Subgroups — C₂²×C₃²×AGL(1,𝔽₅) #10624353

Order 720.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×AGL(1,𝔽₅)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×AGL(1,𝔽₅)	C₃	1	4	yes	maximal, direct product factor
4	C₂²×C₃²×C₅	C₄	1	1	yes	characteristic, complemented
5	C₂×C₃²×D₅	C₄	1	3	yes	complemented
6	C₂×C₃²×D₅	C₂²	1	3	yes	—
7	C₃²×AGL(1,𝔽₅)	C₂²	1	4	yes	direct product factor
8	C₂²×C₃²×C₄	—	5	1	no	maximal, Hall
9	C₂²×C₃×D₅	C₂×C₃	1	4	yes	—
10	C₂×C₃×AGL(1,𝔽₅)	C₂×C₃	1	24	yes	direct product factor
11	C₂×C₃²×C₅	C₂×C₄	1	3	yes	complemented
12	C₃²×D₅	C₂×C₄	1	3	yes	complemented
13	C₃²×D₅	C₂³	1	1	yes	characteristic
14	C₂²×AGL(1,𝔽₅)	C₃²	1	1	yes	characteristic, direct product factor, Hall
15	C₂³×C₃²	—	5	1	no	—
16	C₂×C₃²×C₄	—	5	6	no	—
17	C₂²×C₃×C₅	C₃×C₄	1	4	yes	complemented
18	C₂×C₃×D₅	C₃×C₄	1	12	yes	complemented
19	C₂×C₃×D₅	C₂²×C₃	1	12	yes	—
20	C₃×AGL(1,𝔽₅)	C₂²×C₃	1	16	yes	direct product factor
21	C₂²×C₃×C₄	—	5	4	no	—
22	C₃²×C₅	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
23	C₂²×D₅	C₂×C₃²	1	1	yes	characteristic
24	C₂×AGL(1,𝔽₅)	C₂×C₃²	1	6	yes	direct product factor
25	C₂²×C₃²	—	5	3	no	—
26	C₂²×C₃²	—	5	3	no	—
27	C₂²×C₃²	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
28	C₃²×C₄	—	5	4	no	—
29	C₂×C₃×C₅	C₂×C₃×C₄	1	12	yes	complemented
30	C₃×D₅	C₂×C₃×C₄	1	12	yes	complemented
31	C₃×D₅	C₂³×C₃	1	4	yes	—
32	C₂³×C₃	—	5	4	no	—
33	C₂×C₃×C₄	—	5	24	no	—
34	C₂²×C₅	C₃²×C₄	1	1	yes	characteristic, complemented
35	C₂×D₅	C₃²×C₄	1	3	yes	complemented
36	C₂×D₅	C₂²×C₃²	1	3	yes	—
37	AGL(1,𝔽₅)	C₂²×C₃²	1	4	yes	direct product factor
38	C₂×C₃²	—	5	3	no	—
39	C₂×C₃²	—	5	1	no	—
40	C₂×C₃²	C₂×AGL(1,𝔽₅)	1	3	yes	direct product factor
41	C₂²×C₄	—	5	1	no	Sylow
42	C₃×C₅	C₂²×C₃×C₄	1	4	yes	complemented
43	C₂²×C₃	—	5	12	no	—
44	C₂²×C₃	—	5	12	no	—
45	C₂²×C₃	C₃×AGL(1,𝔽₅)	1	4	yes	direct product factor
46	C₃×C₄	—	5	16	no	—
47	C₂×C₅	C₂×C₃²×C₄	1	3	yes	complemented
48	D₅	C₂×C₃²×C₄	1	3	yes	complemented
49	D₅	C₂³×C₃²	1	1	yes	characteristic
50	C₃²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³	—	5	1	no	—
52	C₂×C₄	—	5	6	no	—
53	C₂×C₃	—	5	4	no	—
54	C₂×C₃	—	5	12	no	—
55	C₂×C₃	C₂×C₃×AGL(1,𝔽₅)	1	12	yes	direct product factor
56	C₅	C₂²×C₃²×C₄	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	5	3	no	—
58	C₂²	—	5	3	no	—
59	C₂²	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
60	C₄	—	5	4	no	—
61	C₃	C₂²×C₃×AGL(1,𝔽₅)	1	4	yes	direct product factor
62	C₂	—	5	1	no	—
63	C₂	—	5	3	no	—
64	C₂	C₂×C₃²×AGL(1,𝔽₅)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃²	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂²×C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₅	C₄
prime radical	C₂²×C₃²×D₅	C₂
cosocle	D₅	C₂³×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₃²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×AGL(1,𝔽₅)	C₂	direct (N × H)
3. C₂²×C₃×AGL(1,𝔽₅)	C₃	direct (N × H)
4. C₂²×C₃²×C₅	C₄	semidirect
5. C₂×C₃²×D₅	C₄	semidirect
7. C₃²×AGL(1,𝔽₅)	C₂²	direct (N × H)
10. C₂×C₃×AGL(1,𝔽₅)	C₂×C₃	direct (N × H)
11. C₂×C₃²×C₅	C₂×C₄	semidirect
12. C₃²×D₅	C₂×C₄	semidirect
14. C₂²×AGL(1,𝔽₅)	C₃²	direct (N × H)
17. C₂²×C₃×C₅	C₃×C₄	semidirect
18. C₂×C₃×D₅	C₃×C₄	semidirect
20. C₃×AGL(1,𝔽₅)	C₂²×C₃	direct (N × H)
22. C₃²×C₅	C₂²×C₄	semidirect
24. C₂×AGL(1,𝔽₅)	C₂×C₃²	direct (N × H)
27. C₂²×C₃²	AGL(1,𝔽₅)	direct (N × H)
29. C₂×C₃×C₅	C₂×C₃×C₄	semidirect
30. C₃×D₅	C₂×C₃×C₄	semidirect
34. C₂²×C₅	C₃²×C₄	semidirect
35. C₂×D₅	C₃²×C₄	semidirect
37. AGL(1,𝔽₅)	C₂²×C₃²	direct (N × H)
40. C₂×C₃²	C₂×AGL(1,𝔽₅)	direct (N × H)
42. C₃×C₅	C₂²×C₃×C₄	semidirect
45. C₂²×C₃	C₃×AGL(1,𝔽₅)	direct (N × H)
47. C₂×C₅	C₂×C₃²×C₄	semidirect
48. D₅	C₂×C₃²×C₄	semidirect
50. C₃²	C₂²×AGL(1,𝔽₅)	direct (N × H)
55. C₂×C₃	C₂×C₃×AGL(1,𝔽₅)	direct (N × H)
56. C₅	C₂²×C₃²×C₄	semidirect
59. C₂²	C₃²×AGL(1,𝔽₅)	direct (N × H)
61. C₃	C₂²×C₃×AGL(1,𝔽₅)	direct (N × H)
64. C₂	C₂×C₃²×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
C₂²×C₃²×D₅	2
C₂×C₃²×AGL(1,𝔽₅)	2
C₂²×C₃×AGL(1,𝔽₅)	3
C₂²×C₃²×C₄	5

Subgroups — C22×C32×AGL(1,𝔽5) #10624353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×AGL(1,𝔽₅) #10624353