Subgroups — C₃×#6661 #10623741

Order 720.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃²×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	C₃×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₃×#689	—	3	1	no	maximal
5	#6661	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₃×C₅×Dic₃	—	2	1	no	—
8	C₃²×Dic₅	—	2	1	no	—
9	C₃×#216	—	5	1	no	maximal, Hall
10	C₃×C₅×Q₈	—	3	1	no	—
11	C₅×Dic₆	C₂×C₃	1	1	yes	characteristic
12	C₃×Dic₁₀	—	2	1	no	—
13	C₃×Dic₁₀	D₃	1	1	yes	characteristic
14	C₃×Dic₁₀	C₂×C₃	1	1	yes	characteristic
15	C₃⋊#168	C₂×C₃	1	1	yes	characteristic
16	C₃×#168	—	3	1	no	—
17	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
18	#689	—	3	1	no	Hall
19	C₃²×Q₈	—	5	1	no	—
20	C₃×Dic₆	D₅	1	1	yes	characteristic
21	C₃×C₃⋊C₈	—	5	1	no	—
22	C₃×C₄×C₅	—	2	1	no	—
23	C₃×C₄×C₅	—	6	1	no	—
24	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
25	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
26	C₃×Dic₅	—	2	2	no	—
27	C₃×Dic₅	—	2	1	no	—
28	C₃×Dic₅	—	2	1	no	—
29	C₅×Dic₃	—	2	1	no	—
30	C₃×Q₁₆	—	15	1	no	—
31	#216	—	5	1	no	—
32	C₃²×C₅	Q₁₆	1	1	yes	characteristic, complemented, Hall
33	C₅×Q₈	—	3	1	no	—
34	Dic₁₀	C₃×D₃	1	1	yes	characteristic
35	#168	—	3	1	no	—
36	C₃²×C₄	—	10	1	no	—
37	C₃²×C₄	C₂×D₅	1	1	yes	characteristic
38	C₃×Dic₃	—	2	1	no	—
39	C₂×C₃×C₅	—	2	1	no	—
40	C₂×C₃×C₅	#67	1	1	yes	characteristic
41	C₂×C₃×C₅	C₃×D₄	1	1	yes	characteristic
42	C₃×Q₈	—	3	1	no	—
43	C₃×Q₈	—	5	1	no	—
44	C₃×Q₈	—	5	1	no	—
45	C₃×Q₈	—	10	1	no	—
46	Dic₆	C₃×D₅	1	1	yes	characteristic
47	C₃×C₈	—	15	1	no	—
48	C₃⋊C₈	—	5	1	no	—
49	C₄×C₅	—	6	1	no	—
50	C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
51	Dic₅	—	2	1	no	—
52	C₂×C₃²	C₂²⋊D₅	1	1	yes	characteristic
53	Q₁₆	—	15	1	no	Sylow
54	C₃×C₅	—	2	1	no	—
55	C₃×C₅	#216	1	1	yes	characteristic, complemented
56	C₃×C₅	C₃×Q₁₆	1	1	yes	characteristic, complemented
57	C₃×C₄	—	2	1	no	—
58	C₃×C₄	—	6	1	no	—
59	C₃×C₄	—	10	2	no	—
60	C₃×C₄	—	10	1	no	—
61	C₃×C₄	—	10	1	no	—
62	C₃×C₄	D₃×D₅	1	1	yes	characteristic
63	C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
64	Dic₃	—	2	1	no	—
65	C₂×C₅	C₃×#67	1	1	yes	characteristic
66	C₃²	#689	1	1	yes	characteristic, complemented, Sylow
67	Q₈	—	3	1	no	—
68	Q₈	—	5	1	no	—
69	C₈	—	15	1	no	—
70	C₂×C₃	—	2	1	no	—
71	C₂×C₃	#1201	1	1	yes	characteristic
72	C₂×C₃	C₃×C₂²⋊D₅	1	1	yes	characteristic
73	C₅	C₃×#216	1	1	yes	characteristic, complemented, Sylow
74	C₄	—	6	1	no	—
75	C₄	—	10	1	no	—
76	C₄	C₃×D₃×D₅	1	1	yes	characteristic
77	C₃	—	2	1	no	—
78	C₃	#6661	1	1	yes	characteristic, direct product factor
79	C₃	C₃×#689	1	1	yes	characteristic, complemented
80	C₂	C₃×#1201	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#1201
commutator, cosocle	C₃×C₄×C₅	C₂²×C₃
Frattini	C₄	C₃×D₃×D₅
Fitting	C₃²×C₄×C₅	C₂²
socle, prime radical	C₂×C₃²×C₅	D₄
hypercenter	C₃×C₄	D₃×D₅
hypocenter	C₃×C₅	C₃×Q₁₆
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

Complements

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

N	H ≅ G/N	type
5. #6661	C₃	direct (N × H)
32. C₃²×C₅	Q₁₆	semidirect
55. C₃×C₅	#216	semidirect
56. C₃×C₅	C₃×Q₁₆	semidirect
66. C₃²	#689	semidirect
73. C₅	C₃×#216	semidirect
78. C₃	#6661	direct (N × H)
79. C₃	C₃×#689	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×Dic₆	2
C₃²×Dic₁₀	2
C₃×C₃⋊#168	2
C₃×#689	3
#6661	3
C₃×#216	5

Subgroups — C3×#6661 #10623741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#6661 #10623741