Subgroups — C₂³×C₃×AGL(1,𝔽₅) #92129

Order 480.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×AGL(1,𝔽₅)	C₂	1	14	yes	maximal, direct product factor
3	C₂³×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₅	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₃×D₅	C₄	1	7	yes	complemented
6	C₂²×C₃×D₅	C₂²	1	7	yes	—
7	C₂×C₃×AGL(1,𝔽₅)	C₂²	1	28	yes	direct product factor
8	C₂³×C₃×C₄	—	5	1	no	maximal, Hall
9	C₂³×D₅	C₂×C₃	1	1	yes	characteristic
10	C₂²×AGL(1,𝔽₅)	C₂×C₃	1	14	yes	direct product factor
11	C₂²×C₃×C₅	C₂×C₄	1	7	yes	complemented
12	C₂×C₃×D₅	C₂×C₄	1	21	yes	complemented
13	C₂×C₃×D₅	C₂³	1	7	yes	—
14	C₃×AGL(1,𝔽₅)	C₂³	1	8	yes	direct product factor
15	C₂⁴×C₃	—	5	1	no	—
16	C₂²×C₃×C₄	—	5	14	no	—
17	C₂³×C₅	C₃×C₄	1	1	yes	characteristic, complemented
18	C₂²×D₅	C₃×C₄	1	7	yes	complemented
19	C₂²×D₅	C₂²×C₃	1	7	yes	—
20	C₂×AGL(1,𝔽₅)	C₂²×C₃	1	28	yes	direct product factor
21	C₂³×C₄	—	5	1	no	Sylow
22	C₂×C₃×C₅	C₂²×C₄	1	7	yes	complemented
23	C₃×D₅	C₂²×C₄	1	7	yes	complemented
24	C₃×D₅	C₂⁴	1	1	yes	characteristic
25	C₂³×C₃	—	5	7	no	—
26	C₂³×C₃	—	5	7	no	—
27	C₂³×C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
28	C₂×C₃×C₄	—	5	28	no	—
29	C₂²×C₅	C₂×C₃×C₄	1	7	yes	complemented
30	C₂×D₅	C₂×C₃×C₄	1	21	yes	complemented
31	C₂×D₅	C₂³×C₃	1	7	yes	—
32	AGL(1,𝔽₅)	C₂³×C₃	1	8	yes	direct product factor
33	C₂⁴	—	5	1	no	—
34	C₂²×C₄	—	5	14	no	—
35	C₃×C₅	C₂³×C₄	1	1	yes	characteristic, complemented, Hall
36	C₂²×C₃	—	5	21	no	—
37	C₂²×C₃	—	5	7	no	—
38	C₂²×C₃	C₂×AGL(1,𝔽₅)	1	7	yes	direct product factor
39	C₃×C₄	—	5	8	no	—
40	C₂×C₅	C₂²×C₃×C₄	1	7	yes	complemented
41	D₅	C₂²×C₃×C₄	1	7	yes	complemented
42	D₅	C₂⁴×C₃	1	1	yes	characteristic
43	C₂³	—	5	7	no	—
44	C₂³	—	5	7	no	—
45	C₂³	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
46	C₂×C₄	—	5	28	no	—
47	C₂×C₃	—	5	1	no	—
48	C₂×C₃	—	5	7	no	—
49	C₂×C₃	C₂²×AGL(1,𝔽₅)	1	7	yes	direct product factor
50	C₅	C₂³×C₃×C₄	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	5	7	no	—
52	C₂²	—	5	21	no	—
53	C₂²	C₂×C₃×AGL(1,𝔽₅)	1	7	yes	direct product factor
54	C₄	—	5	8	no	—
55	C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
56	C₂	—	5	1	no	—
57	C₂	—	5	7	no	—
58	C₂	C₂²×C₃×AGL(1,𝔽₅)	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₃	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂³×C₃×C₄
Frattini	C₁	G
Fitting, socle	C₂³×C₃×C₅	C₄
prime radical	C₂³×C₃×D₅	C₂
cosocle	D₅	C₂⁴×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×AGL(1,𝔽₅)	C₂	direct (N × H)
3. C₂³×AGL(1,𝔽₅)	C₃	direct (N × H)
4. C₂³×C₃×C₅	C₄	semidirect
5. C₂²×C₃×D₅	C₄	semidirect
7. C₂×C₃×AGL(1,𝔽₅)	C₂²	direct (N × H)
10. C₂²×AGL(1,𝔽₅)	C₂×C₃	direct (N × H)
11. C₂²×C₃×C₅	C₂×C₄	semidirect
12. C₂×C₃×D₅	C₂×C₄	semidirect
14. C₃×AGL(1,𝔽₅)	C₂³	direct (N × H)
17. C₂³×C₅	C₃×C₄	semidirect
18. C₂²×D₅	C₃×C₄	semidirect
20. C₂×AGL(1,𝔽₅)	C₂²×C₃	direct (N × H)
22. C₂×C₃×C₅	C₂²×C₄	semidirect
23. C₃×D₅	C₂²×C₄	semidirect
27. C₂³×C₃	AGL(1,𝔽₅)	direct (N × H)
29. C₂²×C₅	C₂×C₃×C₄	semidirect
30. C₂×D₅	C₂×C₃×C₄	semidirect
32. AGL(1,𝔽₅)	C₂³×C₃	direct (N × H)
35. C₃×C₅	C₂³×C₄	semidirect
38. C₂²×C₃	C₂×AGL(1,𝔽₅)	direct (N × H)
40. C₂×C₅	C₂²×C₃×C₄	semidirect
41. D₅	C₂²×C₃×C₄	semidirect
45. C₂³	C₃×AGL(1,𝔽₅)	direct (N × H)
49. C₂×C₃	C₂²×AGL(1,𝔽₅)	direct (N × H)
50. C₅	C₂³×C₃×C₄	semidirect
53. C₂²	C₂×C₃×AGL(1,𝔽₅)	direct (N × H)
55. C₃	C₂³×AGL(1,𝔽₅)	direct (N × H)
58. C₂	C₂²×C₃×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×C₃×D₅	2
C₂²×C₃×AGL(1,𝔽₅)	2
C₂³×AGL(1,𝔽₅)	3
C₂³×C₃×C₄	5

Subgroups — C23×C3×AGL(1,𝔽5) #92129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×AGL(1,𝔽₅) #92129