Subgroups — C₇×#6718 #13132351

Order 1680.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×Dic₃₀	C₂	1	1	yes	maximal, characteristic
3	C₇×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₇×#689	—	3	1	no	maximal, Hall
5	C₃×C₄×C₅×C₇	—	2	1	no	—
6	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₇×Dic₁₅	—	2	1	no	—
8	C₇×#216	—	5	1	no	maximal, Hall
9	C₅×C₇×Q₈	D₃	1	1	yes	characteristic
10	C₇×Dic₁₀	—	3	1	no	—
11	C₇×#168	—	3	1	no	—
12	#6718	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×C₇	D₄	1	1	yes	characteristic
14	C₃×C₇×Q₈	D₅	1	1	yes	characteristic
15	C₇×Dic₆	—	5	1	no	—
16	C₇×C₃⋊C₈	—	5	1	no	—
17	C₄×C₅×C₇	—	2	1	no	—
18	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic
19	C₇×Dic₅	—	6	1	no	—
20	C₃×C₅×Q₈	C₂×C₇	1	1	yes	characteristic
21	Dic₃₀	C₂×C₇	1	1	yes	characteristic
22	C₃⋊#168	C₂×C₇	1	1	yes	characteristic
23	C₇×Q₁₆	—	15	1	no	Hall
24	C₃×C₅×C₇	Q₁₆	1	1	yes	characteristic, complemented, Hall
25	C₃×C₄×C₇	—	2	1	no	—
26	C₃×C₄×C₇	C₂×D₅	1	1	yes	characteristic
27	C₇×Dic₃	—	10	1	no	—
28	#689	—	3	1	no	Hall
29	C₂×C₅×C₇	#67	1	1	yes	characteristic
30	C₃×C₄×C₅	—	2	1	no	—
31	C₃×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
32	Dic₁₅	—	2	1	no	—
33	C₇×Q₈	—	15	1	no	—
34	C₇×Q₈	D₁₅	1	1	yes	characteristic
35	C₇×C₈	—	15	1	no	—
36	#216	—	5	1	no	Hall
37	C₂×C₃×C₇	C₂²⋊D₅	1	1	yes	characteristic
38	C₅×Q₈	D₃×C₇	1	1	yes	characteristic
39	Dic₁₀	—	3	1	no	—
40	#168	—	3	1	no	—
41	C₅×C₇	#216	1	1	yes	characteristic, complemented, Hall
42	C₂×C₃×C₅	C₇×D₄	1	1	yes	characteristic
43	C₄×C₇	—	2	1	no	—
44	C₄×C₇	—	30	1	no	—
45	C₄×C₇	C₂×D₁₅	1	1	yes	characteristic
46	C₃×Q₈	C₇×D₅	1	1	yes	characteristic
47	Dic₆	—	5	1	no	—
48	C₃⋊C₈	—	5	1	no	—
49	C₃×C₇	#689	1	1	yes	characteristic, complemented, Hall
50	C₄×C₅	—	2	1	no	—
51	C₄×C₅	C₂×D₃×C₇	1	1	yes	characteristic
52	Dic₅	—	6	1	no	—
53	Q₁₆	—	15	1	no	Sylow
54	C₃×C₅	C₇×Q₁₆	1	1	yes	characteristic, complemented, Hall
55	C₂×C₇	#1220	1	1	yes	characteristic
56	C₃×C₄	—	2	1	no	—
57	C₃×C₄	C₂×C₇×D₅	1	1	yes	characteristic
58	Dic₃	—	10	1	no	—
59	C₂×C₅	C₇×#67	1	1	yes	characteristic
60	Q₈	—	15	1	no	—
61	Q₈	C₇×D₁₅	1	1	yes	characteristic
62	C₈	—	15	1	no	—
63	C₇	#6718	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃	C₇×C₂²⋊D₅	1	1	yes	characteristic
65	C₅	C₇×#216	1	1	yes	characteristic, complemented, Sylow
66	C₄	—	2	1	no	—
67	C₄	—	30	1	no	—
68	C₄	C₂×C₇×D₁₅	1	1	yes	characteristic
69	C₃	C₇×#689	1	1	yes	characteristic, complemented, Sylow
70	C₂	C₇×#1220	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	#1220
commutator, cosocle	C₃×C₄×C₅	C₂²×C₇
Frattini	C₄	C₂×C₇×D₁₅
Fitting	C₃×C₅×C₇×Q₈	C₂
socle, prime radical	C₂×C₃×C₅×C₇	D₄
hypercenter	C₇×Q₈	D₁₅
hypocenter	C₃×C₅	C₇×Q₁₆
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×Q₈

Complements

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

N	H ≅ G/N	type
12. #6718	C₇	direct (N × H)
24. C₃×C₅×C₇	Q₁₆	semidirect
41. C₅×C₇	#216	semidirect
49. C₃×C₇	#689	semidirect
54. C₃×C₅	C₇×Q₁₆	semidirect
63. C₇	#6718	direct (N × H)
65. C₅	C₇×#216	semidirect
69. C₃	C₇×#689	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₇×Q₈	2
C₇×Dic₃₀	2
C₇×C₃⋊#168	2
C₇×#689	3
C₇×#216	5
#6718	7

Subgroups — C7×#6718 #13132351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6718 #13132351