Subgroups — C₂×C₁₁×#704 #13183342

Order 1760.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#704	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₁×#170	C₂	1	2	yes	maximal, complemented
4	C₂³×C₅×C₁₁	C₄	1	1	yes	characteristic
5	C₂×C₁₁×Dic₅	C₄	1	2	yes	—
6	C₂×C₁₁×Dic₅	C₄	1	1	yes	characteristic
7	C₂×C₁₁×Dic₅	C₂²	1	1	yes	characteristic
8	C₂×C₁₁×Dic₅	C₂²	1	2	yes	—
9	C₁₁×#170	C₂²	1	4	yes	complemented
10	C₂×C₁₁×M₁₆	—	5	1	no	maximal, Hall
11	C₂²×C₅×C₁₁	—	2	2	no	—
12	C₂²×C₅×C₁₁	C₂×C₄	1	2	yes	—
13	C₂²×C₅×C₁₁	C₂×C₄	1	1	yes	characteristic
14	C₁₁×Dic₅	C₂×C₄	1	1	yes	characteristic
15	C₁₁×Dic₅	C₂×C₄	1	2	yes	—
16	C₁₁×Dic₅	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₁₁	—	5	1	no	—
18	C₁₁×M₁₆	—	5	4	no	—
19	C₂×C₈×C₁₁	—	5	2	no	—
20	C₂×#704	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×C₁₁	—	2	2	no	—
22	C₂×C₅×C₁₁	M₁₆	1	2	yes	complemented
23	C₂×C₅×C₁₁	C₂²×C₄	1	1	yes	characteristic
24	C₂³×C₁₁	AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₂×C₄×C₁₁	—	5	2	no	—
26	C₂×C₄×C₁₁	—	5	2	no	—
27	C₂×C₄×C₁₁	—	5	1	no	—
28	C₂×C₄×C₁₁	—	5	1	no	—
29	C₈×C₁₁	—	5	4	no	—
30	C₂²×Dic₅	C₂×C₁₁	1	1	yes	characteristic
31	#704	C₂×C₁₁	1	4	yes	direct product factor
32	C₂×#170	C₂×C₁₁	1	2	yes	complemented
33	C₅×C₁₁	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
34	C₂²×C₁₁	—	2	2	no	—
35	C₂²×C₁₁	C₂×AGL(1,𝔽₅)	1	2	yes	—
36	C₂²×C₁₁	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
37	C₄×C₁₁	—	5	1	no	—
38	C₄×C₁₁	—	5	2	no	—
39	C₄×C₁₁	—	5	1	no	—
40	C₂³×C₅	C₄×C₁₁	1	1	yes	characteristic
41	C₂×Dic₅	C₄×C₁₁	1	2	yes	—
42	C₂×Dic₅	C₄×C₁₁	1	1	yes	characteristic
43	C₂×Dic₅	C₂²×C₁₁	1	1	yes	characteristic
44	C₂×Dic₅	C₂²×C₁₁	1	2	yes	—
45	#170	C₂²×C₁₁	1	4	yes	complemented
46	C₂×M₁₆	—	5	1	no	Sylow
47	C₂×C₁₁	—	2	2	no	—
48	C₂×C₁₁	#704	1	2	yes	direct product factor
49	C₂×C₁₁	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₂²×C₅	—	2	2	no	—
51	C₂²×C₅	C₂×C₄×C₁₁	1	2	yes	—
52	C₂²×C₅	C₂×C₄×C₁₁	1	1	yes	characteristic
53	Dic₅	C₂×C₄×C₁₁	1	1	yes	characteristic
54	Dic₅	C₂×C₄×C₁₁	1	2	yes	—
55	Dic₅	C₂³×C₁₁	1	1	yes	characteristic
56	C₂²×C₄	—	5	1	no	—
57	M₁₆	—	5	4	no	—
58	C₂×C₈	—	5	2	no	—
59	C₁₁	C₂×#704	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₅	—	2	2	no	—
61	C₂×C₅	C₁₁×M₁₆	1	2	yes	complemented
62	C₂×C₅	C₂²×C₄×C₁₁	1	1	yes	characteristic
63	C₂³	C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₂×C₄	—	5	1	no	—
65	C₂×C₄	—	5	2	no	—
66	C₂×C₄	—	5	2	no	—
67	C₂×C₄	—	5	1	no	—
68	C₈	—	5	4	no	—
69	C₅	C₂×C₁₁×M₁₆	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	2	2	no	—
71	C₂²	C₂×C₁₁×AGL(1,𝔽₅)	1	2	yes	—
72	C₂²	C₂×C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₄	—	5	1	no	—
74	C₄	—	5	1	no	—
75	C₄	—	5	2	no	—
76	C₂	—	2	2	no	—
77	C₂	C₁₁×#704	1	2	yes	direct product factor
78	C₂	C₂²×C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₁	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₄×C₁₁
Frattini	C₂	C₂²×C₁₁×AGL(1,𝔽₅)
Fitting, prime radical	C₂³×C₅×C₁₁	C₄
socle	C₂²×C₅×C₁₁	C₂×C₄
cosocle	Dic₅	C₂³×C₁₁
hypercenter	C₂³×C₁₁	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₁₁×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂³×C₁₁

Complements

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

N	H ≅ G/N	type
2. C₁₁×#704	C₂	direct (N × H)
3. C₂×C₁₁×#170	C₂	semidirect
9. C₁₁×#170	C₂²	semidirect
20. C₂×#704	C₁₁	direct (N × H)
22. C₂×C₅×C₁₁	M₁₆	semidirect
31. #704	C₂×C₁₁	direct (N × H)
32. C₂×#170	C₂×C₁₁	semidirect
33. C₅×C₁₁	C₂×M₁₆	semidirect
45. #170	C₂²×C₁₁	semidirect
48. C₂×C₁₁	#704	direct (N × H)
59. C₁₁	C₂×#704	direct (N × H)
61. C₂×C₅	C₁₁×M₁₆	semidirect
69. C₅	C₂×C₁₁×M₁₆	semidirect
77. C₂	C₁₁×#704	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×C₁₁×Dic₅	2
C₁₁×#704	2
C₂×C₁₁×#170	2
C₂×C₁₁×M₁₆	5
C₂×#704	11

Subgroups — C2×C11×#704 #13183342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#704 #13183342