Subgroups — C₅×#3722 #10623789

Order 720.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×Dic₃	C₂	1	2	yes	maximal, complemented
3	C₅×(C₂×D₃)⋊C₄	—	3	2	no	maximal
4	C₂²×C₃²×C₅	C₂²	1	1	yes	characteristic
5	C₂×C₅×C₃⋊D₃	—	2	2	no	—
6	C₂×C₅×C₃⋊D₃	C₄	1	2	yes	complemented
7	C₃×C₅×Dic₃	—	2	2	no	—
8	#3722	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₅×D₃	—	3	2	no	—
10	C₂²×C₅×D₃	—	6	1	no	—
11	C₂×C₃×C₄×C₅	—	3	2	no	—
12	C₂×C₅×Dic₃	D₃	1	2	yes	complemented
13	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃²×C₅	D₄	1	2	yes	—
15	C₅×C₃⋊D₃	—	2	2	no	—
16	C₅×C₂²⋊C₄	—	9	1	no	Hall
17	C₂²×C₃⋊D₃	C₂×C₅	1	1	yes	characteristic
18	C₂×C₃×Dic₃	C₂×C₅	1	2	yes	complemented
19	C₂²×C₃×C₅	—	2	1	no	—
20	C₂²×C₃×C₅	C₂×D₃	1	2	yes	—
21	C₂×C₅×D₃	—	3	4	no	—
22	C₂×C₅×D₃	—	6	2	no	—
23	C₂×C₅×D₃	—	6	4	no	—
24	C₂×C₅×D₃	—	6	2	no	—
25	C₂×C₅×D₃	—	6	2	no	—
26	C₃×C₄×C₅	—	6	2	no	—
27	C₅×Dic₃	—	2	2	no	—
28	(C₂×D₃)⋊C₄	—	3	2	no	—
29	C₃²×C₅	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
30	C₂³×C₅	—	9	1	no	—
31	C₂×C₄×C₅	—	3	2	no	—
32	C₂²×C₃²	C₂²×C₅	1	1	yes	characteristic
33	C₂×C₃⋊D₃	—	2	2	no	—
34	C₂×C₃⋊D₃	C₄×C₅	1	2	yes	complemented
35	C₃×Dic₃	—	2	2	no	—
36	C₂×C₃×C₅	—	2	1	no	—
37	C₂×C₃×C₅	—	2	2	no	—
38	C₂×C₃×C₅	C₄×D₃	1	2	yes	—
39	C₂×C₃×C₅	D₁₂	1	2	yes	—
40	C₂×C₃×C₅	#67	1	2	yes	—
41	C₅×D₃	—	6	4	no	—
42	C₅×D₃	—	6	4	no	—
43	C₂²×D₃	—	3	2	no	—
44	C₂²×D₃	—	6	1	no	—
45	C₂×C₃×C₄	—	3	2	no	—
46	C₂×Dic₃	C₅×D₃	1	2	yes	complemented
47	C₂²×C₅	—	9	2	no	—
48	C₂²×C₅	—	18	2	no	—
49	C₂²×C₅	D₃²	1	1	yes	characteristic
50	C₄×C₅	—	6	2	no	—
51	C₂×C₃²	C₂×C₄×C₅	1	1	yes	characteristic
52	C₂×C₃²	C₅×D₄	1	2	yes	—
53	C₃⋊D₃	—	2	2	no	—
54	C₂²⋊C₄	—	9	1	no	Sylow
55	C₃×C₅	—	2	1	no	—
56	C₃×C₅	(C₂×D₃)⋊C₄	1	2	yes	complemented
57	C₂²×C₃	—	2	1	no	—
58	C₂²×C₃	C₂×C₅×D₃	1	2	yes	—
59	C₂×D₃	—	3	4	no	—
60	C₂×D₃	—	6	2	no	—
61	C₂×D₃	—	6	4	no	—
62	C₂×D₃	—	6	2	no	—
63	C₂×D₃	—	6	2	no	—
64	C₃×C₄	—	6	2	no	—
65	Dic₃	—	2	2	no	—
66	C₂×C₅	—	18	2	no	—
67	C₂×C₅	#624	1	1	yes	characteristic
68	C₂×C₅	#626	1	2	yes	—
69	C₃²	C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
70	C₂³	—	9	1	no	—
71	C₂×C₄	—	3	2	no	—
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	2	2	no	—
74	C₂×C₃	C₄×C₅×D₃	1	2	yes	—
75	C₂×C₃	C₅×D₁₂	1	2	yes	—
76	C₂×C₃	C₅×#67	1	2	yes	—
77	D₃	—	6	4	no	—
78	D₃	—	6	4	no	—
79	C₅	#3722	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²	—	9	2	no	—
81	C₂²	—	18	2	no	—
82	C₂²	C₅×D₃²	1	1	yes	characteristic
83	C₄	—	6	2	no	—
84	C₃	—	2	1	no	—
85	C₃	C₅×(C₂×D₃)⋊C₄	1	2	yes	complemented
86	C₂	—	18	2	no	—
87	C₂	C₅×#624	1	1	yes	characteristic
88	C₂	C₅×#626	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₅	D₃²
commutator	C₂×C₃²	C₂×C₄×C₅
Frattini	C₂²	C₅×D₃²
Fitting, socle	C₂²×C₃²×C₅	C₂²
prime radical	C₂²×C₅×C₃⋊D₃	C₂
cosocle	C₂²×C₃²	C₂²×C₅
hypocenter	C₃²	C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×Dic₃	C₂	semidirect
6. C₂×C₅×C₃⋊D₃	C₄	semidirect
8. #3722	C₅	direct (N × H)
12. C₂×C₅×Dic₃	D₃	semidirect
18. C₂×C₃×Dic₃	C₂×C₅	semidirect
29. C₃²×C₅	C₂²⋊C₄	semidirect
34. C₂×C₃⋊D₃	C₄×C₅	semidirect
46. C₂×Dic₃	C₅×D₃	semidirect
56. C₃×C₅	(C₂×D₃)⋊C₄	semidirect
69. C₃²	C₅×C₂²⋊C₄	semidirect
79. C₅	#3722	direct (N × H)
85. C₃	C₅×(C₂×D₃)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₃⋊D₃	2
C₂×C₃×C₅×Dic₃	2
C₅×(C₂×D₃)⋊C₄	3
#3722	5

Subgroups — C5×#3722 #10623789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3722 #10623789