Subgroups — D₃²×D₅ #67207

Order 360.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₃⋊D₃)⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×D₁₅	C₂	1	2	yes	maximal, complemented
3	C₅×D₃²	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×D₃×D₅	C₂	1	2	yes	maximal, complemented
6	C₂×D₃×D₅	—	3	2	no	maximal
7	C₃⋊D₁₅	C₂²	1	1	yes	characteristic, complemented
8	C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic, complemented
9	C₃×D₁₅	C₂²	1	2	yes	complemented
10	C₃×C₅×D₃	C₂²	1	2	yes	complemented
11	C₃²×D₅	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₃²	—	5	1	no	maximal, Hall
13	C₂×D₁₅	—	3	2	no	—
14	C₂×C₅×D₃	—	3	2	no	—
15	C₂×C₃×D₅	—	3	2	no	—
16	D₃×D₅	—	3	2	no	—
17	D₃×D₅	—	3	2	no	—
18	D₃×D₅	—	3	2	no	—
19	D₃×D₅	—	6	1	no	—
20	D₃×D₅	D₃	1	2	yes	direct product factor
21	C₃²×C₅	C₂³	1	1	yes	characteristic, complemented, Hall
22	C₂²×D₅	—	9	1	no	Hall
23	C₂×C₃⋊D₃	—	5	1	no	—
24	C₂×C₃×D₃	—	5	2	no	—
25	D₃²	—	5	1	no	—
26	D₃²	—	5	2	no	—
27	D₃²	D₅	1	1	yes	characteristic, direct product factor
28	C₂×C₃×C₅	—	3	2	no	—
29	D₁₅	—	3	2	no	—
30	D₁₅	—	6	1	no	—
31	D₁₅	C₂×D₃	1	2	yes	complemented
32	C₃×D₅	—	2	1	no	—
33	C₃×D₅	—	3	2	no	—
34	C₃×D₅	C₂×D₃	1	2	yes	complemented
35	C₅×D₃	—	3	2	no	—
36	C₅×D₃	—	6	1	no	—
37	C₅×D₃	C₂×D₃	1	2	yes	complemented
38	C₂²×D₃	—	15	2	no	—
39	C₂²×C₅	—	9	1	no	—
40	C₂×D₅	—	3	2	no	—
41	C₂×D₅	—	9	1	no	—
42	C₂×D₅	—	9	2	no	—
43	C₂×D₅	—	9	1	no	—
44	C₂×C₃²	—	5	1	no	—
45	C₃⋊D₃	—	5	1	no	—
46	C₃⋊D₃	C₂×D₅	1	1	yes	characteristic, complemented
47	C₃×D₃	—	5	2	no	—
48	C₃×D₃	C₂×D₅	1	2	yes	complemented
49	C₃×C₅	—	2	1	no	—
50	C₃×C₅	C₂²×D₃	1	2	yes	complemented
51	C₂²×C₃	—	15	2	no	—
52	C₂×D₃	—	3	2	no	—
53	C₂×D₃	—	5	2	no	—
54	C₂×D₃	—	15	2	no	—
55	C₂×D₃	—	15	2	no	—
56	C₂×D₃	—	15	2	no	—
57	C₂×D₃	—	15	2	no	—
58	C₂×D₃	—	30	1	no	—
59	C₂×C₅	—	3	2	no	—
60	C₂×C₅	—	9	1	no	—
61	D₅	—	3	2	no	—
62	D₅	—	9	1	no	—
63	D₅	D₃²	1	1	yes	characteristic, direct product factor
64	C₃²	C₂²×D₅	1	1	yes	characteristic, complemented, Sylow
65	C₂³	—	45	1	no	Sylow
66	C₂×C₃	—	3	2	no	—
67	C₂×C₃	—	5	2	no	—
68	C₂×C₃	—	10	1	no	—
69	C₂×C₃	—	15	2	no	—
70	D₃	—	3	2	no	—
71	D₃	—	5	2	no	—
72	D₃	—	6	1	no	—
73	D₃	—	15	2	no	—
74	D₃	—	30	1	no	—
75	D₃	D₃×D₅	1	2	yes	direct product factor
76	C₅	C₂×D₃²	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	9	1	no	—
78	C₂²	—	15	2	no	—
79	C₂²	—	45	2	no	—
80	C₂²	—	45	1	no	—
81	C₂²	—	45	1	no	—
82	C₃	—	2	1	no	—
83	C₃	C₂×D₃×D₅	1	2	yes	complemented
84	C₂	—	3	2	no	—
85	C₂	—	5	1	no	—
86	C₂	—	9	1	no	—
87	C₂	—	15	2	no	—
88	C₂	—	45	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₅	C₂³
LCS	G ▷ C₃²×C₅
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. (C₃⋊D₃)⋊D₅	C₂	semidirect
2. D₃×D₁₅	C₂	semidirect
3. C₅×D₃²	C₂	semidirect
4. D₅×C₃⋊D₃	C₂	semidirect
5. C₃×D₃×D₅	C₂	semidirect
7. C₃⋊D₁₅	C₂²	semidirect
8. C₅×C₃⋊D₃	C₂²	semidirect
9. C₃×D₁₅	C₂²	semidirect
10. C₃×C₅×D₃	C₂²	semidirect
11. C₃²×D₅	C₂²	semidirect
20. D₃×D₅	D₃	direct (N × H)
21. C₃²×C₅	C₂³	semidirect
27. D₃²	D₅	direct (N × H)
31. D₁₅	C₂×D₃	semidirect
34. C₃×D₅	C₂×D₃	semidirect
37. C₅×D₃	C₂×D₃	semidirect
46. C₃⋊D₃	C₂×D₅	semidirect
48. C₃×D₃	C₂×D₅	semidirect
50. C₃×C₅	C₂²×D₃	semidirect
63. D₅	D₃²	direct (N × H)
64. C₃²	C₂²×D₅	semidirect
75. D₃	D₃×D₅	direct (N × H)
76. C₅	C₂×D₃²	semidirect
83. C₃	C₂×D₃×D₅	semidirect

Maximal subgroups

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

subgroup	index
(C₃⋊D₃)⋊D₅	2
D₃×D₁₅	2
C₅×D₃²	2
D₅×C₃⋊D₃	2
C₃×D₃×D₅	2
C₂×D₃×D₅	3
C₂×D₃²	5

Subgroups — D32×D5 #67207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃²×D₅ #67207