Subgroups — C₅×AGL(1,𝔽₅)×3₋¹⁺² #72247409

Order 2700.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₅×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic
3	C₅×C₉×AGL(1,𝔽₅)	C₃	1	3	yes	maximal, complemented
4	C₅²×3₋¹⁺²	C₄	1	1	yes	characteristic, complemented, Hall
5	AGL(1,𝔽₅)×3₋¹⁺²	C₅	1	1	yes	maximal, characteristic, direct product factor
6	C₄×C₅×3₋¹⁺²	—	5	1	no	maximal
7	C₃²×C₅×D₅	C₂×C₃	1	1	yes	characteristic
8	C₅×C₉×D₅	C₂×C₃	1	3	yes	—
9	C₃×C₅×AGL(1,𝔽₅)	—	3	1	no	—
10	C₃×C₅×AGL(1,𝔽₅)	C₃²	1	1	yes	characteristic
11	C₂×C₅×3₋¹⁺²	—	5	1	no	—
12	D₅×3₋¹⁺²	C₂×C₅	1	1	yes	characteristic
13	C₃²×C₅²	C₃×C₄	1	1	yes	characteristic
14	C₅²×C₉	C₃×C₄	1	3	yes	complemented
15	C₃²×AGL(1,𝔽₅)	C₃×C₅	1	1	yes	characteristic
16	C₃²×C₄×C₅	—	5	1	no	—
17	C₉×AGL(1,𝔽₅)	C₃×C₅	1	3	yes	complemented
18	C₄×C₅×C₉	—	5	3	no	—
19	C₃×C₅×D₅	—	3	1	no	—
20	C₃×C₅×D₅	C₂×C₃²	1	1	yes	characteristic
21	C₅×3₋¹⁺²	—	4	1	no	—
22	C₅×3₋¹⁺²	C₄×C₅	1	1	yes	characteristic, complemented
23	C₅×3₋¹⁺²	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
24	C₄×3₋¹⁺²	—	5	1	no	Hall
25	C₅×AGL(1,𝔽₅)	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Hall
26	C₂×C₃²×C₅	—	5	1	no	—
27	C₃²×D₅	C₂×C₃×C₅	1	1	yes	characteristic
28	C₂×C₅×C₉	—	5	3	no	—
29	C₉×D₅	C₂×C₃×C₅	1	3	yes	—
30	C₃×C₅²	—	3	1	no	—
31	C₃×C₅²	C₃²×C₄	1	1	yes	characteristic
32	C₃×AGL(1,𝔽₅)	—	3	1	no	—
33	C₃×AGL(1,𝔽₅)	C₃²×C₅	1	1	yes	characteristic
34	C₃×C₄×C₅	—	5	1	no	—
35	C₃×C₄×C₅	—	15	1	no	—
36	C₂×3₋¹⁺²	—	5	1	no	—
37	C₅×D₅	C₂×3₋¹⁺²	1	1	yes	characteristic
38	C₃²×C₅	—	4	1	no	—
39	C₃²×C₅	C₃×C₄×C₅	1	1	yes	characteristic
40	C₃²×C₅	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
41	C₅×C₉	—	4	3	no	—
42	C₅×C₉	C₃×C₄×C₅	1	3	yes	complemented
43	C₅×C₉	C₃×AGL(1,𝔽₅)	1	3	yes	complemented
44	C₃²×C₄	—	5	1	no	—
45	C₄×C₉	—	5	3	no	—
46	C₂×C₃×C₅	—	5	1	no	—
47	C₂×C₃×C₅	—	15	1	no	—
48	C₃×D₅	—	3	1	no	—
49	C₃×D₅	C₂×C₃²×C₅	1	1	yes	characteristic
50	3₋¹⁺²	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
51	C₅²	C₄×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
52	AGL(1,𝔽₅)	C₅×3₋¹⁺²	1	1	yes	characteristic, direct product factor
53	C₄×C₅	—	5	1	no	—
54	C₂×C₃²	—	5	1	no	—
55	C₂×C₉	—	5	3	no	—
56	C₃×C₅	—	3	1	no	—
57	C₃×C₅	—	3	1	no	—
58	C₃×C₅	—	4	1	no	—
59	C₃×C₅	—	12	1	no	—
60	C₃×C₅	C₃²×C₄×C₅	1	1	yes	characteristic
61	C₃×C₅	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₃×C₄	—	5	1	no	—
63	C₃×C₄	—	15	1	no	—
64	C₂×C₅	—	5	1	no	—
65	D₅	C₂×C₅×3₋¹⁺²	1	1	yes	characteristic
66	C₃²	C₃×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₉	C₃×C₅×AGL(1,𝔽₅)	1	3	yes	complemented
68	C₂×C₃	—	5	1	no	—
69	C₂×C₃	—	15	1	no	—
70	C₅	—	4	1	no	—
71	C₅	C₄×C₅×3₋¹⁺²	1	1	yes	characteristic, complemented
72	C₅	AGL(1,𝔽₅)×3₋¹⁺²	1	1	yes	characteristic, direct product factor
73	C₄	—	5	1	no	Sylow
74	C₃	—	3	1	no	—
75	C₃	C₃²×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₂	—	5	1	no	—

	subgroup	G/H
center	C₃×C₅	C₃²×AGL(1,𝔽₅)
commutator	C₃×C₅	C₃²×C₄×C₅
Frattini	C₃	C₃²×C₅×AGL(1,𝔽₅)
Fitting	C₅²×3₋¹⁺²	C₄
socle	C₃×C₅²	C₃²×C₄
prime radical	C₃²×C₅×D₅	C₂×C₃
cosocle	C₃×D₅	C₂×C₃²×C₅
hypercenter	C₅×3₋¹⁺²	AGL(1,𝔽₅)
hypocenter	C₅	C₄×C₅×3₋¹⁺²
LCS	G ▷ C₃×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₅ ◁ C₅×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
3. C₅×C₉×AGL(1,𝔽₅)	C₃	semidirect
4. C₅²×3₋¹⁺²	C₄	semidirect
5. AGL(1,𝔽₅)×3₋¹⁺²	C₅	direct (N × H)
14. C₅²×C₉	C₃×C₄	semidirect
17. C₉×AGL(1,𝔽₅)	C₃×C₅	semidirect
22. C₅×3₋¹⁺²	C₄×C₅	semidirect
23. C₅×3₋¹⁺²	AGL(1,𝔽₅)	direct (N × H)
25. C₅×AGL(1,𝔽₅)	3₋¹⁺²	direct (N × H)
42. C₅×C₉	C₃×C₄×C₅	semidirect
43. C₅×C₉	C₃×AGL(1,𝔽₅)	semidirect
50. 3₋¹⁺²	C₅×AGL(1,𝔽₅)	direct (N × H)
51. C₅²	C₄×3₋¹⁺²	semidirect
52. AGL(1,𝔽₅)	C₅×3₋¹⁺²	direct (N × H)
67. C₉	C₃×C₅×AGL(1,𝔽₅)	semidirect
71. C₅	C₄×C₅×3₋¹⁺²	semidirect
72. C₅	AGL(1,𝔽₅)×3₋¹⁺²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×D₅×3₋¹⁺²	2
C₃²×C₅×AGL(1,𝔽₅)	3
C₅×C₉×AGL(1,𝔽₅)	3
AGL(1,𝔽₅)×3₋¹⁺²	5
C₄×C₅×3₋¹⁺²	5

Subgroups — C5×AGL(1,𝔽5)×3−1+2 #72247409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×AGL(1,𝔽₅)×3₋¹⁺² #72247409