Subgroups — C₂×C₇×#681 #11765165

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₇×#681	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₇×#168	C₂	1	2	yes	maximal, complemented
4	C₂³×C₅×C₇	C₄	1	1	yes	characteristic
5	C₂×C₄×C₅×C₇	C₄	1	2	yes	—
6	C₂×C₄×C₅×C₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₇	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
9	C₇×#168	C₂²	1	4	yes	complemented
10	C₂×C₇×M₁₆	—	5	1	no	maximal, Hall
11	C₂×#681	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂²×C₅×C₇	—	2	2	no	—
13	C₂²×C₅×C₇	C₂×C₄	1	2	yes	—
14	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
15	C₄×C₅×C₇	C₂×C₄	1	2	yes	—
16	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
17	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₇	D₅	1	1	yes	characteristic
19	C₇×M₁₆	—	5	4	no	—
20	C₂×C₇×C₈	—	5	2	no	—
21	C₂²×C₄×C₅	C₂×C₇	1	1	yes	characteristic
22	#681	C₂×C₇	1	4	yes	direct product factor
23	C₂×#168	C₂×C₇	1	2	yes	complemented
24	C₂×C₅×C₇	—	2	2	no	—
25	C₂×C₅×C₇	M₁₆	1	2	yes	complemented
26	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
27	C₂³×C₇	Dic₅	1	1	yes	characteristic
28	C₂×C₄×C₇	Dic₅	1	1	yes	characteristic
29	C₂×C₄×C₇	Dic₅	1	2	yes	—
30	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
31	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
32	C₇×C₈	—	5	4	no	—
33	C₂³×C₅	C₄×C₇	1	1	yes	characteristic
34	C₂×C₄×C₅	C₄×C₇	1	2	yes	—
35	C₂×C₄×C₅	C₄×C₇	1	1	yes	characteristic
36	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
37	C₂×C₄×C₅	C₂²×C₇	1	2	yes	—
38	#168	C₂²×C₇	1	4	yes	complemented
39	C₅×C₇	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
40	C₂×M₁₆	—	5	1	no	Sylow
41	C₂²×C₇	—	2	2	no	—
42	C₂²×C₇	C₂×Dic₅	1	2	yes	—
43	C₂²×C₇	C₂×Dic₅	1	1	yes	characteristic
44	C₄×C₇	C₂×Dic₅	1	2	yes	—
45	C₄×C₇	C₂×Dic₅	1	1	yes	characteristic
46	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
47	C₂²×C₅	—	2	2	no	—
48	C₂²×C₅	C₂×C₄×C₇	1	2	yes	—
49	C₂²×C₅	C₂×C₄×C₇	1	1	yes	characteristic
50	C₄×C₅	C₂×C₄×C₇	1	1	yes	characteristic
51	C₄×C₅	C₂×C₄×C₇	1	2	yes	—
52	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
53	C₂²×C₄	C₇×D₅	1	1	yes	characteristic
54	M₁₆	—	5	4	no	—
55	C₂×C₈	—	5	2	no	—
56	C₂×C₇	—	2	2	no	—
57	C₂×C₇	#681	1	2	yes	direct product factor
58	C₂×C₇	C₂²×Dic₅	1	1	yes	characteristic
59	C₂×C₅	—	2	2	no	—
60	C₂×C₅	C₇×M₁₆	1	2	yes	complemented
61	C₂×C₅	C₂²×C₄×C₇	1	1	yes	characteristic
62	C₂³	C₇×Dic₅	1	1	yes	characteristic
63	C₂×C₄	C₇×Dic₅	1	2	yes	—
64	C₂×C₄	C₇×Dic₅	1	1	yes	characteristic
65	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
66	C₂×C₄	C₂×C₇×D₅	1	2	yes	—
67	C₈	—	5	4	no	—
68	C₇	C₂×#681	1	1	yes	characteristic, direct product factor, Sylow
69	C₅	C₂×C₇×M₁₆	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	2	2	no	—
71	C₂²	C₂×C₇×Dic₅	1	2	yes	—
72	C₂²	C₂×C₇×Dic₅	1	1	yes	characteristic
73	C₄	C₂×C₇×Dic₅	1	1	yes	characteristic
74	C₄	C₂×C₇×Dic₅	1	2	yes	—
75	C₄	C₂²×C₇×D₅	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₇×#681	1	2	yes	direct product factor
78	C₂	C₂²×C₇×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₅
commutator	C₂×C₅	C₂²×C₄×C₇
Frattini	C₄	C₂²×C₇×D₅
Fitting	C₂²×C₄×C₅×C₇	C₂
socle	C₂²×C₅×C₇	C₂×C₄
prime radical	C₂³×C₅×C₇	C₄
cosocle	C₄×C₅	C₂³×C₇
hypercenter	C₂²×C₄×C₇	D₅
hypocenter	C₅	C₂×C₇×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇

Complements

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

N	H ≅ G/N	type
2. C₇×#681	C₂	direct (N × H)
3. C₂×C₇×#168	C₂	semidirect
9. C₇×#168	C₂²	semidirect
11. C₂×#681	C₇	direct (N × H)
22. #681	C₂×C₇	direct (N × H)
23. C₂×#168	C₂×C₇	semidirect
25. C₂×C₅×C₇	M₁₆	semidirect
38. #168	C₂²×C₇	semidirect
39. C₅×C₇	C₂×M₁₆	semidirect
57. C₂×C₇	#681	direct (N × H)
60. C₂×C₅	C₇×M₁₆	semidirect
68. C₇	C₂×#681	direct (N × H)
69. C₅	C₂×C₇×M₁₆	semidirect
77. C₂	C₇×#681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₇	2
C₇×#681	2
C₂×C₇×#168	2
C₂×C₇×M₁₆	5
C₂×#681	7

Subgroups — C2×C7×#681 #11765165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#681 #11765165