Subgroups — C₅×AGL(1,𝔽₅)×D₁₁ #72276321

Order 2200.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₅×D₁₁	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₁⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₁₁×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₅₅	C₄	1	1	yes	characteristic, complemented
5	C₅×D₅×C₁₁	C₂²	1	1	yes	characteristic
6	C₅²×D₁₁	C₄	1	1	yes	characteristic, complemented
7	AGL(1,𝔽₅)×D₁₁	C₅	1	1	yes	maximal, characteristic, direct product factor
8	C₄×C₅×D₁₁	—	5	1	no	maximal
9	C₅²×C₁₁	C₂×C₄	1	1	yes	characteristic, complemented, Hall
10	C₂×C₅×D₁₁	—	5	1	no	—
11	D₅×D₁₁	C₂×C₅	1	1	yes	characteristic
12	C₁₁⋊AGL(1,𝔽₅)	C₂×C₅	1	1	yes	characteristic, complemented
13	C₁₁×AGL(1,𝔽₅)	C₂×C₅	1	1	yes	characteristic, complemented
14	C₄×C₅×C₁₁	—	5	1	no	—
15	C₅×Dic₁₁	—	5	1	no	—
16	C₂×C₅×AGL(1,𝔽₅)	—	11	1	no	maximal, Hall
17	C₂×C₅×C₁₁	—	5	1	no	—
18	D₅₅	C₄×C₅	1	1	yes	characteristic, complemented
19	C₅×D₁₁	—	4	1	no	—
20	C₅×D₁₁	—	5	1	no	—
21	C₅×D₁₁	C₄×C₅	1	1	yes	characteristic, complemented
22	C₅×D₁₁	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
23	D₅×C₁₁	C₂²×C₅	1	1	yes	characteristic
24	C₂×C₅×D₅	—	11	1	no	—
25	C₅×AGL(1,𝔽₅)	—	11	1	no	—
26	C₅×AGL(1,𝔽₅)	D₁₁	1	1	yes	characteristic, direct product factor
27	C₄×D₁₁	—	5	1	no	Hall
28	C₅×C₁₁	—	4	1	no	—
29	C₅×C₁₁	C₂×C₄×C₅	1	1	yes	characteristic, complemented
30	C₅×C₁₁	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
31	C₂×C₅²	—	11	1	no	—
32	C₅×D₅	—	11	1	no	—
33	C₅×D₅	C₂×D₁₁	1	1	yes	characteristic
34	C₂×D₁₁	—	5	1	no	—
35	C₄×C₁₁	—	5	1	no	—
36	Dic₁₁	—	5	1	no	—
37	C₂×AGL(1,𝔽₅)	—	11	1	no	—
38	C₂×C₄×C₅	—	55	1	no	—
39	C₅²	C₄×D₁₁	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₁₁	—	5	1	no	—
41	D₁₁	—	5	1	no	—
42	D₁₁	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
43	C₂²×C₅	—	55	1	no	—
44	C₂×D₅	—	11	1	no	—
45	AGL(1,𝔽₅)	—	11	1	no	—
46	AGL(1,𝔽₅)	C₅×D₁₁	1	1	yes	characteristic, direct product factor
47	C₄×C₅	—	5	1	no	—
48	C₄×C₅	—	55	1	no	—
49	C₁₁	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₅	—	5	1	no	—
51	C₂×C₅	—	11	1	no	—
52	C₂×C₅	—	11	1	no	—
53	C₂×C₅	—	44	1	no	—
54	C₂×C₅	—	55	1	no	—
55	D₅	—	11	1	no	—
56	D₅	C₂×C₅×D₁₁	1	1	yes	characteristic
57	C₂×C₄	—	55	1	no	Sylow
58	C₅	—	4	1	no	—
59	C₅	C₄×C₅×D₁₁	1	1	yes	characteristic, complemented
60	C₅	AGL(1,𝔽₅)×D₁₁	1	1	yes	characteristic, direct product factor
61	C₂²	—	55	1	no	—
62	C₄	—	5	1	no	—
63	C₄	—	55	1	no	—
64	C₂	—	5	1	no	—
65	C₂	—	11	1	no	—
66	C₂	—	55	1	no	—

	subgroup	G/H
center, hypercenter	C₅	AGL(1,𝔽₅)×D₁₁
commutator, hypocenter	C₅×C₁₁	C₂×C₄×C₅
Frattini	C₁	G
Fitting, socle	C₅²×C₁₁	C₂×C₄
prime radical	C₅×D₅×D₁₁	C₂
cosocle	D₅×C₁₁	C₂²×C₅
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
2. C₅×C₁₁⋊AGL(1,𝔽₅)	C₂	semidirect
3. C₅×C₁₁×AGL(1,𝔽₅)	C₂	semidirect
4. C₅×D₅₅	C₄	semidirect
6. C₅²×D₁₁	C₄	semidirect
7. AGL(1,𝔽₅)×D₁₁	C₅	direct (N × H)
9. C₅²×C₁₁	C₂×C₄	semidirect
12. C₁₁⋊AGL(1,𝔽₅)	C₂×C₅	semidirect
13. C₁₁×AGL(1,𝔽₅)	C₂×C₅	semidirect
18. D₅₅	C₄×C₅	semidirect
21. C₅×D₁₁	C₄×C₅	semidirect
22. C₅×D₁₁	AGL(1,𝔽₅)	direct (N × H)
26. C₅×AGL(1,𝔽₅)	D₁₁	direct (N × H)
29. C₅×C₁₁	C₂×C₄×C₅	semidirect
30. C₅×C₁₁	C₂×AGL(1,𝔽₅)	semidirect
39. C₅²	C₄×D₁₁	semidirect
42. D₁₁	C₅×AGL(1,𝔽₅)	direct (N × H)
46. AGL(1,𝔽₅)	C₅×D₁₁	direct (N × H)
49. C₁₁	C₂×C₅×AGL(1,𝔽₅)	semidirect
59. C₅	C₄×C₅×D₁₁	semidirect
60. C₅	AGL(1,𝔽₅)×D₁₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×D₅×D₁₁	2
C₅×C₁₁⋊AGL(1,𝔽₅)	2
C₅×C₁₁×AGL(1,𝔽₅)	2
AGL(1,𝔽₅)×D₁₁	5
C₄×C₅×D₁₁	5
C₂×C₅×AGL(1,𝔽₅)	11

Subgroups — C5×AGL(1,𝔽5)×D11 #72276321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×AGL(1,𝔽₅)×D₁₁ #72276321