Group #13100533 — subgroups

Subgroups — #13100533

Order 1600.

328 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 68 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	C₅×#4021	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4020	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87772	C₂	1	2	yes	maximal, complemented
4	C₂×#87770	C₂	1	2	yes	maximal, complemented
5	#11716670	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×D₂₀	—	2	2	no	—
7	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
8	C₂×C₅×D₂₀	C₂²	1	2	yes	—
9	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	C₂×#6029	C₂²	1	2	yes	—
12	C₅×#685	—	2	2	no	—
13	#87772	—	2	4	no	—
14	#87770	—	2	4	no	—
15	#65363	—	5	1	no	maximal
16	#65347	—	5	1	no	maximal
17	C₂²×C₅×D₅	—	2	2	no	—
18	C₂²×C₅×D₅	—	2	1	no	—
19	C₂×C₄×C₅²	D₄	1	2	yes	—
20	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
21	C₂×C₅×Dic₅	—	2	1	no	—
22	C₅×D₂₀	—	2	2	no	—
23	C₅×D₂₀	—	2	2	no	—
24	C₅×D₂₀	—	2	2	no	—
25	C₅×D₂₀	—	4	4	no	—
26	C₅×Dic₁₀	—	2	2	no	—
27	#6029	D₄	1	4	yes	complemented
28	C₅×#127	—	5	1	no	—
29	C₅×#124	—	5	1	no	—
30	C₂×#688	—	5	2	no	—
31	C₂×#687	—	5	2	no	—
32	C₂×#686	—	5	2	no	—
33	C₂×#686	—	5	2	no	—
34	#4021	D₅	1	1	yes	characteristic, complemented
35	#4020	D₅	1	1	yes	characteristic, complemented
36	#3935	—	5	1	no	—
37	#3935	—	5	1	no	—
38	#3935	—	10	2	no	—
39	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
40	C₂×C₅×D₅	—	4	1	no	—
41	C₂×C₅×D₅	—	4	2	no	—
42	C₂×C₅×D₅	—	4	4	no	—
43	C₂×C₅×D₅	—	4	2	no	—
44	C₄×C₅²	—	2	2	no	—
45	C₄×C₅²	C₂×D₄	1	2	yes	—
46	C₅×Dic₅	—	4	1	no	—
47	C₂×C₅×Q₈	—	5	1	no	—
48	C₂×C₅×D₄	—	5	1	no	—
49	C₂×C₅×D₄	—	5	2	no	—
50	C₂×C₅×D₄	—	10	2	no	—
51	C₂×D₂₀	—	2	2	no	—
52	C₂×D₂₀	C₂×D₅	1	2	yes	—
53	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
54	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
55	C₅×C₂²⋊C₄	—	10	2	no	—
56	C₄²×C₅	—	2	2	no	—
57	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
58	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
59	#688	—	10	4	no	—
60	#687	—	10	4	no	—
61	#686	—	10	4	no	—
62	#686	—	10	4	no	—
63	#685	—	2	2	no	—
64	C₂×#168	—	5	2	no	—
65	C₂×#168	—	5	2	no	—
66	C₂×#168	—	10	4	no	—
67	#496	—	25	1	no	Sylow
68	C₂×C₅²	#127	1	1	yes	characteristic
69	C₂×C₅²	AGL(1,ℤ/8)	1	2	yes	—
70	C₅×D₅	—	8	1	no	—
71	C₅×D₅	—	8	2	no	—
72	C₂³×C₅	—	10	1	no	—
73	C₂³×C₅	—	10	2	no	—
74	C₂²×D₅	—	2	1	no	—
75	C₂²×D₅	—	2	2	no	—
76	C₅×Q₈	—	10	2	no	—
77	C₅×D₄	—	10	2	no	—
78	C₅×D₄	—	10	2	no	—
79	C₅×D₄	—	10	2	no	—
80	C₅×D₄	—	20	4	no	—
81	C₂×C₄×C₅	—	2	4	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	10	1	no	—
84	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
85	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
86	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
88	C₂×Dic₅	—	2	1	no	—
89	D₂₀	—	2	2	no	—
90	D₂₀	—	2	2	no	—
91	D₂₀	—	2	2	no	—
92	D₂₀	—	4	4	no	—
93	Dic₁₀	—	2	2	no	—
94	#168	—	5	4	no	—
95	#168	—	5	4	no	—
96	#168	—	10	8	no	—
97	C₂×SD₁₆	—	25	2	no	—
98	C₂×D₈	—	25	2	no	—
99	#127	—	5	1	no	—
100	#124	—	5	1	no	—
101	#97	—	25	1	no	—
102	C₅²	#496	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₅	—	2	2	no	—
104	C₂²×C₅	—	20	2	no	—
105	C₂²×C₅	—	20	2	no	—
106	C₂²×C₅	—	20	1	no	—
107	C₂²×C₅	—	20	4	no	—
108	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
109	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
110	C₂×D₅	—	4	4	no	—
111	C₂×D₅	—	4	1	no	—
112	C₂×D₅	—	4	2	no	—
113	C₂×D₅	—	4	2	no	—
114	C₄×C₅	—	2	2	no	—
115	C₄×C₅	—	2	2	no	—
116	C₄×C₅	—	2	4	no	—
117	C₄×C₅	—	4	4	no	—
118	C₄×C₅	—	20	1	no	—
119	C₄×C₅	D₄×D₅	1	2	yes	—
120	C₄×C₅	D₄×D₅	1	2	yes	—
121	Dic₅	—	4	1	no	—
122	C₂×Q₈	—	5	1	no	—
123	C₂×D₄	—	5	1	no	—
124	C₂×D₄	—	5	2	no	—
125	C₂×D₄	—	10	2	no	—
126	SD₁₆	—	50	4	no	—
127	D₈	—	50	4	no	—
128	C₂×C₈	—	25	2	no	—
129	C₂²⋊C₄	—	10	2	no	—
130	C₄²	D₅²	1	1	yes	characteristic
131	C₂×C₅	—	2	4	no	—
132	C₂×C₅	—	2	2	no	—
133	C₂×C₅	—	40	2	no	—
134	C₂×C₅	—	40	1	no	—
135	C₂×C₅	#4078	1	2	yes	—
136	C₂×C₅	#4086	1	1	yes	characteristic
137	C₂×C₅	#4086	1	1	yes	characteristic
138	C₂×C₅	#4095	1	2	yes	—
139	D₅	—	8	1	no	—
140	D₅	—	8	2	no	—
141	C₂³	—	10	1	no	—
142	C₂³	—	10	2	no	—
143	Q₈	—	10	2	no	—
144	D₄	—	10	2	no	—
145	D₄	—	10	2	no	—
146	D₄	—	10	2	no	—
147	D₄	—	20	4	no	—
148	C₂×C₄	—	10	1	no	—
149	C₂×C₄	#6037	1	2	yes	—
150	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
151	C₈	—	25	4	no	—
152	C₅	—	2	2	no	—
153	C₅	#65347	1	1	yes	characteristic, complemented
154	C₅	#65363	1	1	yes	characteristic, complemented
155	C₂²	—	20	4	no	—
156	C₂²	—	20	2	no	—
157	C₂²	—	20	1	no	—
158	C₂²	—	20	2	no	—
159	C₂²	C₂×#6037	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	20	1	no	—
162	C₄	#87877	1	2	yes	—
163	C₂	—	40	2	no	—
164	C₂	—	40	1	no	—
165	C₂	#11716941	1	2	yes	—
166	C₂	#11717032	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#4021

C₂

yes

maximal, characteristic, complemented

C₅×#4020

C₂

yes

maximal, characteristic, complemented

C₂×#87772

C₂

yes

maximal, complemented

C₂×#87770

C₂

yes

maximal, complemented

#11716670

C₂

yes

maximal, characteristic, complemented

C₂×C₅×D₂₀

—

C₂×C₅×D₂₀

C₂²

yes

characteristic

C₂×C₅×D₂₀

C₂²

yes

—

C₂×C₅×Dic₁₀

C₂²

yes

characteristic

C₄²×C₅²

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

—

—

—

—

—

—

C₂²×C₅×D₅

—

C₂²×C₅×D₅

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

C₅×D₂₀

—

C₅×D₂₀

—

C₅×D₂₀

—

C₅×D₂₀

—

C₅×Dic₁₀

—

#6029

D₄

yes

—

—

—

—

—

—

D₅

yes

characteristic, complemented

#4020

D₅

yes

characteristic, complemented

#3935

—

#3935

—

#3935

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅×D₅

—

C₂×C₅×D₅

—

C₂×C₅×D₅

—

C₂×C₅×D₅

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×D₂₀

—

C₂×D₂₀

C₂×D₅

yes

—

C₂×D₂₀

C₂×D₅

yes

characteristic

C₂×Dic₁₀

C₂×D₅

yes

characteristic

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂×D₅

yes

characteristic

C₄²×C₅

C₂×D₅

yes

—

—

—

—

—

—

—

—

—

C₂×C₅²

yes

C₂×C₅²

yes

—

C₅×D₅

—

C₅×D₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂²×D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

#168

—

#168

—

#168

—

C₂×SD₁₆

—

C₂×D₈

—

#127

—

100

#124

—

101

#97

—

102

C₅²

#496

yes

characteristic, complemented, Sylow

103

C₂²×C₅

—

104

C₂²×C₅

—

105

C₂²×C₅

—

106

C₂²×C₅

—

107

C₂²×C₅

—

108

C₂²×C₅

C₂×C₂²⋊D₅

yes

characteristic

109

C₂²×C₅

C₂×C₂²⋊D₅

yes

characteristic

110

C₂×D₅

—

111

C₂×D₅

—

112

C₂×D₅

—

113

C₂×D₅

—

114

C₄×C₅

—

115

C₄×C₅

—

116

C₄×C₅

—

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

D₄×D₅

yes

—

120

C₄×C₅

D₄×D₅

yes

—

121

Dic₅

—

122

C₂×Q₈

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

SD₁₆

—

127

D₈

—

128

C₂×C₈

—

129

C₂²⋊C₄

—

130

C₄²

D₅²

yes

characteristic

131

C₂×C₅

—

132

C₂×C₅

—

133

C₂×C₅

—

134

C₂×C₅

—

135

C₂×C₅

#4078

yes

—

136

C₂×C₅

#4086

yes

characteristic

137

C₂×C₅

#4086

yes

characteristic

138

C₂×C₅

#4095

yes

—

139

D₅

—

140

D₅

—

141

C₂³

—

142

C₂³

—

143

Q₈

—

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

#6037

yes

—

150

C₂×C₄

C₂×D₅²

yes

characteristic

151

C₈

—

152

C₅

—

153

C₅

#65347

yes

characteristic, complemented

154

C₅

#65363

yes

characteristic, complemented

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

C₂×#6037

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

#87877

yes

—

163

C₂

—

164

C₂

—

165

C₂

#11716941

yes

—

166

C₂

#11717032

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6037
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₄²×C₅²	C₂²
socle	C₂²×C₅²	C₂×D₄
hypercenter	C₄²	D₅²
hypocenter	C₅²	#496
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#6037

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₄²×C₅²

C₂²

socle

C₂²×C₅²

C₂×D₄

hypercenter

C₄²

D₅²

hypocenter

C₅²

#496

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
1. C₅×#4021	C₂	semidirect
2. C₅×#4020	C₂	semidirect
3. C₂×#87772	C₂	semidirect
4. C₂×#87770	C₂	semidirect
5. #11716670	C₂	semidirect
27. #6029	D₄	semidirect
34. #4021	D₅	semidirect
35. #4020	D₅	semidirect
102. C₅²	#496	semidirect
153. C₅	#65347	semidirect
154. C₅	#65363	semidirect

H ≅ G/N

type

1. C₅×#4021

C₂

semidirect

2. C₅×#4020

C₂

semidirect

3. C₂×#87772

C₂

semidirect

4. C₂×#87770

C₂

semidirect

5. #11716670

C₂

semidirect

27. #6029

D₄

semidirect

34. #4021

D₅

semidirect

35. #4020

D₅

semidirect

102. C₅²

#496

semidirect

153. C₅

#65347

semidirect

154. C₅

#65363

semidirect

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

subgroup	index
C₅×#4021	2
C₅×#4020	2
C₂×#87772	2
C₂×#87770	2
#11716670	2
#65363	5
#65347	5

subgroup

index

Subgroups — #13100533

Conjugacy classes of subgroups

Complements

Maximal subgroups