Subgroups — C₅×#66629 #13132329

Order 1680.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₂₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×Dic₂₁	C₂	1	1	yes	maximal, characteristic
4	C₅×Dic₇⋊C₄	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₅×C₇	—	2	1	no	—
7	C₅×Dic₂₁	—	2	1	no	—
8	C₅×Dic₂₁	C₄	1	2	yes	complemented
9	#66629	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₄×C₅×C₇	D₃	1	1	yes	characteristic
11	C₂×C₅×Dic₇	—	3	1	no	—
12	C₂×C₅×Dic₇	—	3	1	no	—
13	C₅×Dic₃⋊C₄	—	7	1	no	maximal, Hall
14	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₅×C₇	D₄	1	1	yes	characteristic
16	C₂×C₃×C₅×C₇	Q₈	1	1	yes	characteristic
17	C₂×C₃×C₄×C₇	C₂×C₅	1	1	yes	characteristic
18	C₂×Dic₂₁	C₂×C₅	1	1	yes	characteristic
19	C₂×Dic₂₁	C₂×C₅	1	1	yes	characteristic
20	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
21	C₄×C₅×C₇	—	2	1	no	—
22	C₅×Dic₇	—	3	2	no	—
23	C₅×Dic₇	—	6	1	no	—
24	C₂×C₃×C₄×C₅	D₇	1	1	yes	characteristic
25	C₂×C₅×Dic₃	—	7	1	no	—
26	C₂×C₅×Dic₃	—	7	1	no	—
27	Dic₇⋊C₄	—	3	1	no	Hall
28	C₃×C₅×C₇	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×C₃×C₇	C₂²×C₅	1	1	yes	characteristic
30	C₃×C₄×C₇	—	2	1	no	—
31	Dic₂₁	—	2	1	no	—
32	Dic₂₁	C₄×C₅	1	2	yes	complemented
33	C₅×C₄⋊C₄	—	21	1	no	Hall
34	C₂×C₅×C₇	Dic₆	1	1	yes	characteristic
35	C₂×C₅×C₇	C₄×D₃	1	1	yes	characteristic
36	C₂×C₅×C₇	#67	1	1	yes	characteristic
37	C₂²×C₃×C₅	C₂×D₇	1	1	yes	characteristic
38	C₃×C₄×C₅	—	2	1	no	—
39	C₅×Dic₃	—	7	2	no	—
40	C₅×Dic₃	—	14	1	no	—
41	C₂×C₄×C₇	C₅×D₃	1	1	yes	characteristic
42	C₂×Dic₇	—	3	1	no	—
43	C₂×Dic₇	—	3	1	no	—
44	Dic₃⋊C₄	—	7	1	no	Hall
45	C₂×C₃×C₇	C₂×C₄×C₅	1	1	yes	characteristic
46	C₂×C₃×C₇	C₅×D₄	1	1	yes	characteristic
47	C₂×C₃×C₇	C₅×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₅	—	21	1	no	—
49	C₂×C₄×C₅	—	21	1	no	—
50	C₂×C₄×C₅	D₂₁	1	1	yes	characteristic
51	C₅×C₇	Dic₃⋊C₄	1	1	yes	characteristic, complemented, Hall
52	C₂×C₃×C₅	Dic₁₄	1	1	yes	characteristic
53	C₂×C₃×C₅	C₄×D₇	1	1	yes	characteristic
54	C₂×C₃×C₅	C₂²⋊D₇	1	1	yes	characteristic
55	C₂²×C₇	C₂×C₅×D₃	1	1	yes	characteristic
56	C₄×C₇	—	2	1	no	—
57	Dic₇	—	3	2	no	—
58	Dic₇	—	6	1	no	—
59	C₂×C₃×C₄	C₅×D₇	1	1	yes	characteristic
60	C₂×Dic₃	—	7	1	no	—
61	C₂×Dic₃	—	7	1	no	—
62	C₃×C₇	C₅×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
63	C₂²×C₅	C₂×D₂₁	1	1	yes	characteristic
64	C₄×C₅	—	2	1	no	—
65	C₄×C₅	—	21	2	no	—
66	C₄×C₅	—	42	1	no	—
67	C₄⋊C₄	—	21	1	no	Sylow
68	C₃×C₅	Dic₇⋊C₄	1	1	yes	characteristic, complemented, Hall
69	C₂×C₇	C₅×Dic₆	1	1	yes	characteristic
70	C₂×C₇	C₄×C₅×D₃	1	1	yes	characteristic
71	C₂×C₇	C₅×#67	1	1	yes	characteristic
72	C₂²×C₃	C₂×C₅×D₇	1	1	yes	characteristic
73	C₃×C₄	—	2	1	no	—
74	Dic₃	—	7	2	no	—
75	Dic₃	—	14	1	no	—
76	C₂×C₅	Dic₄₂	1	1	yes	characteristic
77	C₂×C₅	C₄×D₂₁	1	1	yes	characteristic
78	C₂×C₅	#4268	1	1	yes	characteristic
79	C₂×C₄	—	21	1	no	—
80	C₂×C₄	—	21	1	no	—
81	C₂×C₄	C₅×D₂₁	1	1	yes	characteristic
82	C₇	C₅×Dic₃⋊C₄	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	C₅×Dic₁₄	1	1	yes	characteristic
84	C₂×C₃	C₄×C₅×D₇	1	1	yes	characteristic
85	C₂×C₃	C₅×C₂²⋊D₇	1	1	yes	characteristic
86	C₅	#66629	1	1	yes	characteristic, direct product factor, Sylow
87	C₂²	C₂×C₅×D₂₁	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	21	2	no	—
90	C₄	—	42	1	no	—
91	C₃	C₅×Dic₇⋊C₄	1	1	yes	characteristic, complemented, Sylow
92	C₂	C₅×Dic₄₂	1	1	yes	characteristic
93	C₂	C₄×C₅×D₂₁	1	1	yes	characteristic
94	C₂	C₅×#4268	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₁
commutator	C₂×C₃×C₇	C₂×C₄×C₅
Frattini	C₂²	C₂×C₅×D₂₁
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle, prime radical	C₂²×C₃×C₅×C₇	C₂²
cosocle	C₂²×C₃×C₇	C₂²×C₅
hypercenter	C₂×C₄×C₅	D₂₁
hypocenter	C₃×C₇	C₅×C₄⋊C₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
8. C₅×Dic₂₁	C₄	semidirect
9. #66629	C₅	direct (N × H)
28. C₃×C₅×C₇	C₄⋊C₄	semidirect
32. Dic₂₁	C₄×C₅	semidirect
51. C₅×C₇	Dic₃⋊C₄	semidirect
62. C₃×C₇	C₅×C₄⋊C₄	semidirect
68. C₃×C₅	Dic₇⋊C₄	semidirect
82. C₇	C₅×Dic₃⋊C₄	semidirect
86. C₅	#66629	direct (N × H)
91. C₃	C₅×Dic₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×C₇	2
C₂×C₅×Dic₂₁	2
C₅×Dic₇⋊C₄	3
#66629	5
C₅×Dic₃⋊C₄	7

Subgroups — C5×#66629 #13132329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66629 #13132329