Group #13101052 — subgroups

Subgroups — #13101052

Order 1600.

304 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 72 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	#11717017	C₂	1	2	yes	maximal, complemented
2	C₂×#87770	C₂	1	1	yes	maximal, characteristic
3	#11716682	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3953	C₂	1	2	yes	maximal, complemented
5	#11716615	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×D₂₀	C₂²	1	2	yes	—
7	C₂×D₅×Dic₅	—	2	2	no	—
8	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
9	C₄×#1039	C₂²	1	1	yes	characteristic
10	C₂×#6029	C₂²	1	1	yes	characteristic
11	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
12	#87778	—	2	2	no	—
13	#87770	C₄	1	4	yes	complemented
14	#65074	—	5	2	no	maximal
15	C₂²×C₅×D₅	—	2	2	no	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×#1039	D₄	1	2	yes	—
18	C₂×C₅×Dic₅	—	2	2	no	—
19	C₅×D₂₀	—	2	2	no	—
20	C₅×D₂₀	C₂×C₄	1	4	yes	complemented
21	D₅×Dic₅	—	4	4	no	—
22	C₅²×C₈	—	2	1	no	—
23	#6029	C₂×C₄	1	2	yes	complemented
24	D₄×Dic₅	—	5	2	no	—
25	C₂×#686	—	5	2	no	—
26	#4039	—	5	2	no	—
27	C₅×#102	—	5	2	no	—
28	#3953	D₅	1	2	yes	complemented
29	#3947	—	5	2	no	—
30	#3947	—	10	2	no	—
31	#3940	—	5	2	no	—
32	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
33	C₂×C₅×D₅	—	2	4	no	—
34	C₂×C₅×D₅	—	4	2	no	—
35	C₂×C₅×D₅	—	4	2	no	—
36	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₅²	#41	1	1	yes	characteristic
38	#1039	—	2	2	no	—
39	C₅×Dic₅	—	4	2	no	—
40	C₂×C₅×D₄	—	5	2	no	—
41	C₂²×Dic₅	—	10	2	no	—
42	C₂×D₂₀	C₂×D₅	1	2	yes	—
43	C₂×C₄×D₅	—	10	2	no	—
44	C₂×C₅×C₈	—	2	2	no	—
45	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
46	C₅×C₄⋊C₄	—	5	2	no	—
47	C₂²⋊Dic₅	—	10	2	no	—
48	#686	—	5	8	no	—
49	#685	—	10	2	no	—
50	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
51	C₄×Dic₅	—	5	2	no	—
52	C₄×Dic₅	—	10	2	no	—
53	C₂×#168	—	5	2	no	—
54	C₂×#168	—	10	2	no	—
55	#442	—	25	1	no	Sylow
56	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
57	C₂×C₅²	AGL(1,ℤ/8)	1	2	yes	—
58	C₅×D₅	—	4	4	no	—
59	C₂³×C₅	—	10	2	no	—
60	C₂²×D₅	—	2	2	no	—
61	C₅×D₄	—	5	4	no	—
62	C₅×D₄	—	10	2	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	10	2	no	—
65	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
66	C₂×Dic₅	—	2	2	no	—
67	C₂×Dic₅	—	5	2	no	—
68	C₂×Dic₅	—	5	2	no	—
69	C₂×Dic₅	—	10	4	no	—
70	C₂×Dic₅	—	20	4	no	—
71	D₂₀	—	2	2	no	—
72	D₂₀	C₄×D₅	1	4	yes	complemented
73	C₄×D₅	—	20	4	no	—
74	C₅×C₈	—	2	4	no	—
75	C₅×C₈	—	2	2	no	—
76	#168	—	5	4	no	—
77	#168	—	10	4	no	—
78	C₂×D₈	—	25	1	no	—
79	C₄×D₄	—	25	2	no	—
80	#106	—	25	1	no	—
81	#102	—	5	2	no	—
82	#97	—	25	1	no	—
83	C₅²	#442	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₅	—	2	2	no	—
85	C₂²×C₅	—	10	4	no	—
86	C₂²×C₅	—	20	2	no	—
87	C₂²×C₅	—	20	2	no	—
88	C₂²×C₅	D₄×D₅	1	2	yes	—
89	C₂×D₅	—	2	4	no	—
90	C₂×D₅	—	4	2	no	—
91	C₂×D₅	—	4	2	no	—
92	C₄×C₅	—	2	2	no	—
93	C₄×C₅	—	2	2	no	—
94	C₄×C₅	—	20	2	no	—
95	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
96	C₄×C₅	#711	1	2	yes	—
97	Dic₅	—	4	2	no	—
98	Dic₅	—	10	2	no	—
99	Dic₅	—	10	2	no	—
100	Dic₅	—	20	4	no	—
101	C₂×D₄	—	5	2	no	—
102	C₂²×C₄	—	50	2	no	—
103	D₈	—	25	4	no	—
104	C₂×C₈	—	25	1	no	—
105	C₂×C₈	D₅²	1	1	yes	characteristic
106	C₄⋊C₄	—	5	2	no	—
107	C₂²⋊C₄	—	50	2	no	—
108	C₄²	—	25	1	no	—
109	C₂×C₅	—	2	4	no	—
110	C₂×C₅	—	2	2	no	—
111	C₂×C₅	—	20	4	no	—
112	C₂×C₅	#4027	1	2	yes	—
113	C₂×C₅	#4057	1	2	yes	—
114	C₂×C₅	#4060	1	2	yes	—
115	D₅	—	4	4	no	—
116	C₂³	—	10	2	no	—
117	D₄	—	5	4	no	—
118	D₄	—	10	2	no	—
119	C₂×C₄	—	10	2	no	—
120	C₂×C₄	—	25	2	no	—
121	C₂×C₄	—	100	4	no	—
122	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	25	2	no	—
125	C₅	—	2	2	no	—
126	C₅	#65074	1	2	yes	complemented
127	C₂²	—	10	4	no	—
128	C₂²	—	20	2	no	—
129	C₂²	—	20	2	no	—
130	C₂²	#87877	1	1	yes	characteristic
131	C₄	—	20	2	no	—
132	C₄	—	50	2	no	—
133	C₄	C₄×D₅²	1	1	yes	characteristic
134	C₄	#87880	1	1	yes	characteristic
135	C₂	—	20	4	no	—
136	C₂	#11716916	1	2	yes	—
137	C₂	#11717020	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717017

C₂

yes

maximal, complemented

C₂×#87770

C₂

yes

maximal, characteristic

#11716682

C₂

yes

maximal, characteristic, complemented

C₅×#3953

C₂

yes

maximal, complemented

#11716615

C₂

yes

maximal, characteristic, complemented

C₂×C₅×D₂₀

C₂²

yes

—

C₂×D₅×Dic₅

—

C₂×C₅²×C₈

C₂²

yes

characteristic

C₄×#1039

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

#87778

—

#87770

C₄

yes

complemented

#65074

—

maximal

C₂²×C₅×D₅

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

D₄

yes

—

C₂×C₅×Dic₅

—

C₅×D₂₀

—

C₅×D₂₀

C₂×C₄

yes

complemented

D₅×Dic₅

—

C₅²×C₈

—

#6029

C₂×C₄

yes

complemented

D₄×Dic₅

—

—

—

—

D₅

yes

—

—

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅×D₅

—

C₂×C₅×D₅

—

C₂×C₅×D₅

—

C₄×C₅²

C₂²×C₄

yes

C₄×C₅²

yes

—

C₅×Dic₅

—

C₂×C₅×D₄

—

C₂²×Dic₅

—

C₂×D₂₀

C₂×D₅

yes

—

C₂×C₄×D₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₅×C₄⋊C₄

—

C₂²⋊Dic₅

—

#686

—

#685

—

C₄⋊Dic₅

C₂×D₅

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

—

—

—

C₂×C₅²

C₄×D₄

yes

characteristic

C₂×C₅²

AGL(1,ℤ/8)

yes

—

C₅×D₅

—

C₂³×C₅

—

C₂²×D₅

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

C₄×D₅

yes

complemented

C₄×D₅

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂×D₈

—

C₄×D₄

—

#106

—

#102

—

#97

—

C₅²

#442

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×C₄×D₅

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

100

Dic₅

—

101

C₂×D₄

—

102

C₂²×C₄

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

D₅²

yes

characteristic

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₂×C₅

—

110

C₂×C₅

—

111

C₂×C₅

—

112

C₂×C₅

#4027

yes

—

113

C₂×C₅

#4057

yes

—

114

C₂×C₅

#4060

yes

—

115

D₅

—

116

C₂³

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

100

—

122

C₂×C₄

C₂×D₅²

yes

characteristic

123

C₈

—

124

C₈

—

125

C₅

—

126

C₅

#65074

yes

complemented

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

#87877

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

C₄×D₅²

yes

characteristic

134

C₄

#87880

yes

characteristic

135

C₂

—

136

C₂

#11716916

yes

—

137

C₂

#11717020

yes

characteristic

	subgroup	G/H
center	C₂²	#87877
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₂×C₅²×C₈	C₂²
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×#87770	C₂
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂×C₈	D₅²
hypocenter	C₅²	#442
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂²

#87877

commutator

C₄×C₅²

C₂²×C₄

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₂×C₅²×C₈

C₂²

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂×#87770

C₂

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₂×C₈

D₅²

hypocenter

C₅²

#442

LCS

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
1. #11717017	C₂	semidirect
3. #11716682	C₂	semidirect
4. C₅×#3953	C₂	semidirect
5. #11716615	C₂	semidirect
13. #87770	C₄	semidirect
20. C₅×D₂₀	C₂×C₄	semidirect
23. #6029	C₂×C₄	semidirect
28. #3953	D₅	semidirect
72. D₂₀	C₄×D₅	semidirect
83. C₅²	#442	semidirect
126. C₅	#65074	semidirect

H ≅ G/N

type

1. #11717017

C₂

semidirect

3. #11716682

C₂

semidirect

4. C₅×#3953

C₂

semidirect

5. #11716615

C₂

semidirect

13. #87770

C₄

semidirect

20. C₅×D₂₀

C₂×C₄

semidirect

23. #6029

C₂×C₄

semidirect

28. #3953

D₅

semidirect

72. D₂₀

C₄×D₅

semidirect

83. C₅²

#442

semidirect

126. C₅

#65074

semidirect

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

subgroup	index
#11717017	2
C₂×#87770	2
#11716682	2
C₅×#3953	2
#11716615	2
#65074	5

subgroup

index

Subgroups — #13101052

Conjugacy classes of subgroups

Complements

Maximal subgroups