Subgroups — #10623805

Order 720.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#67180	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#6716	—	3	1	no	maximal
5	#6659	—	3	1	no	maximal
6	C₂×C₃×C₅×D₃	—	2	1	no	—
7	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃⋊Dic₁₅	—	2	1	no	—
9	#3716	—	5	1	no	maximal, Hall
10	C₃×C₅×D₄	—	3	1	no	—
11	Dic₃₀	—	3	1	no	—
12	Dic₃₀	—	3	1	no	—
13	Dic₃₀	—	6	1	no	—
14	C₅×D₁₂	D₃	1	1	yes	characteristic
15	C₃⋊#168	D₃	1	1	yes	characteristic, complemented
16	C₃×#168	—	3	1	no	—
17	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
18	C₃×C₅×D₃	—	4	1	no	—
19	#687	—	9	1	no	Hall
20	#634	—	5	1	no	—
21	C₃×D₁₂	D₅	1	1	yes	characteristic
22	C₃×C₃⋊C₈	—	5	1	no	—
23	C₂²×C₃×C₅	—	6	1	no	—
24	C₂×C₅×D₃	—	2	1	no	—
25	C₃×C₄×C₅	—	2	1	no	—
26	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
27	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
28	Dic₁₅	—	6	2	no	—
29	Dic₁₅	—	6	1	no	—
30	Dic₁₅	—	6	1	no	—
31	#214	—	15	1	no	—
32	#204	—	15	1	no	—
33	C₃²×C₅	SD₁₆	1	1	yes	characteristic, complemented, Hall
34	C₅×D₄	—	3	1	no	—
35	Dic₁₀	—	9	1	no	—
36	#168	—	3	1	no	—
37	C₂×C₃×D₃	—	2	1	no	—
38	C₃²×C₄	C₂×D₅	1	1	yes	characteristic
39	C₃⋊Dic₃	—	10	1	no	—
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	#67	1	1	yes	characteristic
44	C₅×D₃	—	4	1	no	—
45	C₃×D₄	—	3	1	no	—
46	D₁₂	D₁₅	1	1	yes	characteristic
47	Dic₆	—	15	1	no	—
48	Dic₆	—	15	1	no	—
49	Dic₆	—	30	1	no	—
50	C₃×C₈	—	15	1	no	—
51	C₃⋊C₈	—	5	1	no	—
52	C₂²×C₅	—	6	1	no	—
53	C₄×C₅	D₃²	1	1	yes	characteristic
54	Dic₅	—	18	1	no	—
55	C₂×C₃²	C₂²⋊D₅	1	1	yes	characteristic
56	C₃×D₃	—	4	1	no	—
57	SD₁₆	—	45	1	no	Sylow
58	C₃×C₅	—	2	1	no	—
59	C₃×C₅	#204	1	1	yes	characteristic, complemented
60	C₃×C₅	#214	1	1	yes	characteristic, complemented
61	C₂²×C₃	—	6	1	no	—
62	C₂×D₃	—	2	1	no	—
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	D₃×D₅	1	1	yes	characteristic
65	C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
66	Dic₃	—	30	2	no	—
67	Dic₃	—	30	1	no	—
68	Dic₃	—	30	1	no	—
69	C₂×C₅	—	12	1	no	—
70	C₂×C₅	#626	1	1	yes	characteristic
71	C₃²	#687	1	1	yes	characteristic, complemented, Sylow
72	Q₈	—	45	1	no	—
73	D₄	—	3	1	no	—
74	C₈	—	15	1	no	—
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	—	12	1	no	—
77	C₂×C₃	#1203	1	1	yes	characteristic
78	C₂×C₃	#1220	1	1	yes	characteristic
79	D₃	—	4	1	no	—
80	C₅	#3716	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	6	1	no	—
82	C₄	—	90	1	no	—
83	C₄	D₃×D₁₅	1	1	yes	characteristic
84	C₃	—	2	1	no	—
85	C₃	#6659	1	1	yes	characteristic, complemented
86	C₃	#6716	1	1	yes	characteristic, complemented
87	C₂	—	12	1	no	—
88	C₂	#67152	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#67152
commutator, Fitting, cosocle	C₃²×C₄×C₅	C₂²
Frattini, hypercenter	C₄	D₃×D₁₅
socle	C₂×C₃²×C₅	D₄
prime radical	C₃×C₅×D₁₂	C₂
hypocenter	C₃²×C₅	SD₁₆
LCS	G ▷ C₃²×C₄×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. #67180	C₂	semidirect
3. C₃×C₃⋊#168	C₂	semidirect
15. C₃⋊#168	D₃	semidirect
33. C₃²×C₅	SD₁₆	semidirect
59. C₃×C₅	#204	semidirect
60. C₃×C₅	#214	semidirect
71. C₃²	#687	semidirect
80. C₅	#3716	semidirect
85. C₃	#6659	semidirect
86. C₃	#6716	semidirect

Maximal subgroups

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

subgroup	index
#67180	2
C₃×C₅×D₁₂	2
C₃×C₃⋊#168	2
#6716	3
#6659	3
#3716	5