Subgroups — C₂×#10587935 #11765366

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic
2	#10587935	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₇⋊#170	C₂	1	2	yes	maximal, complemented
4	C₂³×C₅×C₇	C₄	1	1	yes	characteristic
5	C₂×C₇×Dic₅	C₄	1	2	yes	—
6	C₂×C₇×Dic₅	C₄	1	1	yes	characteristic
7	C₂×C₇×Dic₅	C₂²	1	1	yes	characteristic
8	C₂×C₇×Dic₅	C₂²	1	2	yes	—
9	C₇⋊#170	C₂²	1	4	yes	complemented
10	C₂×#1136	—	5	1	no	maximal, Hall
11	C₂×#704	—	7	1	no	maximal, 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	C₇×Dic₅	C₂×C₄	1	1	yes	characteristic
16	C₇×Dic₅	C₂×C₄	1	2	yes	—
17	C₇×Dic₅	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₇	—	5	1	no	—
19	#1136	—	5	4	no	—
20	C₂×C₇⋊C₈	—	5	2	no	—
21	C₂²×Dic₅	D₇	1	1	yes	characteristic
22	#704	—	7	4	no	—
23	C₂×#170	—	7	2	no	—
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	AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₂×C₄×C₇	—	5	2	no	—
29	C₂×C₄×C₇	—	5	1	no	—
30	C₂×C₄×C₇	—	5	1	no	—
31	C₂×C₄×C₇	—	5	2	no	—
32	C₇⋊C₈	—	5	4	no	—
33	C₂³×C₅	Dic₇	1	1	yes	characteristic
34	C₂×Dic₅	Dic₇	1	2	yes	—
35	C₂×Dic₅	Dic₇	1	1	yes	characteristic
36	C₂×Dic₅	C₂×D₇	1	1	yes	characteristic
37	C₂×Dic₅	C₂×D₇	1	2	yes	—
38	#170	—	7	4	no	—
39	C₅×C₇	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
40	C₂×M₁₆	—	35	1	no	Sylow
41	C₂²×C₇	—	2	2	no	—
42	C₂²×C₇	C₂×AGL(1,𝔽₅)	1	2	yes	—
43	C₂²×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₄×C₇	—	5	1	no	—
45	C₄×C₇	—	5	1	no	—
46	C₄×C₇	—	5	2	no	—
47	C₂²×C₅	—	2	2	no	—
48	C₂²×C₅	C₂×Dic₇	1	2	yes	—
49	C₂²×C₅	C₂×Dic₇	1	1	yes	characteristic
50	Dic₅	C₂×Dic₇	1	1	yes	characteristic
51	Dic₅	C₂×Dic₇	1	2	yes	—
52	Dic₅	C₂²×D₇	1	1	yes	characteristic
53	C₂²×C₄	—	5	1	no	—
54	M₁₆	—	35	4	no	—
55	C₂×C₈	—	35	2	no	—
56	C₂×C₇	—	2	2	no	—
57	C₂×C₇	#704	1	2	yes	complemented
58	C₂×C₇	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
59	C₂×C₅	—	2	2	no	—
60	C₂×C₅	#1136	1	2	yes	complemented
61	C₂×C₅	C₂²×Dic₇	1	1	yes	characteristic
62	C₂³	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₂×C₄	—	5	2	no	—
64	C₂×C₄	—	5	2	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	—	5	1	no	—
67	C₈	—	35	4	no	—
68	C₇	C₂×#704	1	1	yes	characteristic, complemented, Sylow
69	C₅	C₂×#1136	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	2	2	no	—
71	C₂²	C₂×C₇⋊AGL(1,𝔽₅)	1	2	yes	—
72	C₂²	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₄	—	5	1	no	—
74	C₄	—	5	2	no	—
75	C₄	—	5	1	no	—
76	C₂	—	2	2	no	—
77	C₂	#10587935	1	2	yes	direct product factor
78	C₂	C₂²×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₇⋊AGL(1,𝔽₅)
commutator	C₂×C₅×C₇	C₂²×C₄
Frattini	C₂	C₂²×C₇⋊AGL(1,𝔽₅)
Fitting, prime radical	C₂³×C₅×C₇	C₄
socle	C₂²×C₅×C₇	C₂×C₄
cosocle	C₇×Dic₅	C₂³
hypercenter	C₂³	C₇⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	C₂×M₁₆
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. #10587935	C₂	direct (N × H)
3. C₂×C₇⋊#170	C₂	semidirect
9. C₇⋊#170	C₂²	semidirect
25. C₂×C₅×C₇	M₁₆	semidirect
39. C₅×C₇	C₂×M₁₆	semidirect
57. C₂×C₇	#704	semidirect
60. C₂×C₅	#1136	semidirect
68. C₇	C₂×#704	semidirect
69. C₅	C₂×#1136	semidirect
77. C₂	#10587935	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×C₇×Dic₅	2
#10587935	2
C₂×C₇⋊#170	2
C₂×#1136	5
C₂×#704	7

Subgroups — C2×#10587935 #11765366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10587935 #11765366