Subgroups — AGL(1,𝔽₅)×Dic₉ #10623630

Order 720.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	D₅×Dic₉	C₂	1	1	yes	maximal, characteristic
4	Dic₃×AGL(1,𝔽₅)	—	3	1	no	maximal
5	C₂×C₉×D₅	C₂²	1	1	yes	characteristic
6	C₉⋊AGL(1,𝔽₅)	C₄	1	2	yes	complemented
7	C₉×AGL(1,𝔽₅)	C₄	1	2	yes	complemented
8	Dic₄₅	C₄	1	1	yes	characteristic, complemented
9	C₅×Dic₉	C₄	1	1	yes	characteristic, complemented
10	C₄×Dic₉	—	5	1	no	maximal, Hall
11	C₂×ΓL(1,𝔽₁₆)	—	3	1	no	—
12	C₂×C₃×AGL(1,𝔽₅)	D₃	1	1	yes	characteristic
13	D₅×Dic₃	—	3	1	no	—
14	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
15	C₉×D₅	C₂×C₄	1	1	yes	characteristic
16	C₉×D₅	C₂×C₄	1	1	yes	characteristic
17	C₄×AGL(1,𝔽₅)	—	9	1	no	Hall
18	C₂×C₄×C₉	—	5	1	no	—
19	C₂×Dic₉	—	5	1	no	—
20	C₂×Dic₉	—	5	1	no	—
21	C₂×C₃×D₅	C₂×D₃	1	1	yes	characteristic
22	ΓL(1,𝔽₁₆)	—	3	2	no	—
23	C₃×AGL(1,𝔽₅)	Dic₃	1	2	yes	—
24	Dic₁₅	—	3	1	no	—
25	C₅×Dic₃	—	3	1	no	—
26	C₄×Dic₃	—	15	1	no	—
27	C₅×C₉	C₄²	1	1	yes	characteristic, complemented, Hall
28	C₂×AGL(1,𝔽₅)	—	9	1	no	—
29	C₂×AGL(1,𝔽₅)	D₉	1	1	yes	characteristic
30	C₄×D₅	—	9	1	no	—
31	C₂²×C₉	—	5	1	no	—
32	C₄×C₉	—	5	2	no	—
33	Dic₉	—	5	2	no	—
34	Dic₉	—	5	1	no	—
35	Dic₉	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
36	C₂×C₃×C₅	C₄×D₃	1	1	yes	characteristic
37	C₃×D₅	C₄×D₃	1	1	yes	characteristic
38	C₃×D₅	C₂×Dic₃	1	1	yes	characteristic
39	C₂×C₃×C₄	—	5	1	no	—
40	C₂×Dic₃	—	15	1	no	—
41	C₂×Dic₃	—	15	1	no	—
42	C₂×D₅	C₂×D₉	1	1	yes	characteristic
43	AGL(1,𝔽₅)	—	9	2	no	—
44	AGL(1,𝔽₅)	Dic₉	1	2	yes	direct product factor
45	C₄×C₅	—	9	1	no	—
46	Dic₅	—	9	1	no	—
47	C₂×C₉	—	5	1	no	—
48	C₂×C₉	—	5	1	no	—
49	C₂×C₉	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₄²	—	45	1	no	Sylow
51	C₃×C₅	C₄×Dic₃	1	1	yes	characteristic
52	C₂²×C₃	—	5	1	no	—
53	C₃×C₄	—	5	2	no	—
54	Dic₃	—	3	1	no	—
55	Dic₃	—	15	2	no	—
56	Dic₃	—	15	1	no	—
57	C₂×C₅	C₄×D₉	1	1	yes	characteristic
58	D₅	C₄×D₉	1	1	yes	characteristic
59	D₅	C₂×Dic₉	1	1	yes	characteristic
60	C₉	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₄	—	5	1	no	—
62	C₂×C₄	—	45	1	no	—
63	C₂×C₄	—	45	1	no	—
64	C₂×C₃	—	5	1	no	—
65	C₂×C₃	—	5	1	no	—
66	C₂×C₃	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₅	C₄×Dic₉	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	5	1	no	—
69	C₄	—	5	2	no	—
70	C₄	—	9	1	no	—
71	C₄	—	45	2	no	—
72	C₄	—	45	1	no	—
73	C₃	Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₂	—	5	1	no	—
75	C₂	—	5	1	no	—
76	C₂	D₉×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	D₉×AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₉	C₄²
Frattini	C₂×C₃	D₃×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₉	C₂×C₄
socle	C₂×C₃×C₅	C₄×D₃
prime radical	C₂×C₃×D₅	C₂×D₃
cosocle	C₂×C₉×D₅	C₂²
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
6. C₉⋊AGL(1,𝔽₅)	C₄	semidirect
7. C₉×AGL(1,𝔽₅)	C₄	semidirect
8. Dic₄₅	C₄	semidirect
9. C₅×Dic₉	C₄	semidirect
27. C₅×C₉	C₄²	semidirect
35. Dic₉	AGL(1,𝔽₅)	direct (N × H)
44. AGL(1,𝔽₅)	Dic₉	direct (N × H)
60. C₉	C₄×AGL(1,𝔽₅)	semidirect
67. C₅	C₄×Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉⋊AGL(1,𝔽₅)	2
C₂×C₉×AGL(1,𝔽₅)	2
D₅×Dic₉	2
Dic₃×AGL(1,𝔽₅)	3
C₄×Dic₉	5

Subgroups — AGL(1,𝔽5)×Dic9 #10623630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,𝔽₅)×Dic₉ #10623630