Subgroups — C₂²×C₁₁⋊#170 #13183354

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×C₁₁⋊C₈	—	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×C₁₁⋊C₈	—	5	6	no	—
14	C₂²×#170	—	11	1	no	maximal, 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	C₁₁⋊C₈	—	5	4	no	—
21	C₂²×Dic₅	D₁₁	1	1	yes	characteristic
22	C₂×#170	—	11	6	no	—
23	C₅×C₁₁	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
24	C₂²×C₁₁	#170	1	4	yes	complemented
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	Dic₁₁	1	1	yes	characteristic
29	C₂×Dic₅	Dic₁₁	1	3	yes	—
30	C₂×Dic₅	C₂×D₁₁	1	3	yes	—
31	#170	—	11	4	no	—
32	C₂²×C₈	—	55	1	no	Sylow
33	C₂×C₁₁	C₂×#170	1	6	yes	complemented
34	C₂×C₁₁	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
35	C₂²×C₅	C₁₁⋊C₈	1	4	yes	complemented
36	C₂²×C₅	C₂×Dic₁₁	1	3	yes	—
37	Dic₅	C₂×Dic₁₁	1	3	yes	—
38	Dic₅	C₂²×D₁₁	1	1	yes	characteristic
39	C₂²×C₄	—	5	1	no	—
40	C₂×C₈	—	55	6	no	—
41	C₁₁	C₂²×#170	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₅	C₂×C₁₁⋊C₈	1	6	yes	complemented
43	C₂×C₅	C₂²×Dic₁₁	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	—	55	4	no	—
48	C₅	C₂²×C₁₁⋊C₈	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×C₁₁	C₂²×C₈
Frattini	C₂	C₂²×C₁₁⋊AGL(1,𝔽₅)
Fitting, socle, prime radical	C₂³×C₅×C₁₁	C₄
cosocle	C₁₁×Dic₅	C₂³
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₂³

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
15. C₂×C₅×C₁₁	C₂×C₈	semidirect
23. C₅×C₁₁	C₂²×C₈	semidirect
24. C₂²×C₁₁	#170	semidirect
33. C₂×C₁₁	C₂×#170	semidirect
35. C₂²×C₅	C₁₁⋊C₈	semidirect
41. C₁₁	C₂²×#170	semidirect
42. C₂×C₅	C₂×C₁₁⋊C₈	semidirect
48. C₅	C₂²×C₁₁⋊C₈	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×C₁₁⋊C₈	5
C₂²×#170	11

Subgroups — C22×C11⋊#170 #13183354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁⋊#170 #13183354