Subgroups — D₅×#6911 #72247205

Order 2430.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#6911	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×D₅×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×D₅×H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
4	D₅×#732	C₃	1	9	yes	maximal, complemented
5	C₂×#6911	—	5	1	no	maximal, Hall
6	C₃×C₅×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₅×H(3,𝔽₃)	C₂×C₃	1	3	yes	complemented
8	C₅×#732	C₂×C₃	1	9	yes	complemented
9	C₃³×D₅	—	3	3	no	—
10	C₃³×D₅	C₃²	1	1	yes	characteristic
11	D₅×3₋¹⁺²	—	3	3	no	—
12	D₅×H(3,𝔽₃)	—	3	6	no	—
13	D₅×H(3,𝔽₃)	C₃²	1	9	yes	complemented
14	C₃×C₉×D₅	C₃²	1	3	yes	complemented
15	#6911	D₅	1	1	yes	characteristic, direct product factor, Sylow
16	C₂×C₃×3₋¹⁺²	—	5	1	no	—
17	C₂×C₃×H(3,𝔽₃)	—	5	3	no	—
18	C₂×#732	—	5	9	no	—
19	C₃³×C₅	—	3	3	no	—
20	C₃³×C₅	C₂×C₃²	1	1	yes	characteristic
21	C₅×3₋¹⁺²	—	3	3	no	—
22	C₅×H(3,𝔽₃)	—	3	6	no	—
23	C₅×H(3,𝔽₃)	C₂×C₃²	1	9	yes	complemented
24	C₃×C₅×C₉	C₂×C₃²	1	3	yes	complemented
25	C₃²×D₅	—	3	9	no	—
26	C₃²×D₅	—	9	1	no	—
27	C₃²×D₅	—	9	9	no	—
28	C₃²×D₅	H(3,𝔽₃)	1	3	yes	—
29	C₃²×D₅	C₃³	1	1	yes	characteristic
30	C₉×D₅	—	3	3	no	—
31	C₃×3₋¹⁺²	C₃×D₅	1	1	yes	characteristic, complemented
32	C₃×H(3,𝔽₃)	C₃×D₅	1	3	yes	complemented
33	#732	C₃×D₅	1	9	yes	complemented
34	C₂×C₃³	—	5	1	no	—
35	C₂×C₃³	—	15	3	no	—
36	C₂×3₋¹⁺²	—	15	3	no	—
37	C₂×H(3,𝔽₃)	—	5	9	no	—
38	C₂×H(3,𝔽₃)	—	15	6	no	—
39	C₂×C₃×C₉	—	5	3	no	—
40	C₃²×C₅	—	3	9	no	—
41	C₃²×C₅	—	9	1	no	—
42	C₃²×C₅	—	9	9	no	—
43	C₃²×C₅	C₂×H(3,𝔽₃)	1	3	yes	—
44	C₃²×C₅	C₂×C₃³	1	1	yes	characteristic
45	C₅×C₉	—	3	3	no	—
46	C₃×D₅	—	3	3	no	—
47	C₃×D₅	—	3	1	no	—
48	C₃×D₅	—	9	9	no	—
49	C₃×D₅	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃³	—	3	3	no	—
51	C₃³	C₃²×D₅	1	1	yes	characteristic
52	3₋¹⁺²	—	3	3	no	—
53	H(3,𝔽₃)	—	3	6	no	—
54	H(3,𝔽₃)	C₃²×D₅	1	9	yes	complemented
55	C₃×C₉	C₃²×D₅	1	3	yes	complemented
56	C₂×C₃²	—	5	3	no	—
57	C₂×C₃²	—	5	1	no	—
58	C₂×C₃²	—	15	9	no	—
59	C₂×C₃²	—	45	1	no	—
60	C₂×C₃²	—	45	9	no	—
61	C₂×C₉	—	15	3	no	—
62	C₃×C₅	—	3	3	no	—
63	C₃×C₅	—	3	1	no	—
64	C₃×C₅	—	9	9	no	—
65	C₃×C₅	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	D₅	#6911	1	1	yes	characteristic, direct product factor, Hall
67	C₃²	—	3	9	no	—
68	C₃²	—	9	1	no	—
69	C₃²	—	9	9	no	—
70	C₃²	D₅×H(3,𝔽₃)	1	3	yes	—
71	C₃²	C₃³×D₅	1	1	yes	characteristic
72	C₉	—	3	3	no	—
73	C₂×C₃	—	5	1	no	—
74	C₂×C₃	—	15	3	no	—
75	C₂×C₃	—	15	1	no	—
76	C₂×C₃	—	45	9	no	—
77	C₅	C₂×#6911	1	1	yes	characteristic, complemented, Sylow
78	C₃	—	3	3	no	—
79	C₃	—	3	1	no	—
80	C₃	—	9	9	no	—
81	C₃	C₃×D₅×H(3,𝔽₃)	1	1	yes	characteristic
82	C₂	—	5	1	no	Sylow

	subgroup	G/H
center	C₃	C₃×D₅×H(3,𝔽₃)
commutator, cosocle	C₃²×C₅	C₂×C₃³
Frattini	C₃²	C₃³×D₅
Fitting	C₅×#6911	C₂
socle	C₃×C₅	C₂×C₃×H(3,𝔽₃)
hypercenter	#6911	D₅
hypocenter	C₅	C₂×#6911
LCS	G ▷ C₃²×C₅ ▷ C₃×C₅ ▷ C₅
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6911

Complements

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

N	H ≅ G/N	type
1. C₅×#6911	C₂	semidirect
2. C₃×D₅×3₋¹⁺²	C₃	semidirect
3. C₃×D₅×H(3,𝔽₃)	C₃	semidirect
4. D₅×#732	C₃	semidirect
6. C₃×C₅×3₋¹⁺²	C₂×C₃	semidirect
7. C₃×C₅×H(3,𝔽₃)	C₂×C₃	semidirect
8. C₅×#732	C₂×C₃	semidirect
13. D₅×H(3,𝔽₃)	C₃²	semidirect
14. C₃×C₉×D₅	C₃²	semidirect
15. #6911	D₅	direct (N × H)
23. C₅×H(3,𝔽₃)	C₂×C₃²	semidirect
24. C₃×C₅×C₉	C₂×C₃²	semidirect
31. C₃×3₋¹⁺²	C₃×D₅	semidirect
32. C₃×H(3,𝔽₃)	C₃×D₅	semidirect
33. #732	C₃×D₅	semidirect
54. H(3,𝔽₃)	C₃²×D₅	semidirect
55. C₃×C₉	C₃²×D₅	semidirect
66. D₅	#6911	direct (N × H)
77. C₅	C₂×#6911	semidirect

Maximal subgroups

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

subgroup	index
C₅×#6911	2
C₃×D₅×3₋¹⁺²	3
C₃×D₅×H(3,𝔽₃)	3
D₅×#732	3
C₂×#6911	5

Subgroups — D5×#6911 #72247205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#6911 #72247205