Subgroups — C₃×#6658 #10623738

Order 720.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₆₀	C₂	1	1	yes	maximal, characteristic
2	C₃²×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#677	—	3	1	no	maximal
5	#6658	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×D₁₅	—	2	1	no	—
7	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃²×Dic₅	—	2	1	no	—
9	C₃×#215	—	5	1	no	maximal, Hall
10	D₆₀	C₂×C₃	1	1	yes	characteristic
11	C₃×D₂₀	—	3	1	no	—
12	C₃×Dic₁₀	—	2	1	no	—
13	C₃×Dic₁₀	D₃	1	1	yes	characteristic, complemented
14	C₃×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
15	C₃×C₅×C₈	—	3	1	no	—
16	C₅×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
18	C₃×D₁₅	—	4	1	no	—
19	#677	—	3	1	no	Hall
20	C₃²×Q₈	—	5	1	no	—
21	C₃×D₁₂	—	5	1	no	—
22	C₃×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
23	C₂×D₁₅	—	2	1	no	—
24	C₂×C₃×D₅	—	6	1	no	—
25	C₃×C₄×C₅	—	2	1	no	—
26	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
27	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
28	C₃×Dic₅	—	2	2	no	—
29	C₃×Dic₅	—	2	1	no	—
30	C₃×Dic₅	—	2	1	no	—
31	C₃×SD₁₆	—	15	1	no	—
32	#215	—	5	1	no	—
33	C₃²×C₅	SD₁₆	1	1	yes	characteristic, complemented, Hall
34	D₂₀	—	3	1	no	—
35	Dic₁₀	C₃×D₃	1	1	yes	characteristic, complemented
36	C₅×C₈	—	3	1	no	—
37	C₂×C₃×D₃	—	10	1	no	—
38	C₃²×C₄	—	10	1	no	—
39	C₃²×C₄	C₂×D₅	1	1	yes	characteristic
40	C₂×C₃×C₅	—	2	1	no	—
41	C₂×C₃×C₅	#67	1	1	yes	characteristic
42	C₂×C₃×C₅	C₃×D₄	1	1	yes	characteristic
43	D₁₅	—	4	1	no	—
44	C₃×D₅	—	12	1	no	—
45	C₃×Q₈	—	5	1	no	—
46	C₃×Q₈	—	5	1	no	—
47	C₃×Q₈	—	10	1	no	—
48	C₃×D₄	—	15	1	no	—
49	D₁₂	—	5	1	no	—
50	C₃×C₈	—	3	1	no	—
51	C₃⋊C₈	C₃×D₅	1	1	yes	characteristic, complemented
52	C₂×D₅	—	6	1	no	—
53	C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
54	Dic₅	—	2	1	no	—
55	C₂×C₃²	D₂₀	1	1	yes	characteristic
56	C₃×D₃	—	20	1	no	—
57	SD₁₆	—	15	1	no	Sylow
58	C₃×C₅	—	2	1	no	—
59	C₃×C₅	#215	1	1	yes	characteristic, complemented
60	C₃×C₅	C₃×SD₁₆	1	1	yes	characteristic, complemented
61	C₂²×C₃	—	30	1	no	—
62	C₂×D₃	—	10	1	no	—
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	—	10	2	no	—
65	C₃×C₄	—	10	1	no	—
66	C₃×C₄	—	10	1	no	—
67	C₃×C₄	D₃×D₅	1	1	yes	characteristic
68	C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
69	C₂×C₅	C₃×#67	1	1	yes	characteristic
70	D₅	—	12	1	no	—
71	C₃²	#677	1	1	yes	characteristic, complemented, Sylow
72	Q₈	—	5	1	no	—
73	D₄	—	15	1	no	—
74	C₈	—	3	1	no	—
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	—	60	1	no	—
77	C₂×C₃	#1202	1	1	yes	characteristic
78	C₂×C₃	C₃×D₂₀	1	1	yes	characteristic
79	D₃	—	20	1	no	—
80	C₅	C₃×#215	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	30	1	no	—
82	C₄	—	10	1	no	—
83	C₄	C₃×D₃×D₅	1	1	yes	characteristic
84	C₃	—	2	1	no	—
85	C₃	#6658	1	1	yes	characteristic, direct product factor
86	C₃	C₃×#677	1	1	yes	characteristic, complemented
87	C₂	—	60	1	no	—
88	C₂	C₃×#1202	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#1202
commutator, cosocle	C₃×C₄×C₅	C₂²×C₃
Frattini	C₄	C₃×D₃×D₅
Fitting	C₃²×C₄×C₅	C₂²
socle	C₂×C₃²×C₅	D₄
prime radical	C₃×D₆₀	C₂
hypercenter	C₃×C₄	D₃×D₅
hypocenter	C₃×C₅	C₃×SD₁₆
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
2. C₃²×Dic₁₀	C₂	semidirect
3. C₃×C₅×C₃⋊C₈	C₂	semidirect
5. #6658	C₃	direct (N × H)
13. C₃×Dic₁₀	D₃	semidirect
14. C₃×Dic₁₀	C₂×C₃	semidirect
16. C₅×C₃⋊C₈	C₂×C₃	semidirect
22. C₃×C₃⋊C₈	D₅	semidirect
33. C₃²×C₅	SD₁₆	semidirect
35. Dic₁₀	C₃×D₃	semidirect
51. C₃⋊C₈	C₃×D₅	semidirect
59. C₃×C₅	#215	semidirect
60. C₃×C₅	C₃×SD₁₆	semidirect
71. C₃²	#677	semidirect
80. C₅	C₃×#215	semidirect
85. C₃	#6658	direct (N × H)
86. C₃	C₃×#677	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₆₀	2
C₃²×Dic₁₀	2
C₃×C₅×C₃⋊C₈	2
C₃×#677	3
#6658	3
C₃×#215	5

Subgroups — C3×#6658 #10623738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#6658 #10623738