Subgroups — C₇×#6719 #13132352

Order 1680.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₁₅	C₂	1	2	yes	maximal, complemented
3	C₇×C₂²⋊Dic₅	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅×C₇	—	2	2	no	—
5	C₂²×C₃×C₅×C₇	C₄	1	2	yes	complemented
6	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₇×Dic₁₅	—	2	2	no	—
8	C₇×#217	—	5	1	no	maximal, Hall
9	C₂³×C₅×C₇	D₃	1	1	yes	characteristic
10	C₂×C₇×Dic₅	—	3	2	no	—
11	#6719	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₃×C₅×C₇	—	2	2	no	—
13	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₅×C₇	D₄	1	2	yes	—
15	C₂³×C₃×C₇	D₅	1	1	yes	characteristic
16	C₂×C₇×Dic₃	—	5	2	no	—
17	C₂²×C₅×C₇	—	2	2	no	—
18	C₂²×C₅×C₇	Dic₃	1	2	yes	complemented
19	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
20	C₇×Dic₅	—	6	2	no	—
21	C₂³×C₃×C₅	C₂×C₇	1	1	yes	characteristic
22	C₂×Dic₁₅	C₂×C₇	1	2	yes	complemented
23	C₇×C₂²⋊C₄	—	15	1	no	Hall
24	C₃×C₅×C₇	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
25	C₂²×C₃×C₇	—	2	2	no	—
26	C₂²×C₃×C₇	Dic₅	1	2	yes	complemented
27	C₂²×C₃×C₇	C₂×D₅	1	1	yes	characteristic
28	C₇×Dic₃	—	10	2	no	—
29	C₂²⋊Dic₅	—	3	1	no	Hall
30	C₂×C₅×C₇	—	2	2	no	—
31	C₂×C₅×C₇	C₂×Dic₃	1	1	yes	characteristic
32	C₂×C₅×C₇	#67	1	2	yes	—
33	C₂²×C₃×C₅	—	2	2	no	—
34	C₂²×C₃×C₅	C₄×C₇	1	2	yes	complemented
35	C₂²×C₃×C₅	C₂²×C₇	1	1	yes	characteristic
36	Dic₁₅	—	2	2	no	—
37	C₂³×C₇	D₁₅	1	1	yes	characteristic
38	C₂×C₄×C₇	—	15	2	no	—
39	#217	—	5	1	no	Hall
40	C₂×C₃×C₇	—	2	2	no	—
41	C₂×C₃×C₇	C₂×Dic₅	1	1	yes	characteristic
42	C₂×C₃×C₇	C₂²⋊D₅	1	2	yes	—
43	C₂³×C₅	D₃×C₇	1	1	yes	characteristic
44	C₂×Dic₅	—	3	2	no	—
45	C₅×C₇	#217	1	1	yes	characteristic, complemented, Hall
46	C₂×C₃×C₅	—	2	2	no	—
47	C₂×C₃×C₅	C₂×C₄×C₇	1	1	yes	characteristic
48	C₂×C₃×C₅	C₇×D₄	1	2	yes	—
49	C₂²×C₇	—	2	2	no	—
50	C₂²×C₇	Dic₁₅	1	2	yes	complemented
51	C₂²×C₇	C₂×D₁₅	1	1	yes	characteristic
52	C₄×C₇	—	30	2	no	—
53	C₂³×C₃	C₇×D₅	1	1	yes	characteristic
54	C₂×Dic₃	—	5	2	no	—
55	C₃×C₇	C₂²⋊Dic₅	1	1	yes	characteristic, complemented, Hall
56	C₂²×C₅	—	2	2	no	—
57	C₂²×C₅	C₇×Dic₃	1	2	yes	complemented
58	C₂²×C₅	C₂×D₃×C₇	1	1	yes	characteristic
59	Dic₅	—	6	2	no	—
60	C₂²⋊C₄	—	15	1	no	Sylow
61	C₃×C₅	C₇×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
62	C₂×C₇	—	2	2	no	—
63	C₂×C₇	C₂×Dic₁₅	1	1	yes	characteristic
64	C₂×C₇	#1220	1	2	yes	—
65	C₂²×C₃	—	2	2	no	—
66	C₂²×C₃	C₇×Dic₅	1	2	yes	complemented
67	C₂²×C₃	C₂×C₇×D₅	1	1	yes	characteristic
68	Dic₃	—	10	2	no	—
69	C₂×C₅	—	2	2	no	—
70	C₂×C₅	C₂×C₇×Dic₃	1	1	yes	characteristic
71	C₂×C₅	C₇×#67	1	2	yes	—
72	C₂³	C₇×D₁₅	1	1	yes	characteristic
73	C₂×C₄	—	15	2	no	—
74	C₇	#6719	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×C₃	—	2	2	no	—
76	C₂×C₃	C₂×C₇×Dic₅	1	1	yes	characteristic
77	C₂×C₃	C₇×C₂²⋊D₅	1	2	yes	—
78	C₅	C₇×#217	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	2	2	no	—
80	C₂²	C₇×Dic₁₅	1	2	yes	complemented
81	C₂²	C₂×C₇×D₁₅	1	1	yes	characteristic
82	C₄	—	30	2	no	—
83	C₃	C₇×C₂²⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	2	2	no	—
85	C₂	C₂×C₇×Dic₁₅	1	1	yes	characteristic
86	C₂	C₇×#1220	1	2	yes	—

	subgroup	G/H
center	C₂²×C₇	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂×C₄×C₇
Frattini	C₂²	C₂×C₇×D₁₅
Fitting, prime radical	C₂³×C₃×C₅×C₇	C₂
socle	C₂²×C₃×C₅×C₇	C₂²
cosocle	C₂²×C₃×C₅	C₂²×C₇
hypercenter	C₂³×C₇	D₁₅
hypocenter	C₃×C₅	C₇×C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×Dic₁₅	C₂	semidirect
5. C₂²×C₃×C₅×C₇	C₄	semidirect
11. #6719	C₇	direct (N × H)
18. C₂²×C₅×C₇	Dic₃	semidirect
22. C₂×Dic₁₅	C₂×C₇	semidirect
24. C₃×C₅×C₇	C₂²⋊C₄	semidirect
26. C₂²×C₃×C₇	Dic₅	semidirect
34. C₂²×C₃×C₅	C₄×C₇	semidirect
45. C₅×C₇	#217	semidirect
50. C₂²×C₇	Dic₁₅	semidirect
55. C₃×C₇	C₂²⋊Dic₅	semidirect
57. C₂²×C₅	C₇×Dic₃	semidirect
61. C₃×C₅	C₇×C₂²⋊C₄	semidirect
66. C₂²×C₃	C₇×Dic₅	semidirect
74. C₇	#6719	direct (N × H)
78. C₅	C₇×#217	semidirect
80. C₂²	C₇×Dic₁₅	semidirect
83. C₃	C₇×C₂²⋊Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×C₇	2
C₂×C₇×Dic₁₅	2
C₇×C₂²⋊Dic₅	3
C₇×#217	5
#6719	7

Subgroups — C7×#6719 #13132352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6719 #13132352