Subgroups — C₂³×C₁₁×AGL(1,𝔽₅) #13183470

Order 1760.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₁×AGL(1,𝔽₅)	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₅×C₁₁	C₄	1	1	yes	characteristic, complemented
4	C₂²×D₅×C₁₁	C₄	1	7	yes	complemented
5	C₂²×D₅×C₁₁	C₂²	1	7	yes	—
6	C₂×C₁₁×AGL(1,𝔽₅)	C₂²	1	28	yes	direct product factor
7	C₂³×C₄×C₁₁	—	5	1	no	maximal, Hall
8	C₂²×C₅×C₁₁	C₂×C₄	1	7	yes	complemented
9	C₂×D₅×C₁₁	C₂×C₄	1	21	yes	complemented
10	C₂×D₅×C₁₁	C₂³	1	7	yes	—
11	C₁₁×AGL(1,𝔽₅)	C₂³	1	8	yes	direct product factor
12	C₂⁴×C₁₁	—	5	1	no	—
13	C₂²×C₄×C₁₁	—	5	14	no	—
14	C₂³×AGL(1,𝔽₅)	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×C₁₁	C₂²×C₄	1	7	yes	complemented
16	D₅×C₁₁	C₂²×C₄	1	7	yes	complemented
17	D₅×C₁₁	C₂⁴	1	1	yes	characteristic
18	C₂³×C₁₁	—	5	7	no	—
19	C₂³×C₁₁	—	5	7	no	—
20	C₂³×C₁₁	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
21	C₂×C₄×C₁₁	—	5	28	no	—
22	C₂³×D₅	C₂×C₁₁	1	1	yes	characteristic
23	C₂²×AGL(1,𝔽₅)	C₂×C₁₁	1	14	yes	direct product factor
24	C₅×C₁₁	C₂³×C₄	1	1	yes	characteristic, complemented, Hall
25	C₂²×C₁₁	—	5	7	no	—
26	C₂²×C₁₁	—	5	21	no	—
27	C₂²×C₁₁	C₂×AGL(1,𝔽₅)	1	7	yes	direct product factor
28	C₄×C₁₁	—	5	8	no	—
29	C₂³×C₅	C₄×C₁₁	1	1	yes	characteristic, complemented
30	C₂²×D₅	C₄×C₁₁	1	7	yes	complemented
31	C₂²×D₅	C₂²×C₁₁	1	7	yes	—
32	C₂×AGL(1,𝔽₅)	C₂²×C₁₁	1	28	yes	direct product factor
33	C₂³×C₄	—	5	1	no	Sylow
34	C₂×C₁₁	—	5	7	no	—
35	C₂×C₁₁	—	5	1	no	—
36	C₂×C₁₁	C₂²×AGL(1,𝔽₅)	1	7	yes	direct product factor
37	C₂²×C₅	C₂×C₄×C₁₁	1	7	yes	complemented
38	C₂×D₅	C₂×C₄×C₁₁	1	21	yes	complemented
39	C₂×D₅	C₂³×C₁₁	1	7	yes	—
40	AGL(1,𝔽₅)	C₂³×C₁₁	1	8	yes	direct product factor
41	C₂⁴	—	5	1	no	—
42	C₂²×C₄	—	5	14	no	—
43	C₁₁	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×C₅	C₂²×C₄×C₁₁	1	7	yes	complemented
45	D₅	C₂²×C₄×C₁₁	1	7	yes	complemented
46	D₅	C₂⁴×C₁₁	1	1	yes	characteristic
47	C₂³	—	5	7	no	—
48	C₂³	—	5	7	no	—
49	C₂³	C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
50	C₂×C₄	—	5	28	no	—
51	C₅	C₂³×C₄×C₁₁	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	5	7	no	—
53	C₂²	—	5	21	no	—
54	C₂²	C₂×C₁₁×AGL(1,𝔽₅)	1	7	yes	direct product factor
55	C₄	—	5	8	no	—
56	C₂	—	5	1	no	—
57	C₂	—	5	7	no	—
58	C₂	C₂²×C₁₁×AGL(1,𝔽₅)	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₁₁	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂³×C₄×C₁₁
Frattini	C₁	G
Fitting, socle	C₂³×C₅×C₁₁	C₄
prime radical	C₂³×D₅×C₁₁	C₂
cosocle	D₅	C₂⁴×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. C₂²×C₁₁×AGL(1,𝔽₅)	C₂	direct (N × H)
3. C₂³×C₅×C₁₁	C₄	semidirect
4. C₂²×D₅×C₁₁	C₄	semidirect
6. C₂×C₁₁×AGL(1,𝔽₅)	C₂²	direct (N × H)
8. C₂²×C₅×C₁₁	C₂×C₄	semidirect
9. C₂×D₅×C₁₁	C₂×C₄	semidirect
11. C₁₁×AGL(1,𝔽₅)	C₂³	direct (N × H)
14. C₂³×AGL(1,𝔽₅)	C₁₁	direct (N × H)
15. C₂×C₅×C₁₁	C₂²×C₄	semidirect
16. D₅×C₁₁	C₂²×C₄	semidirect
20. C₂³×C₁₁	AGL(1,𝔽₅)	direct (N × H)
23. C₂²×AGL(1,𝔽₅)	C₂×C₁₁	direct (N × H)
24. C₅×C₁₁	C₂³×C₄	semidirect
27. C₂²×C₁₁	C₂×AGL(1,𝔽₅)	direct (N × H)
29. C₂³×C₅	C₄×C₁₁	semidirect
30. C₂²×D₅	C₄×C₁₁	semidirect
32. C₂×AGL(1,𝔽₅)	C₂²×C₁₁	direct (N × H)
36. C₂×C₁₁	C₂²×AGL(1,𝔽₅)	direct (N × H)
37. C₂²×C₅	C₂×C₄×C₁₁	semidirect
38. C₂×D₅	C₂×C₄×C₁₁	semidirect
40. AGL(1,𝔽₅)	C₂³×C₁₁	direct (N × H)
43. C₁₁	C₂³×AGL(1,𝔽₅)	direct (N × H)
44. C₂×C₅	C₂²×C₄×C₁₁	semidirect
45. D₅	C₂²×C₄×C₁₁	semidirect
49. C₂³	C₁₁×AGL(1,𝔽₅)	direct (N × H)
51. C₅	C₂³×C₄×C₁₁	semidirect
54. C₂²	C₂×C₁₁×AGL(1,𝔽₅)	direct (N × H)
58. C₂	C₂²×C₁₁×AGL(1,𝔽₅)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₅×C₁₁	2
C₂²×C₁₁×AGL(1,𝔽₅)	2
C₂³×C₄×C₁₁	5
C₂³×AGL(1,𝔽₅)	11

Subgroups — C23×C11×AGL(1,𝔽5) #13183470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₁×AGL(1,𝔽₅) #13183470