirect
75. C₅	(C₂²⋊A₄)⋊C₉	direct (N × H)
80. C₂²	C₅×C₉×A₄	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₂²⋊A₄	3
C₅×C₂²⋊(C₂²⋊C₉)	3
C₂²×C₅×C₂²⋊C₉	3
C₅×C₉×A₄	4
(C₂²⋊A₄)⋊C₉	5

Subgroups — C5×(C22⋊A4)⋊C9 #72247366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×(C₂²⋊A₄)⋊C₉ #72247366