Subgroups — C₂×C₇×#687 #11765182

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#687	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₅×C₇	—	2	1	no	—
6	C₅×C₇×D₄	—	2	1	no	—
7	C₅×C₇×D₄	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×Dic₅	—	2	1	no	—
10	C₇×Dic₁₀	—	2	1	no	—
11	C₇×Dic₁₀	C₂²	1	2	yes	complemented
12	C₇×#168	C₂²	1	2	yes	complemented
13	C₂×C₇×SD₁₆	—	5	1	no	maximal, Hall
14	C₂×#687	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₅×C₇	—	2	2	no	—
16	C₂²×C₅×C₇	—	4	2	no	—
17	C₂²×C₅×C₇	D₄	1	1	yes	characteristic
18	C₄×C₅×C₇	D₄	1	1	yes	characteristic
19	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
20	C₇×Dic₅	—	2	2	no	—
21	C₂×C₇×Q₈	—	5	1	no	—
22	C₂×C₇×D₄	D₅	1	1	yes	characteristic
23	C₇×SD₁₆	—	5	4	no	—
24	C₂×C₇×C₈	—	5	1	no	—
25	C₂×C₅×D₄	C₂×C₇	1	1	yes	characteristic
26	C₂×Dic₁₀	C₂×C₇	1	1	yes	characteristic, complemented
27	#687	C₂×C₇	1	4	yes	direct product factor
28	C₂×#168	C₂×C₇	1	1	yes	characteristic, complemented
29	C₂×C₅×C₇	—	4	2	no	—
30	C₂×C₅×C₇	SD₁₆	1	2	yes	complemented
31	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
32	C₂³×C₇	—	2	1	no	—
33	C₇×Q₈	—	5	2	no	—
34	C₇×Q₈	—	10	1	no	—
35	C₇×D₄	—	2	1	no	—
36	C₇×D₄	C₂×D₅	1	2	yes	—
37	C₂×C₄×C₇	—	10	1	no	—
38	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
39	C₇×C₈	—	5	2	no	—
40	C₂³×C₅	—	2	1	no	—
41	C₅×D₄	—	2	1	no	—
42	C₅×D₄	C₂²×C₇	1	2	yes	—
43	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
44	C₂×Dic₅	—	2	1	no	—
45	Dic₁₀	—	2	1	no	—
46	Dic₁₀	C₂²×C₇	1	2	yes	complemented
47	#168	C₂²×C₇	1	2	yes	complemented
48	C₅×C₇	C₂×SD₁₆	1	1	yes	characteristic, complemented, Hall
49	C₂×SD₁₆	—	5	1	no	Sylow
50	C₂²×C₇	—	2	2	no	—
51	C₂²×C₇	—	4	2	no	—
52	C₂²×C₇	C₂²⋊D₅	1	1	yes	characteristic
53	C₄×C₇	—	10	2	no	—
54	C₄×C₇	C₂²⋊D₅	1	1	yes	characteristic
55	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
56	C₂²×C₅	—	2	2	no	—
57	C₂²×C₅	—	4	2	no	—
58	C₂²×C₅	C₇×D₄	1	1	yes	characteristic
59	C₄×C₅	C₇×D₄	1	1	yes	characteristic
60	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
61	Dic₅	—	2	2	no	—
62	C₂×Q₈	—	5	1	no	—
63	C₂×D₄	C₇×D₅	1	1	yes	characteristic
64	SD₁₆	—	5	4	no	—
65	C₂×C₈	—	5	1	no	—
66	C₂×C₇	—	4	2	no	—
67	C₂×C₇	#687	1	2	yes	direct product factor
68	C₂×C₇	C₂×C₂²⋊D₅	1	1	yes	characteristic
69	C₂×C₅	—	4	2	no	—
70	C₂×C₅	C₇×SD₁₆	1	2	yes	complemented
71	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
72	C₂³	—	2	1	no	—
73	Q₈	—	5	2	no	—
74	Q₈	—	10	1	no	—
75	D₄	—	2	1	no	—
76	D₄	C₂×C₇×D₅	1	2	yes	—
77	C₂×C₄	—	10	1	no	—
78	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
79	C₈	—	5	2	no	—
80	C₇	C₂×#687	1	1	yes	characteristic, direct product factor, Sylow
81	C₅	C₂×C₇×SD₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	2	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₇×C₂²⋊D₅	1	1	yes	characteristic
85	C₄	—	10	2	no	—
86	C₄	C₇×C₂²⋊D₅	1	1	yes	characteristic
87	C₄	C₂²×C₇×D₅	1	1	yes	characteristic
88	C₂	—	4	2	no	—
89	C₂	C₇×#687	1	2	yes	direct product factor
90	C₂	C₂×C₇×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂³×C₇
Frattini	C₄	C₂²×C₇×D₅
Fitting, prime radical	C₂×C₅×C₇×D₄	C₂
socle	C₂²×C₅×C₇	D₄
hypercenter	C₂×C₇×D₄	D₅
hypocenter	C₅	C₂×C₇×SD₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×Dic₁₀	C₂	semidirect
3. C₇×#687	C₂	direct (N × H)
4. C₂×C₇×#168	C₂	semidirect
11. C₇×Dic₁₀	C₂²	semidirect
12. C₇×#168	C₂²	semidirect
14. C₂×#687	C₇	direct (N × H)
26. C₂×Dic₁₀	C₂×C₇	semidirect
27. #687	C₂×C₇	direct (N × H)
28. C₂×#168	C₂×C₇	semidirect
30. C₂×C₅×C₇	SD₁₆	semidirect
46. Dic₁₀	C₂²×C₇	semidirect
47. #168	C₂²×C₇	semidirect
48. C₅×C₇	C₂×SD₁₆	semidirect
67. C₂×C₇	#687	direct (N × H)
70. C₂×C₅	C₇×SD₁₆	semidirect
80. C₇	C₂×#687	direct (N × H)
81. C₅	C₂×C₇×SD₁₆	semidirect
89. C₂	C₇×#687	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₇×D₄	2
C₂×C₇×Dic₁₀	2
C₇×#687	2
C₂×C₇×#168	2
C₂×C₇×SD₁₆	5
C₂×#687	7

Subgroups — C2×C7×#687 #11765182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#687 #11765182