Subgroups — C₂⁴×C₅×AGL(1,𝔽₅) #13107922

Order 1600.

2791 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×AGL(1,𝔽₅)	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₅²	C₄	1	1	yes	characteristic, complemented
4	C₂³×C₅×D₅	C₄	1	15	yes	complemented
5	C₂³×C₅×D₅	C₂²	1	15	yes	—
6	C₂²×C₅×AGL(1,𝔽₅)	C₂²	1	140	yes	direct product factor
7	C₂⁴×AGL(1,𝔽₅)	C₅	1	1	yes	maximal, characteristic, direct product factor
8	C₂⁴×C₄×C₅	—	5	1	no	maximal
9	C₂³×C₅²	C₂×C₄	1	15	yes	complemented
10	C₂²×C₅×D₅	C₂×C₄	1	105	yes	complemented
11	C₂²×C₅×D₅	C₂³	1	35	yes	—
12	C₂×C₅×AGL(1,𝔽₅)	C₂³	1	120	yes	direct product factor
13	C₂⁵×C₅	—	5	1	no	—
14	C₂⁴×D₅	C₂×C₅	1	1	yes	characteristic
15	C₂³×AGL(1,𝔽₅)	C₂×C₅	1	30	yes	direct product factor
16	C₂³×C₄×C₅	—	5	30	no	—
17	C₂²×C₅²	C₂²×C₄	1	35	yes	complemented
18	C₂×C₅×D₅	C₂²×C₄	1	105	yes	complemented
19	C₂×C₅×D₅	C₂⁴	1	15	yes	—
20	C₅×AGL(1,𝔽₅)	C₂⁴	1	16	yes	direct product factor
21	C₂⁴×C₅	—	4	1	no	—
22	C₂⁴×C₅	—	5	15	no	—
23	C₂⁴×C₅	—	5	15	no	—
24	C₂⁴×C₅	C₄×C₅	1	1	yes	characteristic, complemented
25	C₂⁴×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
26	C₂³×D₅	C₄×C₅	1	15	yes	complemented
27	C₂³×D₅	C₂²×C₅	1	15	yes	—
28	C₂²×AGL(1,𝔽₅)	C₂²×C₅	1	140	yes	direct product factor
29	C₂²×C₄×C₅	—	5	140	no	—
30	C₂⁴×C₄	—	5	1	no	Sylow
31	C₂×C₅²	C₂³×C₄	1	15	yes	complemented
32	C₅×D₅	C₂³×C₄	1	15	yes	complemented
33	C₅×D₅	C₂⁵	1	1	yes	characteristic
34	C₂³×C₅	—	4	15	no	—
35	C₂³×C₅	—	5	105	no	—
36	C₂³×C₅	—	5	35	no	—
37	C₂³×C₅	C₂×C₄×C₅	1	15	yes	complemented
38	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	15	yes	direct product factor
39	C₂²×D₅	C₂×C₄×C₅	1	105	yes	complemented
40	C₂²×D₅	C₂³×C₅	1	35	yes	—
41	C₂×AGL(1,𝔽₅)	C₂³×C₅	1	120	yes	direct product factor
42	C₂×C₄×C₅	—	5	120	no	—
43	C₂⁵	—	5	1	no	—
44	C₂³×C₄	—	5	30	no	—
45	C₅²	C₂⁴×C₄	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₅	—	4	35	no	—
47	C₂²×C₅	—	5	105	no	—
48	C₂²×C₅	—	5	15	no	—
49	C₂²×C₅	C₂²×C₄×C₅	1	35	yes	complemented
50	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	35	yes	direct product factor
51	C₂×D₅	C₂²×C₄×C₅	1	105	yes	complemented
52	C₂×D₅	C₂⁴×C₅	1	15	yes	—
53	AGL(1,𝔽₅)	C₂⁴×C₅	1	16	yes	direct product factor
54	C₄×C₅	—	5	16	no	—
55	C₂⁴	—	5	15	no	—
56	C₂⁴	—	5	15	no	—
57	C₂⁴	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
58	C₂²×C₄	—	5	140	no	—
59	C₂×C₅	—	4	15	no	—
60	C₂×C₅	—	5	1	no	—
61	C₂×C₅	—	5	15	no	—
62	C₂×C₅	C₂³×C₄×C₅	1	15	yes	complemented
63	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	15	yes	direct product factor
64	D₅	C₂³×C₄×C₅	1	15	yes	complemented
65	D₅	C₂⁵×C₅	1	1	yes	characteristic
66	C₂³	—	5	35	no	—
67	C₂³	—	5	105	no	—
68	C₂³	C₂×C₅×AGL(1,𝔽₅)	1	15	yes	direct product factor
69	C₂×C₄	—	5	120	no	—
70	C₅	—	4	1	no	—
71	C₅	C₂⁴×C₄×C₅	1	1	yes	characteristic, complemented
72	C₅	C₂⁴×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
73	C₂²	—	5	15	no	—
74	C₂²	—	5	105	no	—
75	C₂²	C₂²×C₅×AGL(1,𝔽₅)	1	35	yes	direct product factor
76	C₄	—	5	16	no	—
77	C₂	—	5	1	no	—
78	C₂	—	5	15	no	—
79	C₂	C₂³×C₅×AGL(1,𝔽₅)	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴×C₅	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂⁴×C₄×C₅
Frattini	C₁	G
Fitting, socle	C₂⁴×C₅²	C₄
prime radical	C₂⁴×C₅×D₅	C₂
cosocle	D₅	C₂⁵×C₅
LCS	G ▷ C₅
UCS	C₁ ◁ C₂⁴×C₅

Complements

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

N	H ≅ G/N	type
2. C₂³×C₅×AGL(1,𝔽₅)	C₂	direct (N × H)
3. C₂⁴×C₅²	C₄	semidirect
4. C₂³×C₅×D₅	C₄	semidirect
6. C₂²×C₅×AGL(1,𝔽₅)	C₂²	direct (N × H)
7. C₂⁴×AGL(1,𝔽₅)	C₅	direct (N × H)
9. C₂³×C₅²	C₂×C₄	semidirect
10. C₂²×C₅×D₅	C₂×C₄	semidirect
12. C₂×C₅×AGL(1,𝔽₅)	C₂³	direct (N × H)
15. C₂³×AGL(1,𝔽₅)	C₂×C₅	direct (N × H)
17. C₂²×C₅²	C₂²×C₄	semidirect
18. C₂×C₅×D₅	C₂²×C₄	semidirect
20. C₅×AGL(1,𝔽₅)	C₂⁴	direct (N × H)
24. C₂⁴×C₅	C₄×C₅	semidirect
25. C₂⁴×C₅	AGL(1,𝔽₅)	direct (N × H)
26. C₂³×D₅	C₄×C₅	semidirect
28. C₂²×AGL(1,𝔽₅)	C₂²×C₅	direct (N × H)
31. C₂×C₅²	C₂³×C₄	semidirect
32. C₅×D₅	C₂³×C₄	semidirect
37. C₂³×C₅	C₂×C₄×C₅	semidirect
38. C₂³×C₅	C₂×AGL(1,𝔽₅)	direct (N × H)
39. C₂²×D₅	C₂×C₄×C₅	semidirect
41. C₂×AGL(1,𝔽₅)	C₂³×C₅	direct (N × H)
45. C₅²	C₂⁴×C₄	semidirect
49. C₂²×C₅	C₂²×C₄×C₅	semidirect
50. C₂²×C₅	C₂²×AGL(1,𝔽₅)	direct (N × H)
51. C₂×D₅	C₂²×C₄×C₅	semidirect
53. AGL(1,𝔽₅)	C₂⁴×C₅	direct (N × H)
57. C₂⁴	C₅×AGL(1,𝔽₅)	direct (N × H)
62. C₂×C₅	C₂³×C₄×C₅	semidirect
63. C₂×C₅	C₂³×AGL(1,𝔽₅)	direct (N × H)
64. D₅	C₂³×C₄×C₅	semidirect
68. C₂³	C₂×C₅×AGL(1,𝔽₅)	direct (N × H)
71. C₅	C₂⁴×C₄×C₅	semidirect
72. C₅	C₂⁴×AGL(1,𝔽₅)	direct (N × H)
75. C₂²	C₂²×C₅×AGL(1,𝔽₅)	direct (N × H)
79. C₂	C₂³×C₅×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
C₂⁴×C₅×D₅	2
C₂³×C₅×AGL(1,𝔽₅)	2
C₂⁴×AGL(1,𝔽₅)	5
C₂⁴×C₄×C₅	5

Subgroups — C24×C5×AGL(1,𝔽5) #13107922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₅×AGL(1,𝔽₅) #13107922