Subgroups — C₂×C₃×C₃⋊(C₃⋊D₁₅) #13108568

Order 1620.

872 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⁴×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃⋊(C₃⋊D₁₅)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⋊(C₃⋊D₁₅)	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃×C₃⋊D₁₅	—	3	13	no	maximal
5	C₃⁴×C₅	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₃×C₃⋊(C₃⋊D₃)	—	5	1	no	maximal, Hall
7	C₂×C₃³×C₅	—	2	13	no	—
8	C₂×C₃³×C₅	D₃	1	13	yes	complemented
9	C₂×C₃³×C₅	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃⋊(C₃⋊D₁₅)	C₂×C₃	1	2	yes	direct product factor
11	C₃×C₃⋊D₁₅	—	3	26	no	—
12	C₂×C₃⋊D₁₅	—	3	13	no	—
13	C₂×C₃×D₁₅	—	9	13	no	—
14	C₂×C₃⁴	D₅	1	1	yes	characteristic, complemented
15	C₃×C₃⋊(C₃⋊D₃)	—	5	2	no	—
16	C₃³×C₅	—	2	13	no	—
17	C₃³×C₅	C₂×D₃	1	13	yes	complemented
18	C₃³×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
19	C₂×C₃⋊(C₃⋊D₃)	—	5	1	no	—
20	C₂×C₃×C₃⋊D₃	—	15	13	no	—
21	C₂×C₃²×C₅	—	2	52	no	—
22	C₂×C₃²×C₅	C₃×D₃	1	13	yes	complemented
23	C₂×C₃²×C₅	C₃⋊D₃	1	13	yes	complemented
24	C₃⋊D₁₅	—	3	26	no	—
25	C₃×D₁₅	—	9	26	no	—
26	C₃⁴	C₂×D₅	1	1	yes	characteristic, complemented, Sylow
27	C₂×D₁₅	—	9	13	no	—
28	C₂×C₃×D₅	—	27	1	no	—
29	C₂×C₃³	—	2	13	no	—
30	C₂×C₃³	C₃×D₅	1	1	yes	characteristic, complemented
31	C₂×C₃³	D₁₅	1	13	yes	complemented
32	C₃⋊(C₃⋊D₃)	—	5	2	no	—
33	C₃×C₃⋊D₃	—	15	26	no	—
34	C₃²×C₅	—	2	52	no	—
35	C₃²×C₅	C₂×C₃×D₃	1	13	yes	complemented
36	C₃²×C₅	C₂×C₃⋊D₃	1	13	yes	complemented
37	C₂×C₃⋊D₃	—	15	13	no	—
38	C₂×C₃×D₃	—	45	13	no	—
39	C₂×C₃×C₅	—	2	13	no	—
40	C₂×C₃×C₅	C₃×C₃⋊D₃	1	13	yes	complemented
41	C₂×C₃×C₅	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
42	D₁₅	—	9	26	no	—
43	C₃×D₅	—	27	2	no	—
44	C₃³	—	2	13	no	—
45	C₃³	C₂×C₃×D₅	1	1	yes	characteristic, complemented
46	C₃³	C₂×D₁₅	1	13	yes	complemented
47	C₂×D₅	—	27	1	no	Hall
48	C₂×C₃²	—	2	52	no	—
49	C₂×C₃²	C₃×D₁₅	1	13	yes	complemented
50	C₂×C₃²	C₃⋊D₁₅	1	13	yes	complemented
51	C₃⋊D₃	—	15	26	no	—
52	C₃×D₃	—	45	26	no	—
53	C₃×C₅	—	2	13	no	—
54	C₃×C₅	C₂×C₃×C₃⋊D₃	1	13	yes	complemented
55	C₃×C₅	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
56	C₂²×C₃	—	135	1	no	—
57	C₂×D₃	—	45	13	no	—
58	C₂×C₅	C₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
59	D₅	—	27	2	no	—
60	C₃²	—	2	52	no	—
61	C₃²	C₂×C₃×D₁₅	1	13	yes	complemented
62	C₃²	C₂×C₃⋊D₁₅	1	13	yes	complemented
63	C₂×C₃	—	2	13	no	—
64	C₂×C₃	—	135	2	no	—
65	C₂×C₃	C₃×C₃⋊D₁₅	1	13	yes	complemented
66	C₂×C₃	C₃⋊(C₃⋊D₁₅)	1	1	yes	characteristic, direct product factor
67	D₃	—	45	26	no	—
68	C₅	C₂×C₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented, Sylow
69	C₂²	—	135	1	no	Sylow
70	C₃	—	2	13	no	—
71	C₃	C₂×C₃×C₃⋊D₁₅	1	13	yes	complemented
72	C₃	C₂×C₃⋊(C₃⋊D₁₅)	1	1	yes	characteristic, direct product factor
73	C₂	—	135	2	no	—
74	C₂	C₃×C₃⋊(C₃⋊D₁₅)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	C₃⋊(C₃⋊D₁₅)
commutator, cosocle, hypocenter	C₃³×C₅	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃⁴×C₅	C₂
LCS	G ▷ C₃³×C₅
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃⁴×C₅	C₂	semidirect
2. C₃×C₃⋊(C₃⋊D₁₅)	C₂	direct (N × H)
3. C₂×C₃⋊(C₃⋊D₁₅)	C₃	direct (N × H)
5. C₃⁴×C₅	C₂²	semidirect
8. C₂×C₃³×C₅	D₃	semidirect
9. C₂×C₃³×C₅	C₂×C₃	semidirect
10. C₃⋊(C₃⋊D₁₅)	C₂×C₃	direct (N × H)
14. C₂×C₃⁴	D₅	semidirect
17. C₃³×C₅	C₂×D₃	semidirect
18. C₃³×C₅	C₂²×C₃	semidirect
22. C₂×C₃²×C₅	C₃×D₃	semidirect
23. C₂×C₃²×C₅	C₃⋊D₃	semidirect
26. C₃⁴	C₂×D₅	semidirect
30. C₂×C₃³	C₃×D₅	semidirect
31. C₂×C₃³	D₁₅	semidirect
35. C₃²×C₅	C₂×C₃×D₃	semidirect
36. C₃²×C₅	C₂×C₃⋊D₃	semidirect
40. C₂×C₃×C₅	C₃×C₃⋊D₃	semidirect
41. C₂×C₃×C₅	C₃⋊(C₃⋊D₃)	semidirect
45. C₃³	C₂×C₃×D₅	semidirect
46. C₃³	C₂×D₁₅	semidirect
49. C₂×C₃²	C₃×D₁₅	semidirect
50. C₂×C₃²	C₃⋊D₁₅	semidirect
54. C₃×C₅	C₂×C₃×C₃⋊D₃	semidirect
55. C₃×C₅	C₂×C₃⋊(C₃⋊D₃)	semidirect
58. C₂×C₅	C₃×C₃⋊(C₃⋊D₃)	semidirect
61. C₃²	C₂×C₃×D₁₅	semidirect
62. C₃²	C₂×C₃⋊D₁₅	semidirect
65. C₂×C₃	C₃×C₃⋊D₁₅	semidirect
66. C₂×C₃	C₃⋊(C₃⋊D₁₅)	direct (N × H)
68. C₅	C₂×C₃×C₃⋊(C₃⋊D₃)	semidirect
71. C₃	C₂×C₃×C₃⋊D₁₅	semidirect
72. C₃	C₂×C₃⋊(C₃⋊D₁₅)	direct (N × H)
74. C₂	C₃×C₃⋊(C₃⋊D₁₅)	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×C₃⁴×C₅	2
C₃×C₃⋊(C₃⋊D₁₅)	2
C₂×C₃⋊(C₃⋊D₁₅)	3
C₂×C₃×C₃⋊D₁₅	3
C₂×C₃×C₃⋊(C₃⋊D₃)	5

Subgroups — C2×C3×C3⋊(C3⋊D15) #13108568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₃⋊(C₃⋊D₁₅) #13108568