Subgroups — C₂³×C₅×#170 #13107734

Order 1600.

743 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#170	C₂	1	14	yes	maximal, direct product factor
3	C₂⁴×C₅²	C₄	1	1	yes	characteristic
4	C₂²×C₅×Dic₅	C₄	1	7	yes	—
5	C₂²×C₅×Dic₅	C₂²	1	7	yes	—
6	C₂×C₅×#170	C₂²	1	28	yes	direct product factor
7	C₂³×#170	C₅	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₅×C₈	—	5	1	no	maximal
9	C₂³×C₅²	C₈	1	8	yes	complemented
10	C₂³×C₅²	C₂×C₄	1	7	yes	—
11	C₂×C₅×Dic₅	C₂×C₄	1	21	yes	—
12	C₂×C₅×Dic₅	C₂³	1	7	yes	—
13	C₅×#170	C₂³	1	8	yes	direct product factor
14	C₂³×C₄×C₅	—	5	1	no	—
15	C₂³×Dic₅	C₂×C₅	1	1	yes	characteristic
16	C₂²×#170	C₂×C₅	1	14	yes	direct product factor
17	C₂²×C₅×C₈	—	5	14	no	—
18	C₂²×C₅²	C₂×C₈	1	28	yes	complemented
19	C₂²×C₅²	C₂²×C₄	1	7	yes	—
20	C₅×Dic₅	C₂²×C₄	1	7	yes	—
21	C₅×Dic₅	C₂⁴	1	1	yes	characteristic
22	C₂⁴×C₅	—	4	1	no	—
23	C₂⁴×C₅	C₄×C₅	1	1	yes	characteristic
24	C₂⁴×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₂²×C₄×C₅	—	5	7	no	—
26	C₂²×C₄×C₅	—	5	7	no	—
27	C₂²×Dic₅	C₄×C₅	1	7	yes	—
28	C₂²×Dic₅	C₂²×C₅	1	7	yes	—
29	C₂×#170	C₂²×C₅	1	28	yes	direct product factor
30	C₂×C₅×C₈	—	5	28	no	—
31	C₂³×C₈	—	5	1	no	Sylow
32	C₂×C₅²	C₂²×C₈	1	14	yes	complemented
33	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
34	C₂³×C₅	—	4	8	no	—
35	C₂³×C₅	—	4	7	no	—
36	C₂³×C₅	C₅×C₈	1	8	yes	complemented
37	C₂³×C₅	#170	1	8	yes	direct product factor
38	C₂³×C₅	C₂×C₄×C₅	1	7	yes	—
39	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	7	yes	—
40	C₂×C₄×C₅	—	5	21	no	—
41	C₂×C₄×C₅	—	5	7	no	—
42	C₂×Dic₅	C₂×C₄×C₅	1	21	yes	—
43	C₂×Dic₅	C₂³×C₅	1	7	yes	—
44	#170	C₂³×C₅	1	8	yes	direct product factor
45	C₅×C₈	—	5	8	no	—
46	C₂³×C₄	—	5	1	no	—
47	C₂²×C₈	—	5	14	no	—
48	C₅²	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₅	—	4	28	no	—
50	C₂²×C₅	—	4	7	no	—
51	C₂²×C₅	C₂×C₅×C₈	1	28	yes	complemented
52	C₂²×C₅	C₂×#170	1	28	yes	direct product factor
53	C₂²×C₅	C₂²×C₄×C₅	1	7	yes	—
54	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	7	yes	—
55	C₄×C₅	—	5	7	no	—
56	C₄×C₅	—	5	1	no	—
57	Dic₅	C₂²×C₄×C₅	1	7	yes	—
58	Dic₅	C₂⁴×C₅	1	1	yes	characteristic
59	C₂⁴	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₂²×C₄	—	5	7	no	—
61	C₂²×C₄	—	5	7	no	—
62	C₂×C₈	—	5	28	no	—
63	C₂×C₅	—	4	14	no	—
64	C₂×C₅	—	4	1	no	—
65	C₂×C₅	C₂²×C₅×C₈	1	14	yes	complemented
66	C₂×C₅	C₂²×#170	1	14	yes	direct product factor
67	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
68	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂³	C₅×#170	1	8	yes	direct product factor
70	C₂³	C₂×C₅×AGL(1,𝔽₅)	1	7	yes	—
71	C₂×C₄	—	5	7	no	—
72	C₂×C₄	—	5	21	no	—
73	C₈	—	5	8	no	—
74	C₅	—	4	1	no	—
75	C₅	C₂³×C₅×C₈	1	1	yes	characteristic, complemented
76	C₅	C₂³×#170	1	1	yes	characteristic, direct product factor
77	C₂²	C₂×C₅×#170	1	28	yes	direct product factor
78	C₂²	C₂²×C₅×AGL(1,𝔽₅)	1	7	yes	—
79	C₄	—	5	1	no	—
80	C₄	—	5	7	no	—
81	C₂	C₂²×C₅×#170	1	14	yes	direct product factor
82	C₂	C₂³×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₅	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂³×C₅×C₈
Frattini	C₂	C₂³×C₅×AGL(1,𝔽₅)
Fitting, socle, prime radical	C₂⁴×C₅²	C₄
cosocle	Dic₅	C₂⁴×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. C₂²×C₅×#170	C₂	direct (N × H)
6. C₂×C₅×#170	C₂²	direct (N × H)
7. C₂³×#170	C₅	direct (N × H)
9. C₂³×C₅²	C₈	semidirect
13. C₅×#170	C₂³	direct (N × H)
16. C₂²×#170	C₂×C₅	direct (N × H)
18. C₂²×C₅²	C₂×C₈	semidirect
29. C₂×#170	C₂²×C₅	direct (N × H)
32. C₂×C₅²	C₂²×C₈	semidirect
36. C₂³×C₅	C₅×C₈	semidirect
37. C₂³×C₅	#170	direct (N × H)
44. #170	C₂³×C₅	direct (N × H)
48. C₅²	C₂³×C₈	semidirect
51. C₂²×C₅	C₂×C₅×C₈	semidirect
52. C₂²×C₅	C₂×#170	direct (N × H)
65. C₂×C₅	C₂²×C₅×C₈	semidirect
66. C₂×C₅	C₂²×#170	direct (N × H)
69. C₂³	C₅×#170	direct (N × H)
75. C₅	C₂³×C₅×C₈	semidirect
76. C₅	C₂³×#170	direct (N × H)
77. C₂²	C₂×C₅×#170	direct (N × H)
81. C₂	C₂²×C₅×#170	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×C₅×Dic₅	2
C₂²×C₅×#170	2
C₂³×#170	5
C₂³×C₅×C₈	5

Subgroups — C23×C5×#170 #13107734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×#170 #13107734