Subgroups — C₅×Dic₃×C₃⋊D₃ #11762066

Order 1080.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₅×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×C₃⋊D₃	—	3	1	no	maximal
5	C₅×D₃×Dic₃	—	3	4	no	maximal
6	C₂×C₃³×C₅	C₂²	1	1	yes	characteristic
7	C₃×C₅×C₃⋊D₃	C₄	1	2	yes	complemented
8	Dic₃×C₃⋊D₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₅×C₃⋊D₃	D₃	1	1	yes	characteristic
10	C₂×C₃×C₅×D₃	—	3	4	no	—
11	C₃²×C₄×C₅	—	3	1	no	—
12	C₅×C₃⋊Dic₃	—	3	1	no	—
13	C₅×C₃⋊Dic₃	—	3	4	no	—
14	C₅×C₃⋊Dic₃	—	6	4	no	—
15	C₃×C₅×Dic₃	D₃	1	4	yes	complemented
16	C₃³×C₅	C₂×C₄	1	1	yes	characteristic, complemented, Hall
17	C₂×C₅×Dic₃	—	9	1	no	—
18	C₄×C₅×D₃	—	9	4	no	—
19	C₂×C₃×C₃⋊D₃	C₂×C₅	1	1	yes	characteristic
20	C₃⋊(C₃⋊Dic₃)	C₂×C₅	1	1	yes	characteristic, complemented
21	C₃²×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
22	C₂×C₃²×C₅	—	2	4	no	—
23	C₂×C₃²×C₅	C₂×D₃	1	4	yes	—
24	C₂×C₃²×C₅	C₂×D₃	1	1	yes	characteristic
25	C₅×C₃⋊D₃	Dic₃	1	2	yes	direct product factor
26	C₃×C₅×D₃	—	3	8	no	—
27	C₄×C₃⋊D₃	—	3	1	no	—
28	D₃×Dic₃	—	3	4	no	—
29	C₂²×C₃×C₅	—	9	1	no	—
30	C₂×C₅×D₃	—	3	4	no	—
31	C₃×C₄×C₅	—	3	4	no	—
32	C₅×Dic₃	—	9	4	no	—
33	C₅×Dic₃	—	9	1	no	—
34	C₅×Dic₃	—	18	4	no	—
35	C₅×Dic₃	C₃⋊D₃	1	1	yes	characteristic, direct product factor
36	C₂×C₃³	C₂²×C₅	1	1	yes	characteristic
37	C₃×C₃⋊D₃	C₄×C₅	1	2	yes	complemented
38	C₃²×C₅	—	2	4	no	—
39	C₃²×C₅	C₄×D₃	1	4	yes	complemented
40	C₃²×C₅	C₂×Dic₃	1	1	yes	characteristic, complemented
41	C₂×C₄×C₅	—	27	1	no	Hall
42	C₂×C₃⋊D₃	C₅×D₃	1	1	yes	characteristic
43	C₂×C₃×D₃	—	3	4	no	—
44	C₃²×C₄	—	3	1	no	—
45	C₃⋊Dic₃	—	3	1	no	—
46	C₃⋊Dic₃	—	3	4	no	—
47	C₃⋊Dic₃	—	6	4	no	—
48	C₃×Dic₃	C₅×D₃	1	4	yes	complemented
49	C₂×C₃×C₅	—	2	4	no	—
50	C₂×C₃×C₅	—	9	2	no	—
51	C₂×C₃×C₅	D₃²	1	4	yes	—
52	C₂×C₃×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₅×D₃	—	3	8	no	—
54	C₃³	C₂×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
55	C₂×Dic₃	—	9	1	no	—
56	C₄×D₃	—	9	4	no	—
57	C₂²×C₅	—	9	1	no	—
58	C₄×C₅	—	3	1	no	—
59	C₄×C₅	—	27	1	no	—
60	C₂×C₃²	—	2	4	no	—
61	C₂×C₃²	C₂×C₅×D₃	1	4	yes	—
62	C₂×C₃²	C₂×C₅×D₃	1	1	yes	characteristic
63	C₃⋊D₃	C₅×Dic₃	1	2	yes	direct product factor
64	C₃×D₃	—	3	8	no	—
65	C₃×C₅	—	2	4	no	—
66	C₃×C₅	D₃×Dic₃	1	4	yes	complemented
67	C₃×C₅	C₄×C₃⋊D₃	1	1	yes	characteristic, complemented
68	C₂²×C₃	—	9	1	no	—
69	C₂×D₃	—	3	4	no	—
70	C₃×C₄	—	3	4	no	—
71	Dic₃	—	9	4	no	—
72	Dic₃	—	9	1	no	—
73	Dic₃	—	18	4	no	—
74	Dic₃	C₅×C₃⋊D₃	1	1	yes	characteristic, direct product factor
75	C₂×C₅	—	9	2	no	—
76	C₂×C₅	D₃×C₃⋊D₃	1	1	yes	characteristic
77	C₃²	—	2	4	no	—
78	C₃²	C₄×C₅×D₃	1	4	yes	complemented
79	C₃²	C₂×C₅×Dic₃	1	1	yes	characteristic, complemented
80	C₂×C₄	—	27	1	no	Sylow
81	C₂×C₃	—	2	4	no	—
82	C₂×C₃	—	9	2	no	—
83	C₂×C₃	C₅×D₃²	1	4	yes	—
84	C₂×C₃	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic
85	D₃	—	3	8	no	—
86	C₅	Dic₃×C₃⋊D₃	1	1	yes	characteristic, direct product factor, Sylow
87	C₂²	—	9	1	no	—
88	C₄	—	3	1	no	—
89	C₄	—	27	1	no	—
90	C₃	—	2	4	no	—
91	C₃	C₅×D₃×Dic₃	1	4	yes	complemented
92	C₃	C₄×C₅×C₃⋊D₃	1	1	yes	characteristic, complemented
93	C₂	—	9	2	no	—
94	C₂	C₅×D₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	D₃×C₃⋊D₃
commutator, hypocenter	C₃³	C₂×C₄×C₅
Frattini	C₂	C₅×D₃×C₃⋊D₃
Fitting, socle	C₂×C₃³×C₅	C₂²
prime radical	C₂×C₃×C₅×C₃⋊D₃	C₂
cosocle	C₂×C₃³	C₂²×C₅
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
3. C₃²×C₅×Dic₃	C₂	semidirect
7. C₃×C₅×C₃⋊D₃	C₄	semidirect
8. Dic₃×C₃⋊D₃	C₅	direct (N × H)
15. C₃×C₅×Dic₃	D₃	semidirect
16. C₃³×C₅	C₂×C₄	semidirect
20. C₃⋊(C₃⋊Dic₃)	C₂×C₅	semidirect
21. C₃²×Dic₃	C₂×C₅	semidirect
25. C₅×C₃⋊D₃	Dic₃	direct (N × H)
35. C₅×Dic₃	C₃⋊D₃	direct (N × H)
37. C₃×C₃⋊D₃	C₄×C₅	semidirect
39. C₃²×C₅	C₄×D₃	semidirect
40. C₃²×C₅	C₂×Dic₃	semidirect
48. C₃×Dic₃	C₅×D₃	semidirect
54. C₃³	C₂×C₄×C₅	semidirect
63. C₃⋊D₃	C₅×Dic₃	direct (N × H)
66. C₃×C₅	D₃×Dic₃	semidirect
67. C₃×C₅	C₄×C₃⋊D₃	semidirect
74. Dic₃	C₅×C₃⋊D₃	direct (N × H)
78. C₃²	C₄×C₅×D₃	semidirect
79. C₃²	C₂×C₅×Dic₃	semidirect
86. C₅	Dic₃×C₃⋊D₃	direct (N × H)
91. C₃	C₅×D₃×Dic₃	semidirect
92. C₃	C₄×C₅×C₃⋊D₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₃⋊D₃	2
C₅×C₃⋊(C₃⋊Dic₃)	2
C₃²×C₅×Dic₃	2
C₄×C₅×C₃⋊D₃	3
C₅×D₃×Dic₃	3
Dic₃×C₃⋊D₃	5

Subgroups — C5×Dic3×C3⋊D3 #11762066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×Dic₃×C₃⋊D₃ #11762066