Subgroups — C₂×C₇×#689 #11765189

Order 1120.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	C₇×#689	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×#168	C₂	1	1	yes	maximal, characteristic
5	C₅×C₇×Q₈	—	2	1	no	—
6	C₅×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₇	—	2	1	no	—
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	—
12	C₇×#168	C₂²	1	2	yes	—
13	C₂×C₇×Q₁₆	—	5	1	no	maximal, Hall
14	C₂×#689	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₅×C₇	D₄	1	1	yes	characteristic
16	C₄×C₅×C₇	—	2	2	no	—
17	C₄×C₅×C₇	D₄	1	1	yes	characteristic
18	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
19	C₇×Dic₅	—	2	2	no	—
20	C₂×C₇×Q₈	—	5	1	no	—
21	C₂×C₇×Q₈	D₅	1	1	yes	characteristic
22	C₇×Q₁₆	—	5	4	no	—
23	C₂×C₇×C₈	—	5	1	no	—
24	C₂×C₅×Q₈	C₂×C₇	1	1	yes	characteristic
25	C₂×Dic₁₀	C₂×C₇	1	1	yes	characteristic
26	#689	C₂×C₇	1	4	yes	direct product factor
27	C₂×#168	C₂×C₇	1	1	yes	characteristic
28	C₂×C₅×C₇	Q₁₆	1	2	yes	complemented
29	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
30	C₇×Q₈	—	2	1	no	—
31	C₇×Q₈	—	5	2	no	—
32	C₇×Q₈	—	10	1	no	—
33	C₇×Q₈	C₂×D₅	1	2	yes	—
34	C₂×C₄×C₇	—	2	1	no	—
35	C₂×C₄×C₇	—	10	1	no	—
36	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
37	C₇×C₈	—	5	2	no	—
38	C₅×Q₈	—	2	1	no	—
39	C₅×Q₈	C₂²×C₇	1	2	yes	—
40	C₂×C₄×C₅	—	2	1	no	—
41	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
42	C₂×Dic₅	—	2	1	no	—
43	Dic₁₀	—	2	1	no	—
44	Dic₁₀	C₂²×C₇	1	2	yes	—
45	#168	C₂²×C₇	1	2	yes	—
46	C₅×C₇	C₂×Q₁₆	1	1	yes	characteristic, complemented, Hall
47	C₂×Q₁₆	—	5	1	no	Sylow
48	C₂²×C₇	C₂²⋊D₅	1	1	yes	characteristic
49	C₄×C₇	—	2	2	no	—
50	C₄×C₇	—	10	2	no	—
51	C₄×C₇	C₂²⋊D₅	1	1	yes	characteristic
52	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
53	C₂²×C₅	C₇×D₄	1	1	yes	characteristic
54	C₄×C₅	—	2	2	no	—
55	C₄×C₅	C₇×D₄	1	1	yes	characteristic
56	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
57	Dic₅	—	2	2	no	—
58	C₂×Q₈	—	5	1	no	—
59	C₂×Q₈	C₇×D₅	1	1	yes	characteristic
60	Q₁₆	—	5	4	no	—
61	C₂×C₈	—	5	1	no	—
62	C₂×C₇	#689	1	2	yes	direct product factor
63	C₂×C₇	C₂×C₂²⋊D₅	1	1	yes	characteristic
64	C₂×C₅	C₇×Q₁₆	1	2	yes	complemented
65	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
66	Q₈	—	2	1	no	—
67	Q₈	—	5	2	no	—
68	Q₈	—	10	1	no	—
69	Q₈	C₂×C₇×D₅	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	10	1	no	—
72	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
73	C₈	—	5	2	no	—
74	C₇	C₂×#689	1	1	yes	characteristic, direct product factor, Sylow
75	C₅	C₂×C₇×Q₁₆	1	1	yes	characteristic, complemented, Sylow
76	C₂²	C₇×C₂²⋊D₅	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	10	2	no	—
79	C₄	C₇×C₂²⋊D₅	1	1	yes	characteristic
80	C₄	C₂²×C₇×D₅	1	1	yes	characteristic
81	C₂	C₇×#689	1	2	yes	direct product factor
82	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	C₂×C₅×C₇×Q₈	C₂
socle, prime radical	C₂²×C₅×C₇	D₄
hypercenter	C₂×C₇×Q₈	D₅
hypocenter	C₅	C₂×C₇×Q₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×Q₈

Complements

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

N	H ≅ G/N	type
3. C₇×#689	C₂	direct (N × H)
14. C₂×#689	C₇	direct (N × H)
26. #689	C₂×C₇	direct (N × H)
28. C₂×C₅×C₇	Q₁₆	semidirect
46. C₅×C₇	C₂×Q₁₆	semidirect
62. C₂×C₇	#689	direct (N × H)
64. C₂×C₅	C₇×Q₁₆	semidirect
74. C₇	C₂×#689	direct (N × H)
75. C₅	C₂×C₇×Q₁₆	semidirect
81. C₂	C₇×#689	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×Q₈	2
C₂×C₇×Dic₁₀	2
C₇×#689	2
C₂×C₇×#168	2
C₂×C₇×Q₁₆	5
C₂×#689	7

Subgroups — C2×C7×#689 #11765189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#689 #11765189