Group #13102239 — subgroups

Subgroups — C₅×#65269 #13102239

Order 1600.

392 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 122 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×C₅²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#683	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#683	C₂	1	2	yes	maximal
4	C₅×#3963	C₂	1	2	yes	maximal
5	C₅×#3963	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₅²	C₂²	1	2	yes	—
7	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
10	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
11	C₅²×C₄⋊C₄	—	2	2	no	—
12	C₅×#683	—	2	2	no	—
13	C₅×#683	—	2	4	no	—
14	C₅×#398	—	5	1	no	maximal
15	#65269	C₅	1	1	yes	maximal, characteristic, direct product factor
16	C₂³×C₅²	C₂³	1	1	yes	characteristic
17	C₂×C₄×C₅²	—	2	2	no	—
18	C₂×C₄×C₅²	—	2	2	no	—
19	C₂×C₄×C₅²	—	2	2	no	—
20	C₂×C₄×C₅²	—	2	2	no	—
21	C₂×C₄×C₅²	D₄	1	2	yes	—
22	C₂×C₅×Dic₅	—	2	1	no	—
23	C₂×C₅×Dic₅	—	2	2	no	—
24	C₂×C₅×Dic₅	—	2	1	no	—
25	C₂×C₅×Dic₅	—	2	2	no	—
26	C₂×C₅×Dic₅	—	2	1	no	—
27	C₂×C₅×Dic₅	—	2	2	no	—
28	C₂×C₅×Dic₅	D₄	1	2	yes	—
29	C₂×C₅×Dic₅	Q₈	1	4	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	—	5	2	no	—
32	C₂×C₅×C₄⋊C₄	—	5	1	no	—
33	C₂×C₅×C₄⋊C₄	D₅	1	1	yes	characteristic
34	C₂×C₅×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
35	C₂×#683	C₂×C₅	1	1	yes	characteristic
36	C₂×#683	C₂×C₅	1	2	yes	—
37	C₅×#95	—	5	1	no	—
38	C₅×#95	—	5	2	no	—
39	#3963	C₂×C₅	1	2	yes	—
40	#3963	C₂×C₅	1	1	yes	characteristic
41	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₅²	C₂×Q₈	1	2	yes	—
44	C₂²×C₅²	#41	1	1	yes	characteristic
45	C₂²×C₅²	#41	1	2	yes	—
46	C₄×C₅²	—	2	2	no	—
47	C₄×C₅²	—	4	2	no	—
48	C₅×Dic₅	—	2	2	no	—
49	C₅×Dic₅	—	2	4	no	—
50	C₅×Dic₅	—	4	1	no	—
51	C₂²×C₄×C₅	—	2	2	no	—
52	C₂²×C₄×C₅	—	2	4	no	—
53	C₂²×C₄×C₅	—	5	1	no	—
54	C₂²×C₄×C₅	—	5	2	no	—
55	C₂²×C₄×C₅	—	5	1	no	—
56	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
57	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
59	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
60	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
61	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
62	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
63	C₅×C₄⋊C₄	—	2	8	no	—
64	C₅×C₄⋊C₄	—	2	2	no	—
65	C₅×C₄⋊C₄	—	2	2	no	—
66	C₅×C₄⋊C₄	—	10	4	no	—
67	C₅×C₄⋊C₄	—	10	2	no	—
68	#683	—	2	2	no	—
69	#683	—	2	4	no	—
70	#398	—	5	1	no	Sylow
71	C₂×C₅²	#121	1	1	yes	characteristic
72	C₂×C₅²	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₅²	#123	1	1	yes	characteristic
74	C₂×C₅²	#125	1	2	yes	—
75	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
76	C₂³×C₅	—	2	2	no	—
77	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
78	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
79	C₂×C₄×C₅	—	2	8	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	8	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	4	4	no	—
91	C₂×C₄×C₅	—	4	4	no	—
92	C₂×C₄×C₅	—	5	2	no	—
93	C₂×C₄×C₅	—	5	4	no	—
94	C₂×C₄×C₅	—	10	2	no	—
95	C₂×C₄×C₅	—	10	2	no	—
96	C₂×C₄×C₅	—	10	1	no	—
97	C₂×C₄×C₅	—	10	1	no	—
98	C₂×C₄×C₅	—	10	2	no	—
99	C₂×C₄×C₅	—	10	1	no	—
100	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
101	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
102	C₂×Dic₅	—	2	2	no	—
103	C₂×Dic₅	—	2	1	no	—
104	C₂×Dic₅	—	2	2	no	—
105	C₂×Dic₅	—	2	1	no	—
106	C₂×Dic₅	—	2	1	no	—
107	C₂×Dic₅	—	2	2	no	—
108	C₂×Dic₅	C₅×D₄	1	2	yes	—
109	C₂×Dic₅	C₅×Q₈	1	4	yes	—
110	C₂×C₄⋊C₄	—	5	2	no	—
111	C₂×C₄⋊C₄	—	5	1	no	—
112	C₂×C₄⋊C₄	C₅×D₅	1	1	yes	characteristic
113	#95	—	5	1	no	—
114	#95	—	5	2	no	—
115	C₅²	#398	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₅	—	2	4	no	—
117	C₂²×C₅	—	2	2	no	—
118	C₂²×C₅	—	2	2	no	—
119	C₂²×C₅	—	2	4	no	—
120	C₂²×C₅	—	2	2	no	—
121	C₂²×C₅	#709	1	2	yes	—
122	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
123	C₂²×C₅	#711	1	1	yes	characteristic
124	C₂²×C₅	Q₈×D₅	1	2	yes	—
125	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
126	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
127	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
128	C₂²×C₅	C₂×C₅×Q₈	1	2	yes	—
129	C₂²×C₅	C₅×#41	1	2	yes	—
130	C₂²×C₅	C₅×#41	1	1	yes	characteristic
131	C₄×C₅	—	2	8	no	—
132	C₄×C₅	—	2	2	no	—
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	—	4	8	no	—
135	C₄×C₅	—	4	2	no	—
136	C₄×C₅	—	4	2	no	—
137	C₄×C₅	—	10	4	no	—
138	C₄×C₅	—	10	2	no	—
139	C₄×C₅	—	20	1	no	—
140	Dic₅	—	2	2	no	—
141	Dic₅	—	2	4	no	—
142	Dic₅	—	4	1	no	—
143	C₂²×C₄	—	5	2	no	—
144	C₂²×C₄	—	5	1	no	—
145	C₂²×C₄	—	5	1	no	—
146	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
147	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	10	4	no	—
150	C₄⋊C₄	—	10	2	no	—
151	C₂×C₅	—	2	4	no	—
152	C₂×C₅	—	2	4	no	—
153	C₂×C₅	—	2	2	no	—
154	C₂×C₅	—	2	2	no	—
155	C₂×C₅	—	2	2	no	—
156	C₂×C₅	#4035	1	2	yes	—
157	C₂×C₅	#4042	1	2	yes	—
158	C₂×C₅	#4075	1	1	yes	characteristic
159	C₂×C₅	#4084	1	1	yes	characteristic
160	C₂×C₅	#4090	1	1	yes	characteristic
161	C₂×C₅	C₅×#121	1	1	yes	characteristic
162	C₂×C₅	C₅×C₂²⋊Q₈	1	2	yes	—
163	C₂×C₅	C₅×#123	1	1	yes	characteristic
164	C₂×C₅	C₅×#125	1	2	yes	—
165	C₂×C₅	C₅×C₄⋊Q₈	1	1	yes	characteristic
166	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	5	2	no	—
172	C₂×C₄	—	5	4	no	—
173	C₂×C₄	—	10	2	no	—
174	C₂×C₄	—	10	2	no	—
175	C₂×C₄	—	10	1	no	—
176	C₂×C₄	—	10	1	no	—
177	C₂×C₄	—	10	2	no	—
178	C₂×C₄	—	10	1	no	—
179	C₂×C₄	C₅×C₂²⋊D₅	1	2	yes	—
180	C₅	—	2	2	no	—
181	C₅	#65269	1	1	yes	characteristic, direct product factor
182	C₅	C₅×#398	1	1	yes	characteristic, complemented
183	C₂²	C₅×#709	1	2	yes	—
184	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
185	C₂²	C₅×#711	1	1	yes	characteristic
186	C₂²	C₅×Q₈×D₅	1	2	yes	—
187	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	4	2	no	—
190	C₄	—	10	4	no	—
191	C₄	—	10	2	no	—
192	C₄	—	20	1	no	—
193	C₂	C₅×#4035	1	2	yes	—
194	C₂	C₅×#4042	1	2	yes	—
195	C₂	C₅×#4075	1	1	yes	characteristic
196	C₂	C₅×#4084	1	1	yes	characteristic
197	C₂	C₅×#4090	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅²×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×C₅×#683

C₂

yes

maximal, characteristic

C₂×C₅×#683

C₂

yes

maximal

C₅×#3963

C₂

yes

maximal

C₅×#3963

C₂

yes

maximal, characteristic

C₂²×C₄×C₅²

C₂²

yes

—

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂²×C₅×Dic₅

C₂²

yes

characteristic

C₂²×C₅×Dic₅

C₂²

yes

characteristic

C₂²×C₅×Dic₅

C₂²

yes

—

C₅²×C₄⋊C₄

—

—

—

—

C₅

yes

maximal, characteristic, direct product factor

C₂³×C₅²

C₂³

yes

characteristic

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

D₄

yes

—

C₂×C₅×Dic₅

Q₈

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

D₅

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂×C₅

yes

characteristic

C₂×#683

C₂×C₅

yes

characteristic

C₂×#683

C₂×C₅

yes

—

C₅×#95

—

C₅×#95

—

#3963

C₂×C₅

yes

—

#3963

C₂×C₅

yes

characteristic

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂²×C₅²

C₂×Q₈

yes

—

C₂²×C₅²

#41

yes

characteristic

C₂²×C₅²

#41

yes

—

C₄×C₅²

—

C₄×C₅²

—

C₅×Dic₅

—

C₅×Dic₅

—

C₅×Dic₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×C₄×C₅

C₂×D₅

yes

characteristic

C₂²×C₄×C₅

C₂²×C₅

yes

—

C₂²×C₄×C₅

C₂²×C₅

yes

characteristic

C₂²×Dic₅

C₂²×C₅

yes

characteristic

C₂²×Dic₅

C₂²×C₅

yes

characteristic

C₂²×Dic₅

C₂²×C₅

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

—

—

—

C₂×C₅²

yes

C₂×C₅²

C₂²⋊Q₈

yes

—

C₂×C₅²

#123

yes

characteristic

C₂×C₅²

#125

yes

—

C₂×C₅²

C₄⋊Q₈

yes

characteristic

C₂³×C₅

—

C₂³×C₅

C₂²×D₅

yes

characteristic

C₂³×C₅

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

100

C₂×C₄×C₅

C₂²⋊D₅

yes

—

101

C₂×C₄×C₅

C₅×D₄

yes

—

102

C₂×Dic₅

—

103

C₂×Dic₅

—

104

C₂×Dic₅

—

105

C₂×Dic₅

—

106

C₂×Dic₅

—

107

C₂×Dic₅

—

108

C₂×Dic₅

C₅×D₄

yes

—

109

C₂×Dic₅

C₅×Q₈

yes

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₅×D₅

yes

characteristic

113

#95

—

114

#95

—

115

C₅²

#398

yes

characteristic, complemented, Sylow

116

C₂²×C₅

—

117

C₂²×C₅

—

118

C₂²×C₅

—

119

C₂²×C₅

—

120

C₂²×C₅

—

121

C₂²×C₅

#709

yes

—

122

C₂²×C₅

D₄×D₅

yes

characteristic

123

C₂²×C₅

#711

yes

characteristic

124

C₂²×C₅

Q₈×D₅

yes

—

125

C₂²×C₅

C₂×C₂²⋊D₅

yes

characteristic

126

C₂²×C₅

C₂×C₅×D₄

yes

characteristic

127

C₂²×C₅

C₂×C₅×D₄

yes

characteristic

128

C₂²×C₅

C₂×C₅×Q₈

yes

—

129

C₂²×C₅

C₅×#41

yes

—

130

C₂²×C₅

C₅×#41

yes

characteristic

131

C₄×C₅

—

132

C₄×C₅

—

133

C₄×C₅

—

134

C₄×C₅

—

135

C₄×C₅

—

136

C₄×C₅

—

137

C₄×C₅

—

138

C₄×C₅

—

139

C₄×C₅

—

140

Dic₅

—

141

Dic₅

—

142

Dic₅

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×C₅×D₅

yes

—

147

C₂²×C₄

C₂×C₅×D₅

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₂×C₅

—

152

C₂×C₅

—

153

C₂×C₅

—

154

C₂×C₅

—

155

C₂×C₅

—

156

C₂×C₅

#4035

yes

—

157

C₂×C₅

#4042

yes

—

158

C₂×C₅

#4075

yes

characteristic

159

C₂×C₅

#4084

yes

characteristic

160

C₂×C₅

#4090

yes

characteristic

161

C₂×C₅

C₅×#121

yes

characteristic

162

C₂×C₅

C₅×C₂²⋊Q₈

yes

—

163

C₂×C₅

C₅×#123

yes

characteristic

164

C₂×C₅

C₅×#125

yes

—

165

C₂×C₅

C₅×C₄⋊Q₈

yes

characteristic

166

C₂³

C₂²×C₅×D₅

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

C₅×C₂²⋊D₅

yes

—

180

C₅

—

181

C₅

#65269

yes

characteristic, direct product factor

182

C₅

C₅×#398

yes

characteristic, complemented

183

C₂²

C₅×#709

yes

—

184

C₂²

C₅×D₄×D₅

yes

characteristic

185

C₂²

C₅×#711

yes

characteristic

186

C₂²

C₅×Q₈×D₅

yes

—

187

C₂²

C₂×C₅×C₂²⋊D₅

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₂

C₅×#4035

yes

—

194

C₂

C₅×#4042

yes

—

195

C₂

C₅×#4075

yes

characteristic

196

C₂

C₅×#4084

yes

characteristic

197

C₂

C₅×#4090

yes

characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator, cosocle	C₂³×C₅	C₂³×C₅
Frattini	C₂³	C₂²×C₅×D₅
Fitting	C₂×C₅²×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₅²	C₂³
hypercenter	C₂×C₅×C₄⋊C₄	D₅
hypocenter	C₅	C₅×#398
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄

subgroup

G/H

center

C₂³×C₅

C₂²×D₅

commutator, cosocle

C₂³×C₅

Frattini

C₂³

C₂²×C₅×D₅

Fitting

C₂×C₅²×C₄⋊C₄

C₂

socle, prime radical

C₂³×C₅²

C₂³

hypercenter

C₂×C₅×C₄⋊C₄

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄

N	H ≅ G/N	type
15. #65269	C₅	direct (N × H)
115. C₅²	#398	semidirect
181. C₅	#65269	direct (N × H)
182. C₅	C₅×#398	semidirect

H ≅ G/N

type

15. #65269

C₅

direct (N × H)

115. C₅²

#398

semidirect

181. C₅

#65269

direct (N × H)

182. C₅

C₅×#398

semidirect

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

subgroup	index
C₂×C₅²×C₄⋊C₄	2
C₂×C₅×#683	2
C₅×#3963	2
C₅×#398	5
#65269	5

subgroup

index

C₂×C₅²×C₄⋊C₄

C₂×C₅×#683

C₅×#3963

C₅×#398

#65269

Subgroups — C₅×#65269 #13102239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#65269 #13102239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65269 #13102239