Group #13106486 — subgroups

Subgroups — #13106486

Order 1600.

542 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 130 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	Q₈×#1039	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4093	C₂	1	2	yes	maximal, complemented
3	C₄×#6039	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717033	C₂	1	2	yes	maximal, complemented
5	#11717032	C₂	1	1	yes	maximal, characteristic
6	#11717020	C₂	1	2	yes	maximal, complemented
7	Dic₅×D₂₀	C₂	1	2	yes	maximal, complemented
8	#11717003	C₂	1	4	yes	maximal, complemented
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×D₂₀	C₂²	1	2	yes	—
11	C₂×#6039	C₂²	1	1	yes	characteristic
12	C₂×#6037	C₂²	1	2	yes	—
13	C₂×#6037	C₂²	1	2	yes	—
14	C₂×D₅×Dic₅	C₂²	1	4	yes	—
15	C₄⋊#1039	C₂²	1	2	yes	—
16	C₄⋊#1039	C₂²	1	1	yes	characteristic
17	C₄×#1039	C₂²	1	2	yes	—
18	C₄×#1039	C₂²	1	1	yes	characteristic
19	C₅×#685	C₂²	1	4	yes	—
20	C₅×#685	C₂²	1	4	yes	—
21	C₄×C₅×Dic₅	C₂²	1	2	yes	—
22	#87781	C₂²	1	1	yes	characteristic
23	#87780	C₂²	1	2	yes	—
24	#87778	C₂²	1	4	yes	—
25	#66015	—	5	2	no	maximal
26	C₂²×C₅×D₅	C₂³	1	4	yes	—
27	C₅²×Q₈	—	2	2	no	—
28	C₂×C₄×C₅²	C₂³	1	2	yes	—
29	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
30	C₂×#1039	C₂³	1	2	yes	—
31	C₂×#1039	C₂³	1	1	yes	characteristic
32	C₂×#1039	C₂³	1	1	yes	characteristic
33	C₂×C₅×Dic₅	C₂³	1	2	yes	—
34	C₂×C₅×Dic₅	C₂³	1	2	yes	—
35	C₅×D₂₀	—	2	4	no	—
36	#6039	—	2	2	no	—
37	#6037	—	2	4	no	—
38	#6037	—	4	2	no	—
39	D₅×Dic₅	—	4	4	no	—
40	C₅×#124	—	5	2	no	—
41	#4093	D₅	1	2	yes	complemented
42	Q₈×Dic₅	—	5	2	no	—
43	Q₈×Dic₅	—	10	2	no	—
44	#4086	—	5	2	no	—
45	D₄×Dic₅	—	5	2	no	—
46	#4032	—	5	4	no	—
47	#4031	—	5	4	no	—
48	#4029	—	5	4	no	—
49	#4027	—	5	4	no	—
50	C₄×D₂₀	—	5	2	no	—
51	C₄×Dic₁₀	—	5	2	no	—
52	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
53	C₂×C₅×D₅	—	2	4	no	—
54	C₂×C₅×D₅	—	2	4	no	—
55	C₂×C₅×D₅	—	2	4	no	—
56	C₄×C₅²	—	2	2	no	—
57	C₄×C₅²	#41	1	2	yes	—
58	#1039	—	2	2	no	—
59	#1039	—	2	1	no	—
60	#1039	#41	1	2	yes	—
61	C₅×Dic₅	—	2	2	no	—
62	C₅×Dic₅	—	2	2	no	—
63	C₂×C₅×Q₈	—	2	2	no	—
64	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
65	C₂×C₅×D₄	—	5	2	no	—
66	C₂×C₂²⋊D₅	—	5	4	no	—
67	C₂×C₂²⋊D₅	—	5	4	no	—
68	C₂²×Dic₅	—	5	4	no	—
69	C₂×D₂₀	C₂×D₅	1	2	yes	—
70	C₂×C₄×D₅	—	5	4	no	—
71	C₂×Dic₁₀	—	5	2	no	—
72	C₅×C₂²⋊C₄	—	5	4	no	—
73	C₅×C₂²⋊C₄	—	5	4	no	—
74	C₄²×C₅	—	5	2	no	—
75	C₂²⋊Dic₅	—	5	4	no	—
76	#685	—	5	4	no	—
77	#685	C₂×D₅	1	4	yes	—
78	#685	C₂×D₅	1	4	yes	—
79	C₄⋊Dic₅	—	5	2	no	—
80	C₄⋊Dic₅	—	5	4	no	—
81	C₄⋊Dic₅	—	5	2	no	—
82	C₄⋊Dic₅	—	10	2	no	—
83	C₄⋊Dic₅	—	10	4	no	—
84	#683	—	5	2	no	—
85	#683	—	5	2	no	—
86	C₄×Dic₅	—	5	2	no	—
87	C₄×Dic₅	—	5	4	no	—
88	C₄×Dic₅	—	10	4	no	—
89	C₄×Dic₅	—	10	2	no	—
90	C₄×Dic₅	C₂×D₅	1	2	yes	—
91	#559	—	25	1	no	Sylow
92	C₂×C₅²	C₂×#41	1	1	yes	characteristic
93	C₂×C₅²	C₂×#41	1	1	yes	characteristic
94	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
95	C₅×D₅	—	4	4	no	—
96	C₂³×C₅	—	5	4	no	—
97	C₂²×D₅	C₂²×D₅	1	4	yes	—
98	C₅×Q₈	—	2	8	no	—
99	C₅×Q₈	—	2	4	no	—
100	C₅×D₄	—	10	4	no	—
101	C₂×C₄×C₅	—	2	4	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	5	2	no	—
104	C₂×C₄×C₅	—	5	2	no	—
105	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
106	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
107	C₂²⋊D₅	—	10	8	no	—
108	C₂²⋊D₅	—	20	4	no	—
109	C₂×Dic₅	—	5	2	no	—
110	C₂×Dic₅	—	5	2	no	—
111	C₂×Dic₅	—	5	4	no	—
112	C₂×Dic₅	—	10	2	no	—
113	C₂×Dic₅	—	10	4	no	—
114	C₂×Dic₅	—	10	2	no	—
115	C₂×Dic₅	—	20	4	no	—
116	C₂×Dic₅	C₂²×D₅	1	2	yes	—
117	C₂×Dic₅	C₂²×D₅	1	2	yes	—
118	D₂₀	—	2	4	no	—
119	C₄×D₅	—	20	4	no	—
120	Dic₁₀	—	10	4	no	—
121	#127	—	25	1	no	—
122	#124	—	5	2	no	—
123	#124	—	25	2	no	—
124	#123	—	25	4	no	—
125	C₄×Q₈	—	25	1	no	—
126	C₄×Q₈	—	25	1	no	—
127	C₄×D₄	—	25	2	no	—
128	C₄×D₄	—	25	2	no	—
129	C₅²	#559	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	—	10	4	no	—
132	C₂²×C₅	—	10	4	no	—
133	C₂²×C₅	—	10	4	no	—
134	C₂²×C₅	C₂³×D₅	1	2	yes	—
135	C₂×D₅	—	2	4	no	—
136	C₂×D₅	—	2	4	no	—
137	C₂×D₅	—	2	4	no	—
138	C₄×C₅	—	2	8	no	—
139	C₄×C₅	—	2	4	no	—
140	C₄×C₅	—	2	4	no	—
141	C₄×C₅	—	10	2	no	—
142	C₄×C₅	—	10	2	no	—
143	C₄×C₅	#711	1	4	yes	—
144	Dic₅	—	2	2	no	—
145	Dic₅	—	2	2	no	—
146	Dic₅	—	5	4	no	—
147	Dic₅	—	10	4	no	—
148	Dic₅	—	10	4	no	—
149	Dic₅	—	10	2	no	—
150	Dic₅	—	20	4	no	—
151	Dic₅	—	20	2	no	—
152	C₂×Q₈	—	25	1	no	—
153	C₂×Q₈	D₅²	1	1	yes	characteristic
154	C₂×D₄	—	5	2	no	—
155	C₂×D₄	—	25	2	no	—
156	C₂×D₄	—	25	2	no	—
157	C₂²×C₄	—	25	4	no	—
158	C₄⋊C₄	—	25	1	no	—
159	C₄⋊C₄	—	25	2	no	—
160	C₄⋊C₄	—	25	1	no	—
161	C₄⋊C₄	—	25	2	no	—
162	C₂²⋊C₄	—	5	4	no	—
163	C₂²⋊C₄	—	5	4	no	—
164	C₂²⋊C₄	—	25	4	no	—
165	C₄²	—	5	2	no	—
166	C₄²	—	25	2	no	—
167	C₄²	—	25	1	no	—
168	C₂×C₅	—	2	2	no	—
169	C₂×C₅	—	2	2	no	—
170	C₂×C₅	—	2	2	no	—
171	C₂×C₅	—	20	4	no	—
172	C₂×C₅	C₂×#711	1	2	yes	—
173	C₂×C₅	D₅×#41	1	2	yes	—
174	C₂×C₅	#4149	1	2	yes	—
175	D₅	—	4	4	no	—
176	C₂³	—	5	4	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	50	2	no	—
179	D₄	—	10	4	no	—
180	D₄	—	50	4	no	—
181	D₄	—	100	2	no	—
182	C₂×C₄	—	5	2	no	—
183	C₂×C₄	—	5	2	no	—
184	C₂×C₄	—	25	1	no	—
185	C₂×C₄	—	25	2	no	—
186	C₂×C₄	—	25	1	no	—
187	C₂×C₄	—	100	4	no	—
188	C₂×C₄	C₂×D₅²	1	2	yes	—
189	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
190	C₅	—	2	2	no	—
191	C₅	#66015	1	2	yes	complemented
192	C₂²	—	10	4	no	—
193	C₂²	—	10	4	no	—
194	C₂²	—	10	4	no	—
195	C₂²	C₂²×D₅²	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	10	2	no	—
198	C₄	—	10	2	no	—
199	C₄	—	25	2	no	—
200	C₄	—	50	2	no	—
201	C₄	—	50	1	no	—
202	C₄	#87880	1	2	yes	—
203	C₂	—	20	4	no	—
204	C₂	#11717484	1	1	yes	characteristic
205	C₂	#11717485	1	1	yes	characteristic
206	C₂	C₂×#87880	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#1039

C₂

yes

maximal, characteristic, complemented

C₅×#4093

C₂

yes

maximal, complemented

C₄×#6039

C₂

yes

maximal, characteristic, complemented

#11717033

C₂

yes

maximal, complemented

#11717032

C₂

yes

maximal, characteristic

#11717020

C₂

yes

maximal, complemented

Dic₅×D₂₀

C₂

yes

maximal, complemented

#11717003

C₂

yes

maximal, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₅×D₂₀

C₂²

yes

—

C₂×#6039

C₂²

yes

characteristic

C₂×#6037

C₂²

yes

—

C₂×#6037

C₂²

yes

—

C₂×D₅×Dic₅

C₂²

yes

—

C₄⋊#1039

C₂²

yes

—

C₄⋊#1039

C₂²

yes

characteristic

C₄×#1039

C₂²

yes

—

C₄×#1039

C₂²

yes

characteristic

C₅×#685

C₂²

yes

—

C₅×#685

C₂²

yes

—

C₄×C₅×Dic₅

C₂²

yes

—

#87781

C₂²

yes

characteristic

#87780

C₂²

yes

—

#87778

C₂²

yes

—

#66015

—

maximal

C₂²×C₅×D₅

C₂³

yes

—

C₅²×Q₈

—

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

—

C₂×#1039

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

characteristic

C₂×C₅×Dic₅

C₂³

yes

—

C₂×C₅×Dic₅

C₂³

yes

—

C₅×D₂₀

—

#6039

—

#6037

—

#6037

—

D₅×Dic₅

—

C₅×#124

—

#4093

D₅

yes

complemented

Q₈×Dic₅

—

Q₈×Dic₅

—

#4086

—

D₄×Dic₅

—

—

—

—

—

C₄×D₂₀

—

C₄×Dic₁₀

—

C₂²×C₅²

C₂⁴

yes

characteristic

C₂×C₅×D₅

—

C₂×C₅×D₅

—

C₂×C₅×D₅

—

C₄×C₅²

—

C₄×C₅²

#41

yes

—

—

—

yes

—

C₅×Dic₅

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×C₅×D₄

—

C₂×C₂²⋊D₅

—

C₂×C₂²⋊D₅

—

C₂²×Dic₅

—

C₂×D₂₀

C₂×D₅

yes

—

C₂×C₄×D₅

—

C₂×Dic₁₀

—

C₅×C₂²⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₂²⋊Dic₅

—

#685

—

#685

C₂×D₅

yes

—

#685

C₂×D₅

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₂×D₅

yes

—

#559

—

Sylow

C₂×C₅²

C₂×#41

yes

characteristic

C₂×C₅²

C₂×#41

yes

characteristic

C₂×C₅²

H(5,𝔽₂)

yes

characteristic

C₅×D₅

—

C₂³×C₅

—

C₂²×D₅

yes

—

C₅×Q₈

—

C₅×Q₈

—

100

C₅×D₄

—

101

C₂×C₄×C₅

—

102

C₂×C₄×C₅

—

103

C₂×C₄×C₅

—

104

C₂×C₄×C₅

—

105

C₂×C₄×C₅

C₂²×D₅

yes

—

106

C₂×C₄×C₅

C₂²×D₅

yes

—

107

C₂²⋊D₅

—

108

C₂²⋊D₅

—

109

C₂×Dic₅

—

110

C₂×Dic₅

—

111

C₂×Dic₅

—

112

C₂×Dic₅

—

113

C₂×Dic₅

—

114

C₂×Dic₅

—

115

C₂×Dic₅

—

116

C₂×Dic₅

C₂²×D₅

yes

—

117

C₂×Dic₅

C₂²×D₅

yes

—

118

D₂₀

—

119

C₄×D₅

—

120

Dic₁₀

—

121

#127

—

122

#124

—

123

#124

—

124

#123

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×D₄

—

128

C₄×D₄

—

129

C₅²

#559

yes

characteristic, complemented, Sylow

130

C₂²×C₅

—

131

C₂²×C₅

—

132

C₂²×C₅

—

133

C₂²×C₅

—

134

C₂²×C₅

C₂³×D₅

yes

—

135

C₂×D₅

—

136

C₂×D₅

—

137

C₂×D₅

—

138

C₄×C₅

—

139

C₄×C₅

—

140

C₄×C₅

—

141

C₄×C₅

—

142

C₄×C₅

—

143

C₄×C₅

#711

yes

—

144

Dic₅

—

145

Dic₅

—

146

Dic₅

—

147

Dic₅

—

148

Dic₅

—

149

Dic₅

—

150

Dic₅

—

151

Dic₅

—

152

C₂×Q₈

—

153

C₂×Q₈

D₅²

yes

characteristic

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂²×C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₂×C₅

—

169

C₂×C₅

—

170

C₂×C₅

—

171

C₂×C₅

—

172

C₂×C₅

C₂×#711

yes

—

173

C₂×C₅

D₅×#41

yes

—

174

C₂×C₅

#4149

yes

—

175

D₅

—

176

C₂³

—

177

Q₈

—

178

Q₈

—

179

D₄

—

180

D₄

—

181

D₄

—

100

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

100

—

188

C₂×C₄

C₂×D₅²

yes

—

189

C₂×C₄

C₂×D₅²

yes

characteristic

190

C₅

—

191

C₅

#66015

yes

complemented

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

C₂²×D₅²

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

#87880

yes

—

203

C₂

—

204

C₂

#11717484

yes

characteristic

205

C₂

#11717485

yes

characteristic

206

C₂

C₂×#87880

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₂×C₅²×Q₈	C₂²
prime radical	#11717032	C₂
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	#559
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂²×D₅²

commutator, socle, cosocle

C₂²×C₅²

C₂⁴

Fitting

C₂×C₅²×Q₈

C₂²

prime radical

#11717032

C₂

hypercenter

C₂×Q₈

D₅²

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₂×Q₈

N	H ≅ G/N	type
1. Q₈×#1039	C₂	semidirect
2. C₅×#4093	C₂	semidirect
3. C₄×#6039	C₂	semidirect
4. #11717033	C₂	semidirect
6. #11717020	C₂	semidirect
7. Dic₅×D₂₀	C₂	semidirect
8. #11717003	C₂	semidirect
41. #4093	D₅	semidirect
129. C₅²	#559	semidirect
191. C₅	#66015	semidirect

H ≅ G/N

type

1. Q₈×#1039

C₂

semidirect

2. C₅×#4093

C₂

semidirect

3. C₄×#6039

C₂

semidirect

4. #11717033

C₂

semidirect

6. #11717020

C₂

semidirect

7. Dic₅×D₂₀

C₂

semidirect

8. #11717003

C₂

semidirect

41. #4093

D₅

semidirect

129. C₅²

#559

semidirect

191. C₅

#66015

semidirect

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

subgroup	index
Q₈×#1039	2
C₅×#4093	2
C₄×#6039	2
#11717033	2
#11717032	2
#11717020	2
Dic₅×D₂₀	2
#11717003	2
#66015	5

subgroup

index

Dic₅×D₂₀

Subgroups — #13106486

Conjugacy classes of subgroups

Complements

Maximal subgroups