Group #16016296 — subgroups

Subgroups — #16016296

Order 3328.

646 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₂²×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic
2	#13123703	C₂	1	2	yes	maximal
3	#13116706	C₂	1	4	yes	maximal, complemented
4	#13116694	C₂	1	4	yes	maximal, complemented
5	#13116555	C₂	1	2	yes	maximal
6	#13116432	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₅₂	C₂²	1	2	yes	—
9	#11719221	—	2	2	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₁₃×#427	C₂²	1	2	yes	—
12	#11718762	C₂²	1	1	yes	characteristic
13	C₂×#88052	C₂²	1	1	yes	characteristic
14	#11718665	—	2	2	no	—
15	C₂×Dic₁₀₄	C₄	1	8	yes	complemented
16	C₂×Dic₁₀₄	C₂²	1	2	yes	—
17	C₂×Dic₁₀₄	C₂²	1	2	yes	—
18	#11718534	C₂²	1	8	yes	—
19	#11718514	—	2	4	no	—
20	C₁₃×#367	C₂²	1	2	yes	—
21	C₁₃×#366	C₂²	1	2	yes	—
22	#11718094	C₂²	1	2	yes	—
23	C₁₆×Dic₁₃	C₂²	1	2	yes	—
24	#11718084	C₂²	1	8	yes	—
25	C₂²×Dic₂₆	—	2	2	no	—
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×#117	—	2	2	no	—
28	#88180	D₄	1	1	yes	characteristic
29	C₂×#6099	D₄	1	1	yes	characteristic
30	C₂×Dic₅₂	—	2	2	no	—
31	C₂×Dic₅₂	—	2	4	no	—
32	C₂×Dic₅₂	C₂×C₄	1	8	yes	—
33	C₂×Dic₅₂	C₂³	1	4	yes	—
34	#88141	—	2	8	no	—
35	#88131	—	4	4	no	—
36	C₄×Dic₂₆	—	4	4	no	—
37	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
38	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
39	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
40	C₁₃×#107	C₂³	1	4	yes	—
41	C₁₃×#106	—	2	2	no	—
42	#88055	D₄	1	2	yes	—
43	C₈×Dic₁₃	C₂³	1	2	yes	—
44	#88052	—	2	2	no	—
45	C₂×#6090	C₂³	1	2	yes	—
46	#88050	—	2	8	no	—
47	Dic₁₀₄	C₂×C₄	1	8	yes	—
48	Dic₁₀₄	C₂×C₄	1	8	yes	—
49	#33458	—	13	1	no	maximal, Sylow
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	—	4	2	no	—
52	C₂×Dic₂₆	—	2	8	no	—
53	C₂×Dic₂₆	—	2	4	no	—
54	C₂×Dic₂₆	—	4	2	no	—
55	C₂×Dic₂₆	—	4	4	no	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	#41	1	2	yes	—
59	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	—	2	4	no	—
61	C₁₃×C₂²⋊C₄	—	4	2	no	—
62	C₄²×C₁₃	—	4	2	no	—
63	C₂²⋊Dic₁₃	D₈	1	2	yes	—
64	C₄⋊Dic₁₃	D₈	1	2	yes	—
65	#6101	—	4	8	no	—
66	C₄×Dic₁₃	—	4	8	no	—
67	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
68	#6099	—	2	2	no	—
69	C₂×#1055	C₂×D₄	1	2	yes	—
70	Dic₅₂	—	2	8	no	—
71	Dic₅₂	—	2	4	no	—
72	Dic₅₂	C₂²×C₄	1	8	yes	—
73	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
74	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
75	#6090	—	2	2	no	—
76	C₂²×Q₃₂	—	13	1	no	—
77	#2950	—	13	2	no	—
78	#2176	—	13	4	no	—
79	C₄×Q₃₂	—	13	4	no	—
80	#2158	D₁₃	1	1	yes	characteristic
81	#2141	—	13	2	no	—
82	#2108	—	13	1	no	—
83	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	—	4	4	no	—
85	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#135	1	2	yes	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×Dic₁₃	—	4	4	no	—
90	C₂×Dic₁₃	—	4	8	no	—
91	C₂×Dic₁₃	C₂×D₈	1	4	yes	—
92	Dic₂₆	—	2	8	no	—
93	Dic₂₆	—	4	4	no	—
94	Dic₂₆	—	4	8	no	—
95	C₈×C₁₃	C₄×D₄	1	2	yes	—
96	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
97	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
98	#1055	—	2	2	no	—
99	C₂²×Q₁₆	—	13	2	no	—
100	#519	—	26	2	no	—
101	C₂×Q₃₂	—	13	8	no	—
102	C₂×Q₃₂	—	13	2	no	—
103	C₂×Q₃₂	—	13	2	no	—
104	C₂×M₃₂	—	13	1	no	—
105	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
106	#441	—	26	4	no	—
107	C₄×Q₁₆	—	13	8	no	—
108	#427	C₂×D₁₃	1	2	yes	—
109	#419	—	26	2	no	—
110	#405	—	13	1	no	—
111	#367	C₂×D₁₃	1	2	yes	—
112	#366	C₂×D₁₃	1	2	yes	—
113	#358	—	13	8	no	—
114	#346	—	13	2	no	—
115	C₄×C₁₆	—	13	2	no	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	C₄×D₈	1	2	yes	—
118	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
119	C₄×C₁₃	—	8	4	no	—
120	C₄×C₁₃	C₄×D₈	1	2	yes	—
121	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
123	Dic₁₃	—	2	4	no	—
124	Dic₁₃	—	4	8	no	—
125	C₂²×Q₈	—	26	2	no	—
126	C₂×Q₁₆	—	13	8	no	—
127	C₂×Q₁₆	—	13	4	no	—
128	C₂×Q₁₆	—	26	2	no	—
129	C₂×Q₁₆	—	26	4	no	—
130	C₂×M₁₆	—	13	1	no	—
131	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
132	C₄×Q₈	—	26	8	no	—
133	C₄×Q₈	—	52	4	no	—
134	#117	—	2	2	no	—
135	#117	—	13	1	no	—
136	#117	—	52	4	no	—
137	Q₃₂	—	13	8	no	—
138	Q₃₂	—	13	8	no	—
139	M₃₂	—	26	2	no	—
140	C₂×C₁₆	—	13	2	no	—
141	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
142	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
143	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
144	#107	C₂²×D₁₃	1	4	yes	—
145	#106	—	2	2	no	—
146	#103	—	26	8	no	—
147	#97	—	13	2	no	—
148	C₄×C₈	—	13	2	no	—
149	C₂×C₁₃	—	2	2	no	—
150	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
151	C₂×C₁₃	#3417	1	2	yes	—
152	C₂×Q₈	—	26	8	no	—
153	C₂×Q₈	—	26	4	no	—
154	C₂×Q₈	—	52	2	no	—
155	C₂×Q₈	—	52	4	no	—
156	C₂²×C₄	—	52	2	no	—
157	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
158	Q₁₆	—	13	8	no	—
159	Q₁₆	—	26	8	no	—
160	Q₁₆	—	26	4	no	—
161	M₁₆	—	26	2	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
164	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
165	C₂×C₈	#6131	1	2	yes	—
166	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	13	2	no	—
169	C₄⋊C₄	—	52	8	no	—
170	C₂²⋊C₄	—	4	2	no	—
171	C₂²⋊C₄	—	13	2	no	—
172	C₄²	—	4	2	no	—
173	C₄²	—	13	2	no	—
174	C₄²	—	52	8	no	—
175	C₁₆	—	26	2	no	—
176	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
177	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
178	C₁₃	#33458	1	1	yes	characteristic, complemented, Sylow
179	C₂³	D₈×D₁₃	1	1	yes	characteristic
180	Q₈	—	26	8	no	—
181	Q₈	—	52	4	no	—
182	Q₈	—	52	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	13	4	no	—
185	C₂×C₄	—	52	4	no	—
186	C₂×C₄	—	52	8	no	—
187	C₂×C₄	#88147	1	2	yes	—
188	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
189	C₂×C₄	#88173	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
191	C₈	—	26	2	no	—
192	C₈	#88147	1	2	yes	—
193	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
194	C₈	C₂×#6131	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	#11718533	1	2	yes	—
197	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
198	C₄	—	8	4	no	—
199	C₄	—	26	4	no	—
200	C₄	—	52	8	no	—
201	C₄	#11718533	1	2	yes	—
202	C₄	C₂×#88147	1	1	yes	characteristic
203	C₄	C₂×#88173	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×#11718533	1	1	yes	characteristic
206	C₂	#13127986	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₀₄

C₂

yes

maximal, characteristic

#13123703

C₂

yes

maximal

#13116706

C₂

yes

maximal, complemented

#13116694

C₂

yes

maximal, complemented

#13116555

C₂

yes

maximal

#13116432

C₂

yes

maximal, characteristic

C₁₃×#2158

C₂

yes

maximal, characteristic

C₂²×Dic₅₂

C₂²

yes

—

#11719221

—

C₂²×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#427

C₂²

yes

—

#11718762

C₂²

yes

characteristic

C₂×#88052

C₂²

yes

characteristic

#11718665

—

C₂×Dic₁₀₄

C₄

yes

complemented

C₂×Dic₁₀₄

C₂²

yes

—

C₂×Dic₁₀₄

C₂²

yes

—

#11718534

C₂²

yes

—

#11718514

—

C₁₃×#367

C₂²

yes

—

C₁₃×#366

C₂²

yes

—

#11718094

C₂²

yes

—

C₁₆×Dic₁₃

C₂²

yes

—

#11718084

C₂²

yes

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

—

#88180

D₄

yes

characteristic

C₂×#6099

D₄

yes

characteristic

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

C₂×C₄

yes

—

C₂×Dic₅₂

C₂³

yes

—

#88141

—

#88131

—

C₄×Dic₂₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

C₂³

yes

characteristic

C₂×C₁₃×C₁₆

C₂³

yes

characteristic

C₁₃×#107

C₂³

yes

—

C₁₃×#106

—

#88055

D₄

yes

—

C₈×Dic₁₃

C₂³

yes

—

#88052

—

C₂×#6090

C₂³

yes

—

#88050

—

Dic₁₀₄

C₂×C₄

yes

—

Dic₁₀₄

C₂×C₄

yes

—

#33458

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

C₂²×C₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

C₂⁴

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₂²⋊Dic₁₃

D₈

yes

—

C₄⋊Dic₁₃

D₈

yes

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×D₄

yes

—

#6099

—

C₂×#1055

C₂×D₄

yes

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

C₂²×C₄

yes

—

C₁₃×C₁₆

C₂²×C₄

yes

—

C₁₃×C₁₆

C₂²×C₄

yes

—

#6090

—

C₂²×Q₃₂

—

#2950

—

#2176

—

C₄×Q₃₂

—

#2158

D₁₃

yes

characteristic

#2141

—

#2108

—

C₂³×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₈

yes

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

C₄×D₄

yes

—

C₈×C₁₃

C₂³×C₄

yes

characteristic

C₈×C₁₃

C₂×#41

yes

characteristic

#1055

—

C₂²×Q₁₆

—

100

#519

—

101

C₂×Q₃₂

—

102

C₂×Q₃₂

—

103

C₂×Q₃₂

—

104

C₂×M₃₂

—

105

C₂²×C₁₆

C₂×D₁₃

yes

characteristic

106

#441

—

107

C₄×Q₁₆

—

108

#427

C₂×D₁₃

yes

—

109

#419

—

110

#405

—

111

#367

C₂×D₁₃

yes

—

112

#366

C₂×D₁₃

yes

—

113

#358

—

114

#346

—

115

C₄×C₁₆

—

116

C₂²×C₁₃

—

117

C₂²×C₁₃

C₄×D₈

yes

—

118

C₂²×C₁₃

C₂²×D₈

yes

characteristic

119

C₄×C₁₃

—

120

C₄×C₁₃

C₄×D₈

yes

—

121

C₄×C₁₃

C₂×C₄×D₄

yes

characteristic

122

C₄×C₁₃

C₂×#135

yes

characteristic

123

Dic₁₃

—

124

Dic₁₃

—

125

C₂²×Q₈

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×M₁₆

—

131

C₂²×C₈

C₂²×D₁₃

yes

characteristic

132

C₄×Q₈

—

133

C₄×Q₈

—

134

#117

—

135

#117

—

136

#117

—

137

Q₃₂

—

138

Q₃₂

—

139

M₃₂

—

140

C₂×C₁₆

—

141

C₂×C₁₆

C₄×D₁₃

yes

—

142

C₂×C₁₆

C₂²×D₁₃

yes

characteristic

143

C₂×C₁₆

C₂²×D₁₃

yes

characteristic

144

#107

C₂²×D₁₃

yes

—

145

#106

—

146

#103

—

147

#97

—

148

C₄×C₈

—

149

C₂×C₁₃

—

150

C₂×C₁₃

C₂×C₄×D₈

yes

characteristic

151

C₂×C₁₃

#3417

yes

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂²×C₄

—

157

C₂²×C₄

D₄×D₁₃

yes

characteristic

158

Q₁₆

—

159

Q₁₆

—

160

Q₁₆

—

161

M₁₆

—

162

C₂×C₈

—

163

C₂×C₈

C₂×C₄×D₁₃

yes

—

164

C₂×C₈

D₄×D₁₃

yes

characteristic

165

C₂×C₈

#6131

yes

—

166

C₂×C₈

C₂³×D₁₃

yes

characteristic

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₂²⋊C₄

—

171

C₂²⋊C₄

—

172

C₄²

—

173

C₄²

—

174

C₄²

—

175

C₁₆

—

176

C₁₆

C₂×C₄×D₁₃

yes

—

177

C₁₆

C₂×C₄×D₁₃

yes

—

178

C₁₃

#33458

yes

characteristic, complemented, Sylow

179

C₂³

D₈×D₁₃

yes

characteristic

180

Q₈

—

181

Q₈

—

182

Q₈

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

#88147

yes

—

188

C₂×C₄

D₈×D₁₃

yes

characteristic

189

C₂×C₄

#88173

yes

—

190

C₂×C₄

C₂×D₄×D₁₃

yes

characteristic

191

C₈

—

192

C₈

#88147

yes

—

193

C₈

C₂²×C₄×D₁₃

yes

characteristic

194

C₈

C₂×#6131

yes

characteristic

195

C₂²

—

196

C₂²

#11718533

yes

—

197

C₂²

C₂×D₈×D₁₃

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

#11718533

yes

—

202

C₄

C₂×#88147

yes

characteristic

203

C₄

C₂×#88173

yes

characteristic

204

C₂

—

205

C₂

C₂×#11718533

yes

characteristic

206

C₂

#13127986

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2158	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2158	D₁₃
hypocenter	C₁₃	#33458
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂²

C₂×D₈×D₁₃

commutator

C₈×C₁₃

C₂³×C₄

Frattini

C₂×C₈

C₂³×D₁₃

Fitting

C₁₃×#2158

C₂

socle

C₂²×C₁₃

C₂²×D₈

prime radical

C₂³×C₁₃

C₂×D₈

cosocle

C₂×C₈×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
3. #13116706	C₂	semidirect
4. #13116694	C₂	semidirect
15. C₂×Dic₁₀₄	C₄	semidirect
178. C₁₃	#33458	semidirect

H ≅ G/N

type

3. #13116706

C₂

semidirect

4. #13116694

C₂

semidirect

15. C₂×Dic₁₀₄

C₄

semidirect

178. C₁₃

#33458

semidirect

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

subgroup	index
C₂²×Dic₁₀₄	2
#13123703	2
#13116706	2
#13116694	2
#13116555	2
#13116432	2
C₁₃×#2158	2
#33458	13

subgroup

index

C₂²×Dic₁₀₄

C₁₃×#2158

Subgroups — #16016296

Conjugacy classes of subgroups

Complements

Maximal subgroups