Subgroups — C₂×C₃×D₃×D₅ #67221

Order 360.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₃×D₅	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₃×D₅	—	3	1	no	maximal
6	C₂×D₃×D₅	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic, complemented
8	C₃×D₁₅	C₂²	1	2	yes	complemented
9	C₃×C₅×D₃	C₂²	1	2	yes	complemented
10	C₃²×D₅	C₂²	1	2	yes	complemented
11	C₂²×C₃×D₃	—	5	1	no	maximal, Hall
12	C₂²×C₃×C₅	—	3	1	no	—
13	C₂×D₁₅	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂×C₅×D₃	C₂×C₃	1	1	yes	characteristic, complemented
15	C₂×C₃×D₅	—	2	1	no	—
16	C₂×C₃×D₅	—	3	1	no	—
17	C₂×C₃×D₅	—	3	4	no	—
18	C₂×C₃×D₅	D₃	1	1	yes	characteristic, direct product factor
19	C₂×C₃×D₅	C₂×C₃	1	1	yes	characteristic, complemented
20	D₃×D₅	C₂×C₃	1	4	yes	direct product factor
21	C₃²×C₅	C₂³	1	1	yes	characteristic, complemented, Hall
22	C₂²×D₅	—	3	1	no	Hall
23	C₂²×C₃²	—	5	1	no	—
24	C₂×C₃×D₃	—	5	1	no	—
25	C₂×C₃×D₃	—	5	4	no	—
26	C₂×C₃×D₃	D₅	1	1	yes	characteristic, direct product factor
27	C₂×C₃×C₅	—	2	1	no	—
28	C₂×C₃×C₅	—	3	2	no	—
29	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic, complemented
30	C₂×C₃×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
31	D₁₅	C₂²×C₃	1	2	yes	complemented
32	C₃×D₅	—	2	2	no	—
33	C₃×D₅	—	3	2	no	—
34	C₃×D₅	C₂×D₃	1	2	yes	direct product factor
35	C₃×D₅	C₂²×C₃	1	2	yes	complemented
36	C₅×D₃	C₂²×C₃	1	2	yes	complemented
37	C₂³×C₃	—	15	1	no	—
38	C₂²×D₃	—	5	1	no	—
39	C₂²×C₅	—	3	1	no	—
40	C₂×D₅	—	3	1	no	—
41	C₂×D₅	—	3	4	no	—
42	C₂×D₅	C₃×D₃	1	1	yes	characteristic, direct product factor
43	C₂×C₃²	—	5	2	no	—
44	C₂×C₃²	C₂×D₅	1	1	yes	characteristic, complemented
45	C₃×D₃	—	5	2	no	—
46	C₃×D₃	C₂×D₅	1	2	yes	direct product factor
47	C₃×C₅	—	2	1	no	—
48	C₃×C₅	C₂²×D₃	1	1	yes	characteristic, complemented
49	C₃×C₅	C₂³×C₃	1	1	yes	characteristic, complemented
50	C₂²×C₃	—	3	1	no	—
51	C₂²×C₃	—	5	1	no	—
52	C₂²×C₃	—	5	1	no	—
53	C₂²×C₃	—	10	1	no	—
54	C₂²×C₃	—	15	4	no	—
55	C₂²×C₃	—	15	1	no	—
56	C₂×D₃	—	5	1	no	—
57	C₂×D₃	—	5	4	no	—
58	C₂×D₃	C₃×D₅	1	1	yes	characteristic, direct product factor
59	C₂×C₅	—	3	2	no	—
60	C₂×C₅	C₂×C₃×D₃	1	1	yes	characteristic, complemented
61	D₅	—	3	2	no	—
62	D₅	C₂×C₃×D₃	1	2	yes	direct product factor
63	C₃²	C₂²×D₅	1	1	yes	characteristic, complemented, Sylow
64	C₂³	—	15	1	no	Sylow
65	C₂×C₃	—	2	1	no	—
66	C₂×C₃	—	3	2	no	—
67	C₂×C₃	—	5	2	no	—
68	C₂×C₃	—	5	2	no	—
69	C₂×C₃	—	10	2	no	—
70	C₂×C₃	—	15	2	no	—
71	C₂×C₃	D₃×D₅	1	1	yes	characteristic, direct product factor
72	C₂×C₃	C₂×C₃×D₅	1	1	yes	characteristic, complemented
73	D₃	—	5	2	no	—
74	D₃	C₂×C₃×D₅	1	2	yes	direct product factor
75	C₅	C₂²×C₃×D₃	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	3	1	no	—
77	C₂²	—	5	1	no	—
78	C₂²	—	15	4	no	—
79	C₂²	—	15	1	no	—
80	C₃	—	2	1	no	—
81	C₃	C₂×D₃×D₅	1	1	yes	characteristic, direct product factor
82	C₃	C₂²×C₃×D₅	1	1	yes	characteristic, complemented
83	C₂	—	3	2	no	—
84	C₂	—	5	2	no	—
85	C₂	—	15	2	no	—
86	C₂	C₃×D₃×D₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×D₅
commutator, cosocle, hypocenter	C₃×C₅	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₁₅	C₂	semidirect
2. C₂×C₃×C₅×D₃	C₂	semidirect
3. C₂×C₃²×D₅	C₂	semidirect
4. C₃×D₃×D₅	C₂	direct (N × H)
6. C₂×D₃×D₅	C₃	direct (N × H)
7. C₂×C₃²×C₅	C₂²	semidirect
8. C₃×D₁₅	C₂²	semidirect
9. C₃×C₅×D₃	C₂²	semidirect
10. C₃²×D₅	C₂²	semidirect
13. C₂×D₁₅	C₂×C₃	semidirect
14. C₂×C₅×D₃	C₂×C₃	semidirect
18. C₂×C₃×D₅	D₃	direct (N × H)
19. C₂×C₃×D₅	C₂×C₃	semidirect
20. D₃×D₅	C₂×C₃	direct (N × H)
21. C₃²×C₅	C₂³	semidirect
26. C₂×C₃×D₃	D₅	direct (N × H)
29. C₂×C₃×C₅	C₂×D₃	semidirect
30. C₂×C₃×C₅	C₂²×C₃	semidirect
31. D₁₅	C₂²×C₃	semidirect
34. C₃×D₅	C₂×D₃	direct (N × H)
35. C₃×D₅	C₂²×C₃	semidirect
36. C₅×D₃	C₂²×C₃	semidirect
42. C₂×D₅	C₃×D₃	direct (N × H)
44. C₂×C₃²	C₂×D₅	semidirect
46. C₃×D₃	C₂×D₅	direct (N × H)
48. C₃×C₅	C₂²×D₃	semidirect
49. C₃×C₅	C₂³×C₃	semidirect
58. C₂×D₃	C₃×D₅	direct (N × H)
60. C₂×C₅	C₂×C₃×D₃	semidirect
62. D₅	C₂×C₃×D₃	direct (N × H)
63. C₃²	C₂²×D₅	semidirect
71. C₂×C₃	D₃×D₅	direct (N × H)
72. C₂×C₃	C₂×C₃×D₅	semidirect
74. D₃	C₂×C₃×D₅	direct (N × H)
75. C₅	C₂²×C₃×D₃	semidirect
81. C₃	C₂×D₃×D₅	direct (N × H)
82. C₃	C₂²×C₃×D₅	semidirect
86. C₂	C₃×D₃×D₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×D₁₅	2
C₂×C₃×C₅×D₃	2
C₂×C₃²×D₅	2
C₃×D₃×D₅	2
C₂²×C₃×D₅	3
C₂×D₃×D₅	3
C₂²×C₃×D₃	5

Subgroups — C2×C3×D3×D5 #67221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₃×D₅ #67221