Group #13104473 — subgroups

Subgroups — C₂×C₅×D₅⋊C₁₆ #13104473

Order 1600.

207 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#674	C₂	1	2	yes	maximal, complemented
3	C₅×D₅⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic
5	C₂×C₅²×C₈	C₄	1	1	yes	characteristic
6	C₂×C₅×#168	C₂²	1	1	yes	characteristic
7	C₅×C₈×D₅	C₄	1	2	yes	—
8	C₅×C₈×D₅	C₂²	1	2	yes	—
9	C₅×#674	C₂²	1	4	yes	complemented
10	C₂×D₅⋊C₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor
11	C₂²×C₅×C₁₆	—	5	1	no	maximal
12	C₂²×C₅×D₅	C₈	1	1	yes	characteristic
13	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
14	C₂×C₅×Dic₅	C₈	1	1	yes	characteristic
15	C₄×C₅×D₅	C₈	1	2	yes	—
16	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
17	C₅²×C₈	C₂×C₄	1	2	yes	—
18	C₅×#168	C₂×C₄	1	1	yes	characteristic
19	C₅×#168	C₂³	1	1	yes	characteristic
20	C₂²×C₅×C₈	—	5	1	no	—
21	C₂×C₈×D₅	C₂×C₅	1	1	yes	characteristic
22	C₂×#674	C₂×C₅	1	2	yes	complemented
23	D₅⋊C₁₆	C₂×C₅	1	4	yes	direct product factor
24	C₂×C₅×C₁₆	—	5	4	no	—
25	C₂×C₅×C₁₆	—	5	2	no	—
26	C₂²×C₅²	C₂×C₈	1	1	yes	characteristic
27	C₂×C₅×D₅	C₁₆	1	4	yes	complemented
28	C₂×C₅×D₅	C₂×C₈	1	2	yes	—
29	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
30	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
31	C₅×Dic₅	C₂×C₈	1	2	yes	—
32	C₂²×C₄×C₅	—	5	1	no	—
33	C₂×C₄×D₅	C₄×C₅	1	1	yes	characteristic
34	C₂×C₅×C₈	—	4	1	no	—
35	C₂×C₅×C₈	—	5	1	no	—
36	C₂×C₅×C₈	—	5	2	no	—
37	C₂×C₅×C₈	—	5	2	no	—
38	C₂×C₅×C₈	C₄×C₅	1	1	yes	characteristic
39	C₂×C₅×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₂×#168	C₂²×C₅	1	1	yes	characteristic
41	C₈×D₅	C₄×C₅	1	2	yes	—
42	C₈×D₅	C₂²×C₅	1	2	yes	—
43	#674	C₂²×C₅	1	4	yes	complemented
44	C₅×C₁₆	—	5	4	no	—
45	C₂²×C₁₆	—	5	1	no	Sylow
46	C₂×C₅²	C₂×C₁₆	1	2	yes	complemented
47	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
48	C₅×D₅	C₂×C₁₆	1	4	yes	complemented
49	C₂³×C₅	—	5	1	no	—
50	C₂²×D₅	C₅×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅	—	4	1	no	—
52	C₂×C₄×C₅	—	5	1	no	—
53	C₂×C₄×C₅	—	5	2	no	—
54	C₂×C₄×C₅	—	5	2	no	—
55	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂×Dic₅	C₅×C₈	1	1	yes	characteristic
58	C₄×D₅	C₅×C₈	1	2	yes	—
59	C₄×D₅	C₂×C₄×C₅	1	2	yes	—
60	C₅×C₈	—	4	2	no	—
61	C₅×C₈	—	5	1	no	—
62	C₅×C₈	—	5	1	no	—
63	C₅×C₈	C₂×C₄×C₅	1	2	yes	—
64	C₅×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
65	#168	C₂×C₄×C₅	1	1	yes	characteristic
66	#168	C₂³×C₅	1	1	yes	characteristic
67	C₂²×C₈	—	5	1	no	—
68	C₂×C₁₆	—	5	4	no	—
69	C₂×C₁₆	—	5	2	no	—
70	C₅²	C₂²×C₁₆	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₅	—	4	1	no	—
72	C₂²×C₅	—	5	2	no	—
73	C₂²×C₅	—	5	4	no	—
74	C₂²×C₅	C₂×C₅×C₈	1	1	yes	characteristic
75	C₂²×C₅	D₅⋊C₈	1	1	yes	characteristic
76	C₂×D₅	C₅×C₁₆	1	4	yes	complemented
77	C₂×D₅	C₂×C₅×C₈	1	2	yes	—
78	C₄×C₅	—	4	1	no	—
79	C₄×C₅	—	4	1	no	—
80	C₄×C₅	—	5	2	no	—
81	C₄×C₅	C₂×C₅×C₈	1	1	yes	characteristic
82	C₄×C₅	D₅⋊C₈	1	1	yes	characteristic
83	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
84	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
85	Dic₅	C₂×C₅×C₈	1	2	yes	—
86	C₂²×C₄	—	5	1	no	—
87	C₂×C₈	—	5	1	no	—
88	C₂×C₈	—	5	2	no	—
89	C₂×C₈	—	5	2	no	—
90	C₂×C₈	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
91	C₁₆	—	5	4	no	—
92	C₂×C₅	—	4	1	no	—
93	C₂×C₅	—	4	2	no	—
94	C₂×C₅	—	5	4	no	—
95	C₂×C₅	C₂×C₅×C₁₆	1	2	yes	complemented
96	C₂×C₅	D₅⋊C₁₆	1	2	yes	direct product factor
97	C₂×C₅	C₂²×C₅×C₈	1	1	yes	characteristic
98	C₂×C₅	C₂×D₅⋊C₈	1	1	yes	characteristic
99	D₅	C₂×C₅×C₁₆	1	4	yes	complemented
100	C₂³	—	5	1	no	—
101	C₂×C₄	—	5	1	no	—
102	C₂×C₄	—	5	2	no	—
103	C₂×C₄	—	5	2	no	—
104	C₂×C₄	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₈	—	5	1	no	—
106	C₈	—	5	1	no	—
107	C₈	C₂×C₅×AGL(1,𝔽₅)	1	2	yes	—
108	C₅	—	4	1	no	—
109	C₅	C₂²×C₅×C₁₆	1	1	yes	characteristic, complemented
110	C₅	C₂×D₅⋊C₁₆	1	1	yes	characteristic, direct product factor
111	C₂²	—	5	2	no	—
112	C₂²	—	5	4	no	—
113	C₂²	C₅×D₅⋊C₈	1	1	yes	characteristic
114	C₄	—	5	2	no	—
115	C₄	C₅×D₅⋊C₈	1	1	yes	characteristic
116	C₄	C₂²×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂	—	5	4	no	—
118	C₂	C₅×D₅⋊C₁₆	1	2	yes	direct product factor
119	C₂	C₂×C₅×D₅⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×C₈×D₅

C₂

yes

maximal, characteristic

C₂×C₅×#674

C₂

yes

maximal, complemented

C₅×D₅⋊C₁₆

C₂

yes

maximal, direct product factor

C₂×C₄×C₅×D₅

C₄

yes

characteristic

C₂×C₅²×C₈

C₄

yes

characteristic

C₂×C₅×#168

C₂²

yes

characteristic

C₅×C₈×D₅

C₄

yes

—

C₅×C₈×D₅

C₂²

yes

—

C₅×#674

C₂²

yes

complemented

C₂×D₅⋊C₁₆

C₅

yes

maximal, characteristic, direct product factor

C₂²×C₅×C₁₆

—

maximal

C₂²×C₅×D₅

C₈

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₅×Dic₅

C₈

yes

characteristic

C₄×C₅×D₅

C₈

yes

—

C₄×C₅×D₅

C₂×C₄

yes

—

C₅²×C₈

C₂×C₄

yes

—

C₅×#168

C₂×C₄

yes

characteristic

C₅×#168

C₂³

yes

characteristic

C₂²×C₅×C₈

—

C₂×C₈×D₅

C₂×C₅

yes

characteristic

C₂×#674

C₂×C₅

yes

complemented

D₅⋊C₁₆

C₂×C₅

yes

direct product factor

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

—

C₂²×C₅²

C₂×C₈

yes

characteristic

C₂×C₅×D₅

C₁₆

yes

complemented

C₂×C₅×D₅

C₂×C₈

yes

—

C₄×C₅²

C₂×C₈

yes

characteristic

C₄×C₅²

C₂²×C₄

yes

characteristic

C₅×Dic₅

C₂×C₈

yes

—

C₂²×C₄×C₅

—

C₂×C₄×D₅

C₄×C₅

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄×C₅

yes

C₂×C₅×C₈

yes

C₂²×C₅

yes

characteristic

C₈×D₅

C₄×C₅

yes

—

C₈×D₅

C₂²×C₅

yes

—

#674

C₂²×C₅

yes

complemented

C₅×C₁₆

—

C₂²×C₁₆

—

Sylow

C₂×C₅²

C₂×C₁₆

yes

complemented

C₂×C₅²

C₂²×C₈

yes

characteristic

C₅×D₅

C₂×C₁₆

yes

complemented

C₂³×C₅

—

C₂²×D₅

C₅×C₈

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

yes

characteristic

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×Dic₅

C₅×C₈

yes

characteristic

C₄×D₅

C₅×C₈

yes

—

C₄×D₅

C₂×C₄×C₅

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₂×C₄×C₅

yes

—

C₅×C₈

C₂×AGL(1,𝔽₅)

yes

—

#168

C₂×C₄×C₅

yes

characteristic

#168

C₂³×C₅

yes

characteristic

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₅²

C₂²×C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₅×C₈

yes

characteristic

C₂²×C₅

D₅⋊C₈

yes

characteristic

C₂×D₅

C₅×C₁₆

yes

complemented

C₂×D₅

C₂×C₅×C₈

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×C₅×C₈

yes

characteristic

C₄×C₅

D₅⋊C₈

yes

characteristic

C₄×C₅

C₂²×C₄×C₅

yes

characteristic

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

characteristic

Dic₅

C₂×C₅×C₈

yes

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₅×AGL(1,𝔽₅)

yes

characteristic

C₁₆

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂×C₅×C₁₆

yes

complemented

C₂×C₅

D₅⋊C₁₆

yes

direct product factor

C₂×C₅

C₂²×C₅×C₈

yes

characteristic

C₂×C₅

C₂×D₅⋊C₈

yes

characteristic

D₅

C₂×C₅×C₁₆

yes

complemented

100

C₂³

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₅×AGL(1,𝔽₅)

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

C₂×C₅×AGL(1,𝔽₅)

yes

—

108

C₅

—

109

C₅

C₂²×C₅×C₁₆

yes

characteristic, complemented

110

C₅

C₂×D₅⋊C₁₆

yes

characteristic, direct product factor

111

C₂²

—

112

C₂²

—

113

C₂²

C₅×D₅⋊C₈

yes

characteristic

114

C₄

—

115

C₄

C₅×D₅⋊C₈

yes

characteristic

116

C₄

C₂²×C₅×AGL(1,𝔽₅)

yes

characteristic

117

C₂

—

118

C₂

C₅×D₅⋊C₁₆

yes

direct product factor

119

C₂

C₂×C₅×D₅⋊C₈

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅×C₈	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂²×C₅×C₁₆
Frattini	C₄	C₂²×C₅×AGL(1,𝔽₅)
Fitting	C₂×C₅²×C₈	C₄
socle	C₂²×C₅²	C₂×C₈
prime radical	C₂²×C₅×D₅	C₈
cosocle	#168	C₂³×C₅
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₅×C₈

subgroup

G/H

center, hypercenter

C₂×C₅×C₈

AGL(1,𝔽₅)

commutator, hypocenter

C₅

C₂²×C₅×C₁₆

Frattini

C₄

C₂²×C₅×AGL(1,𝔽₅)

Fitting

C₂×C₅²×C₈

C₄

socle

C₂²×C₅²

C₂×C₈

prime radical

C₂²×C₅×D₅

C₈

C₂³×C₅

G ▷ C₅

C₁ ◁ C₂×C₅×C₈

N	H ≅ G/N	type
2. C₂×C₅×#674	C₂	semidirect
3. C₅×D₅⋊C₁₆	C₂	direct (N × H)
9. C₅×#674	C₂²	semidirect
10. C₂×D₅⋊C₁₆	C₅	direct (N × H)
22. C₂×#674	C₂×C₅	semidirect
23. D₅⋊C₁₆	C₂×C₅	direct (N × H)
27. C₂×C₅×D₅	C₁₆	semidirect
43. #674	C₂²×C₅	semidirect
46. C₂×C₅²	C₂×C₁₆	semidirect
48. C₅×D₅	C₂×C₁₆	semidirect
70. C₅²	C₂²×C₁₆	semidirect
76. C₂×D₅	C₅×C₁₆	semidirect
95. C₂×C₅	C₂×C₅×C₁₆	semidirect
96. C₂×C₅	D₅⋊C₁₆	direct (N × H)
99. D₅	C₂×C₅×C₁₆	semidirect
109. C₅	C₂²×C₅×C₁₆	semidirect
110. C₅	C₂×D₅⋊C₁₆	direct (N × H)
118. C₂	C₅×D₅⋊C₁₆	direct (N × H)

H ≅ G/N

type

2. C₂×C₅×#674

C₂

semidirect

3. C₅×D₅⋊C₁₆

C₂

direct (N × H)

9. C₅×#674

C₂²

semidirect

10. C₂×D₅⋊C₁₆

C₅

direct (N × H)

22. C₂×#674

C₂×C₅

semidirect

23. D₅⋊C₁₆

C₂×C₅

direct (N × H)

27. C₂×C₅×D₅

C₁₆

semidirect

43. #674

C₂²×C₅

semidirect

46. C₂×C₅²

C₂×C₁₆

semidirect

48. C₅×D₅

C₂×C₁₆

semidirect

70. C₅²

C₂²×C₁₆

semidirect

76. C₂×D₅

C₅×C₁₆

semidirect

95. C₂×C₅

C₂×C₅×C₁₆

semidirect

96. C₂×C₅

D₅⋊C₁₆

direct (N × H)

99. D₅

C₂×C₅×C₁₆

semidirect

109. C₅

C₂²×C₅×C₁₆

semidirect

110. C₅

C₂×D₅⋊C₁₆

direct (N × H)

118. C₂

C₅×D₅⋊C₁₆

direct (N × H)

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

subgroup	index
C₂×C₅×C₈×D₅	2
C₂×C₅×#674	2
C₅×D₅⋊C₁₆	2
C₂×D₅⋊C₁₆	5
C₂²×C₅×C₁₆	5

subgroup

index

C₂×C₅×C₈×D₅

C₂×C₅×#674

C₅×D₅⋊C₁₆

C₂×D₅⋊C₁₆

C₂²×C₅×C₁₆

Subgroups — C₂×C₅×D₅⋊C₁₆ #13104473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C5×D5⋊C16 #13104473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₅⋊C₁₆ #13104473