Subgroups — C₃×#6654 #10623734

Order 720.

96 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, complemented
2	C₃×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#686	—	3	1	no	maximal
5	#6654	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×D₁₅	—	2	1	no	—
7	C₂×C₃×C₅×D₃	—	2	1	no	—
8	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₃×D₂₄	—	5	1	no	maximal, Hall
10	C₃×C₅×D₄	—	3	1	no	—
11	D₆₀	C₂×C₃	1	1	yes	characteristic, complemented
12	C₅×D₁₂	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃×D₂₀	—	3	1	no	—
14	C₃×#168	—	2	1	no	—
15	C₃×#168	D₃	1	1	yes	characteristic, complemented
16	C₃×#168	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
18	C₃×D₁₅	—	4	1	no	—
19	C₃×C₅×D₃	—	4	1	no	—
20	#686	—	3	1	no	Hall
21	C₃×D₁₂	—	5	1	no	—
22	C₃×D₁₂	D₅	1	1	yes	characteristic, complemented
23	C₃²×C₈	—	5	1	no	—
24	C₂²×C₃×C₅	—	6	1	no	—
25	C₂×D₁₅	—	2	1	no	—
26	C₂×C₅×D₃	—	2	1	no	—
27	C₂×C₃×D₅	—	6	1	no	—
28	C₃×C₄×C₅	—	2	1	no	—
29	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
30	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
31	C₃×D₈	—	15	1	no	—
32	D₂₄	—	5	1	no	—
33	C₃²×C₅	D₈	1	1	yes	characteristic, complemented, Hall
34	C₅×D₄	—	3	1	no	—
35	D₂₀	—	3	1	no	—
36	#168	C₃×D₃	1	1	yes	characteristic, complemented
37	C₂×C₃×D₃	—	2	1	no	—
38	C₂×C₃×D₃	—	10	1	no	—
39	C₃²×C₄	C₂×D₅	1	1	yes	characteristic
40	C₂×C₃×C₅	—	2	1	no	—
41	C₂×C₃×C₅	—	12	1	no	—
42	C₂×C₃×C₅	D₁₂	1	1	yes	characteristic
43	C₂×C₃×C₅	C₃×D₄	1	1	yes	characteristic
44	D₁₅	—	4	1	no	—
45	C₃×D₅	—	12	1	no	—
46	C₅×D₃	—	4	1	no	—
47	C₃×D₄	—	3	1	no	—
48	C₃×D₄	—	15	1	no	—
49	D₁₂	—	5	1	no	—
50	D₁₂	C₃×D₅	1	1	yes	characteristic, complemented
51	C₃×C₈	—	5	1	no	—
52	C₃×C₈	—	5	1	no	—
53	C₃×C₈	—	10	1	no	—
54	C₂²×C₅	—	6	1	no	—
55	C₂×D₅	—	6	1	no	—
56	C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
57	C₂×C₃²	C₂²⋊D₅	1	1	yes	characteristic
58	C₃×D₃	—	4	1	no	—
59	C₃×D₃	—	20	1	no	—
60	D₈	—	15	1	no	Sylow
61	C₃×C₅	—	2	1	no	—
62	C₃×C₅	D₂₄	1	1	yes	characteristic, complemented
63	C₃×C₅	C₃×D₈	1	1	yes	characteristic, complemented
64	C₂²×C₃	—	6	1	no	—
65	C₂²×C₃	—	30	1	no	—
66	C₂×D₃	—	2	1	no	—
67	C₂×D₃	—	10	1	no	—
68	C₃×C₄	—	2	1	no	—
69	C₃×C₄	D₃×D₅	1	1	yes	characteristic
70	C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
71	C₂×C₅	—	12	1	no	—
72	C₂×C₅	C₃×D₁₂	1	1	yes	characteristic
73	D₅	—	12	1	no	—
74	C₃²	#686	1	1	yes	characteristic, complemented, Sylow
75	D₄	—	3	1	no	—
76	D₄	—	15	1	no	—
77	C₈	—	5	1	no	—
78	C₂×C₃	—	2	1	no	—
79	C₂×C₃	—	12	1	no	—
80	C₂×C₃	—	60	1	no	—
81	C₂×C₃	#1203	1	1	yes	characteristic
82	C₂×C₃	C₃×C₂²⋊D₅	1	1	yes	characteristic
83	D₃	—	4	1	no	—
84	D₃	—	20	1	no	—
85	C₅	C₃×D₂₄	1	1	yes	characteristic, complemented, Sylow
86	C₂²	—	6	1	no	—
87	C₂²	—	30	1	no	—
88	C₄	C₃×D₃×D₅	1	1	yes	characteristic
89	C₃	—	2	1	no	—
90	C₃	#6654	1	1	yes	characteristic, direct product factor
91	C₃	C₃×#686	1	1	yes	characteristic, complemented
92	C₂	—	12	1	no	—
93	C₂	—	60	1	no	—
94	C₂	C₃×#1203	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#1203
commutator, cosocle	C₃×C₄×C₅	C₂²×C₃
Frattini	C₄	C₃×D₃×D₅
Fitting	C₃²×C₄×C₅	C₂²
socle	C₂×C₃²×C₅	D₄
hypercenter	C₃×C₄	D₃×D₅
hypocenter	C₃×C₅	C₃×D₈
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
1. C₃×D₆₀	C₂	semidirect
2. C₃×C₅×D₁₂	C₂	semidirect
3. C₃²×#168	C₂	semidirect
5. #6654	C₃	direct (N × H)
11. D₆₀	C₂×C₃	semidirect
12. C₅×D₁₂	C₂×C₃	semidirect
15. C₃×#168	D₃	semidirect
16. C₃×#168	C₂×C₃	semidirect
22. C₃×D₁₂	D₅	semidirect
33. C₃²×C₅	D₈	semidirect
36. #168	C₃×D₃	semidirect
50. D₁₂	C₃×D₅	semidirect
62. C₃×C₅	D₂₄	semidirect
63. C₃×C₅	C₃×D₈	semidirect
74. C₃²	#686	semidirect
85. C₅	C₃×D₂₄	semidirect
90. C₃	#6654	direct (N × H)
91. C₃	C₃×#686	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
C₃×C₅×D₁₂	2
C₃²×#168	2
C₃×#686	3
#6654	3
C₃×D₂₄	5

Subgroups — C3×#6654 #10623734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#6654 #10623734