Subgroups — C₂²×C₁₁×#170 #13183341

Order 1760.

152 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	C₂×C₁₁×#170	C₂	1	6	yes	maximal, direct product factor
3	C₂³×C₅×C₁₁	C₄	1	1	yes	characteristic
4	C₂×C₁₁×Dic₅	C₄	1	3	yes	—
5	C₂×C₁₁×Dic₅	C₂²	1	3	yes	—
6	C₁₁×#170	C₂²	1	4	yes	direct product factor
7	C₂²×C₈×C₁₁	—	5	1	no	maximal, Hall
8	C₂²×C₅×C₁₁	C₈	1	4	yes	complemented
9	C₂²×C₅×C₁₁	C₂×C₄	1	3	yes	—
10	C₁₁×Dic₅	C₂×C₄	1	3	yes	—
11	C₁₁×Dic₅	C₂³	1	1	yes	characteristic
12	C₂²×C₄×C₁₁	—	5	1	no	—
13	C₂×C₈×C₁₁	—	5	6	no	—
14	C₂²×#170	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×C₁₁	C₂×C₈	1	6	yes	complemented
16	C₂×C₅×C₁₁	C₂²×C₄	1	1	yes	characteristic
17	C₂³×C₁₁	AGL(1,𝔽₅)	1	1	yes	characteristic
18	C₂×C₄×C₁₁	—	5	3	no	—
19	C₂×C₄×C₁₁	—	5	3	no	—
20	C₈×C₁₁	—	5	4	no	—
21	C₂²×Dic₅	C₂×C₁₁	1	1	yes	characteristic
22	C₂×#170	C₂×C₁₁	1	6	yes	direct product factor
23	C₅×C₁₁	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
24	C₂²×C₁₁	#170	1	4	yes	direct product factor
25	C₂²×C₁₁	C₂×AGL(1,𝔽₅)	1	3	yes	—
26	C₄×C₁₁	—	5	3	no	—
27	C₄×C₁₁	—	5	1	no	—
28	C₂³×C₅	C₄×C₁₁	1	1	yes	characteristic
29	C₂×Dic₅	C₄×C₁₁	1	3	yes	—
30	C₂×Dic₅	C₂²×C₁₁	1	3	yes	—
31	#170	C₂²×C₁₁	1	4	yes	direct product factor
32	C₂²×C₈	—	5	1	no	Sylow
33	C₂×C₁₁	C₂×#170	1	6	yes	direct product factor
34	C₂×C₁₁	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
35	C₂²×C₅	C₈×C₁₁	1	4	yes	complemented
36	C₂²×C₅	C₂×C₄×C₁₁	1	3	yes	—
37	Dic₅	C₂×C₄×C₁₁	1	3	yes	—
38	Dic₅	C₂³×C₁₁	1	1	yes	characteristic
39	C₂²×C₄	—	5	1	no	—
40	C₂×C₈	—	5	6	no	—
41	C₁₁	C₂²×#170	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₅	C₂×C₈×C₁₁	1	6	yes	complemented
43	C₂×C₅	C₂²×C₄×C₁₁	1	1	yes	characteristic
44	C₂³	C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic
45	C₂×C₄	—	5	3	no	—
46	C₂×C₄	—	5	3	no	—
47	C₈	—	5	4	no	—
48	C₅	C₂²×C₈×C₁₁	1	1	yes	characteristic, complemented, Sylow
49	C₂²	C₁₁×#170	1	4	yes	direct product factor
50	C₂²	C₂×C₁₁×AGL(1,𝔽₅)	1	3	yes	—
51	C₄	—	5	1	no	—
52	C₄	—	5	3	no	—
53	C₂	C₂×C₁₁×#170	1	6	yes	direct product factor
54	C₂	C₂²×C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₁₁	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂²×C₈×C₁₁
Frattini	C₂	C₂²×C₁₁×AGL(1,𝔽₅)
Fitting, socle, prime radical	C₂³×C₅×C₁₁	C₄
cosocle	Dic₅	C₂³×C₁₁
LCS	G ▷ C₅
UCS	C₁ ◁ C₂³×C₁₁

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₁×#170	C₂	direct (N × H)
6. C₁₁×#170	C₂²	direct (N × H)
8. C₂²×C₅×C₁₁	C₈	semidirect
14. C₂²×#170	C₁₁	direct (N × H)
15. C₂×C₅×C₁₁	C₂×C₈	semidirect
22. C₂×#170	C₂×C₁₁	direct (N × H)
23. C₅×C₁₁	C₂²×C₈	semidirect
24. C₂²×C₁₁	#170	direct (N × H)
31. #170	C₂²×C₁₁	direct (N × H)
33. C₂×C₁₁	C₂×#170	direct (N × H)
35. C₂²×C₅	C₈×C₁₁	semidirect
41. C₁₁	C₂²×#170	direct (N × H)
42. C₂×C₅	C₂×C₈×C₁₁	semidirect
48. C₅	C₂²×C₈×C₁₁	semidirect
49. C₂²	C₁₁×#170	direct (N × H)
53. C₂	C₂×C₁₁×#170	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
C₂×C₁₁×#170	2
C₂²×C₈×C₁₁	5
C₂²×#170	11

Subgroups — C22×C11×#170 #13183341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#170 #13183341