Subgroups — C₃×C₅×#67 #67169

Order 360.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×D₄	—	3	1	no	maximal
5	C₅×#67	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×C₅	—	2	1	no	—
7	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×D₃	—	2	1	no	—
9	C₃×#67	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₃×C₅	—	2	1	no	—
11	C₂²×C₃×C₅	—	3	1	no	—
12	C₂²×C₃×C₅	D₃	1	1	yes	characteristic, complemented
13	C₂²×C₃×C₅	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂×C₅×D₃	C₂×C₃	1	1	yes	characteristic, complemented
15	C₃×C₄×C₅	—	3	1	no	—
16	C₅×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
17	C₃²×C₅	D₄	1	1	yes	characteristic, complemented, Hall
18	C₅×D₄	—	3	1	no	Hall
19	C₂²×C₃²	C₂×C₅	1	1	yes	characteristic, complemented
20	C₂×C₃×D₃	C₂×C₅	1	1	yes	characteristic, complemented
21	C₃×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
22	C₂×C₃×C₅	—	2	2	no	—
23	C₂×C₃×C₅	—	2	1	no	—
24	C₂×C₃×C₅	—	2	1	no	—
25	C₂×C₃×C₅	—	2	1	no	—
26	C₂×C₃×C₅	—	6	1	no	—
27	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
29	C₅×D₃	—	2	1	no	—
30	C₃×D₄	—	3	1	no	—
31	#67	C₃×C₅	1	1	yes	characteristic, direct product factor
32	C₂²×C₅	—	3	1	no	—
33	C₂²×C₅	C₃×D₃	1	1	yes	characteristic, complemented
34	C₄×C₅	—	3	1	no	—
35	C₂×C₃²	—	2	1	no	—
36	C₂×C₃²	C₂²×C₅	1	1	yes	characteristic
37	C₃×D₃	—	2	1	no	—
38	C₃×C₅	—	2	1	no	—
39	C₃×C₅	#67	1	1	yes	characteristic, direct product factor
40	C₃×C₅	C₃×D₄	1	1	yes	characteristic, complemented
41	C₂²×C₃	—	2	1	no	—
42	C₂²×C₃	—	3	1	no	—
43	C₂²×C₃	C₅×D₃	1	1	yes	characteristic, complemented
44	C₂²×C₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
45	C₂×D₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
46	C₃×C₄	—	3	1	no	—
47	Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
48	C₂×C₅	—	2	1	no	—
49	C₂×C₅	—	6	1	no	—
50	C₂×C₅	C₂×C₃×D₃	1	1	yes	characteristic
51	C₃²	C₅×D₄	1	1	yes	characteristic, complemented, Sylow
52	D₄	—	3	1	no	Sylow
53	C₂×C₃	—	2	2	no	—
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	—	2	1	no	—
56	C₂×C₃	—	2	1	no	—
57	C₂×C₃	—	6	1	no	—
58	C₂×C₃	C₂×C₅×D₃	1	1	yes	characteristic
59	C₂×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
60	D₃	—	2	1	no	—
61	C₅	C₃×#67	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²	—	3	1	no	—
63	C₂²	C₃×C₅×D₃	1	1	yes	characteristic, complemented
64	C₄	—	3	1	no	—
65	C₃	—	2	1	no	—
66	C₃	C₅×#67	1	1	yes	characteristic, direct product factor
67	C₃	C₃×C₅×D₄	1	1	yes	characteristic, complemented
68	C₂	—	2	1	no	—
69	C₂	—	6	1	no	—
70	C₂	C₂×C₃×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅	C₂×D₃
commutator, cosocle	C₂×C₃	C₂²×C₃×C₅
Frattini	C₂	C₂×C₃×C₅×D₃
Fitting	C₂²×C₃²×C₅	C₂
socle	C₂×C₃²×C₅	C₂²
hypercenter	C₂²×C₃×C₅	D₃
hypocenter	C₃	C₃×C₅×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂²×C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₅	C₂	semidirect
2. C₂×C₃×C₅×D₃	C₂	semidirect
3. C₃×C₅×Dic₃	C₂	semidirect
5. C₅×#67	C₃	direct (N × H)
9. C₃×#67	C₅	direct (N × H)
12. C₂²×C₃×C₅	D₃	semidirect
13. C₂²×C₃×C₅	C₂×C₃	semidirect
14. C₂×C₅×D₃	C₂×C₃	semidirect
16. C₅×Dic₃	C₂×C₃	semidirect
17. C₃²×C₅	D₄	semidirect
19. C₂²×C₃²	C₂×C₅	semidirect
20. C₂×C₃×D₃	C₂×C₅	semidirect
21. C₃×Dic₃	C₂×C₅	semidirect
31. #67	C₃×C₅	direct (N × H)
33. C₂²×C₅	C₃×D₃	semidirect
39. C₃×C₅	#67	direct (N × H)
40. C₃×C₅	C₃×D₄	semidirect
43. C₂²×C₃	C₅×D₃	semidirect
44. C₂²×C₃	C₂×C₃×C₅	semidirect
45. C₂×D₃	C₂×C₃×C₅	semidirect
47. Dic₃	C₂×C₃×C₅	semidirect
51. C₃²	C₅×D₄	semidirect
61. C₅	C₃×#67	direct (N × H)
63. C₂²	C₃×C₅×D₃	semidirect
66. C₃	C₅×#67	direct (N × H)
67. C₃	C₃×C₅×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₅	2
C₂×C₃×C₅×D₃	2
C₃×C₅×Dic₃	2
C₃×C₅×D₄	3
C₅×#67	3
C₃×#67	5

Subgroups — C3×C5×#67 #67169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#67 #67169