Group #15781151 — subgroups

Subgroups — #15781151

Order 3328.

590 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 163 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×#11718356	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718101	C₂	1	4	yes	maximal, complemented
3	#13115000	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₀₄	—	2	1	no	—
6	C₂²×D₁₀₄	C₂²	1	2	yes	—
7	C₂×#88128	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88058	C₂²	1	2	yes	—
11	C₂×#88057	—	2	1	no	—
12	#11718597	—	2	1	no	—
13	#11718356	—	2	2	no	—
14	#11718356	C₄	1	4	yes	complemented
15	C₁₃×#346	—	2	2	no	—
16	#11718101	—	2	8	no	—
17	C₂²×D₅₂	—	2	2	no	—
18	C₂²×D₅₂	—	4	2	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	—	2	2	no	—
23	C₂×D₁₀₄	—	2	4	no	—
24	C₂×D₁₀₄	—	2	4	no	—
25	C₂×D₁₀₄	—	2	4	no	—
26	C₂×D₁₀₄	—	2	2	no	—
27	C₂×D₁₀₄	—	4	2	no	—
28	C₂×D₁₀₄	—	4	2	no	—
29	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
30	#88128	—	2	4	no	—
31	#88128	—	4	2	no	—
32	C₂×C₁₃×C₁₆	—	2	2	no	—
33	C₂×C₁₃×C₁₆	—	2	2	no	—
34	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
35	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₃	D₄	1	2	yes	—
37	#88071	—	4	2	no	—
38	#88058	—	2	4	no	—
39	#88057	—	4	2	no	—
40	#13206	—	13	1	no	maximal, Sylow
41	C₂³×D₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	D₈	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	4	2	no	—
45	C₂×D₅₂	—	2	4	no	—
46	C₂×D₅₂	—	4	2	no	—
47	C₂×D₅₂	—	4	2	no	—
48	C₂×D₅₂	—	4	2	no	—
49	C₂×D₅₂	—	4	2	no	—
50	C₂×D₅₂	—	4	8	no	—
51	C₂×D₅₂	—	4	2	no	—
52	C₂×D₅₂	—	8	8	no	—
53	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₃	D₈	1	2	yes	—
55	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	#41	1	2	yes	—
60	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₁₃	SD₁₆	1	2	yes	—
62	C₄⋊Dic₁₃	—	4	2	no	—
63	C₄⋊Dic₁₃	—	4	2	no	—
64	D₁₀₄	—	2	8	no	—
65	D₁₀₄	—	4	8	no	—
66	D₁₀₄	—	4	4	no	—
67	C₁₃×C₁₆	—	2	4	no	—
68	C₂×#493	—	13	1	no	—
69	C₂×#357	—	13	4	no	—
70	C₂×#346	D₁₃	1	1	yes	characteristic, complemented
71	#1846	—	13	1	no	—
72	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
73	C₂²×D₁₃	—	4	4	no	—
74	C₂²×D₁₃	—	8	2	no	—
75	C₂²×D₁₃	—	8	2	no	—
76	C₂²×D₁₃	—	8	8	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	#102	1	4	yes	—
79	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×Dic₁₃	—	8	2	no	—
86	C₂×Dic₁₃	—	8	4	no	—
87	D₅₂	—	4	4	no	—
88	D₅₂	—	4	4	no	—
89	D₅₂	—	8	16	no	—
90	D₅₂	—	8	2	no	—
91	D₅₂	—	8	2	no	—
92	C₈×C₁₃	#102	1	4	yes	—
93	C₈×C₁₃	C₄×D₄	1	2	yes	—
94	C₈×C₁₃	#121	1	2	yes	—
95	C₂²×D₈	—	13	2	no	—
96	C₂²×D₈	—	26	1	no	—
97	C₂×#127	—	26	1	no	—
98	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
99	#493	—	13	4	no	—
100	#493	—	26	2	no	—
101	C₂×#107	—	13	2	no	—
102	C₂×#106	—	26	1	no	—
103	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
104	#382	—	26	1	no	—
105	#357	—	26	8	no	—
106	#346	—	2	2	no	—
107	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
108	C₂²×C₁₃	#486	1	1	yes	characteristic
109	C₂²×C₁₃	#493	1	1	yes	characteristic
110	C₂²×C₁₃	#509	1	4	yes	—
111	C₂×D₁₃	—	8	8	no	—
112	C₂×D₁₃	—	8	4	no	—
113	C₂×D₁₃	—	16	4	no	—
114	C₂×D₁₃	—	16	4	no	—
115	C₄×C₁₃	—	2	4	no	—
116	C₄×C₁₃	#390	1	1	yes	characteristic
117	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
118	C₄×C₁₃	#486	1	1	yes	characteristic
119	C₄×C₁₃	#492	1	1	yes	characteristic
120	Dic₁₃	—	16	2	no	—
121	C₂²×D₄	—	26	2	no	—
122	C₂²×D₄	—	52	2	no	—
123	C₂×D₈	—	13	4	no	—
124	C₂×D₈	—	26	4	no	—
125	C₂×D₈	—	26	4	no	—
126	C₂×D₈	—	26	4	no	—
127	C₂×D₈	—	26	2	no	—
128	C₂×D₈	—	52	2	no	—
129	C₂×D₈	—	52	2	no	—
130	C₂²×C₈	D₅₂	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
132	#127	—	26	4	no	—
133	#127	—	52	2	no	—
134	C₂×C₄⋊C₄	—	26	2	no	—
135	C₂×C₄²	D₅₂	1	1	yes	characteristic
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	D₅₂	1	4	yes	complemented
139	#107	—	26	4	no	—
140	#106	—	52	2	no	—
141	C₄×C₈	C₄×D₁₃	1	2	yes	—
142	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
143	#95	—	52	2	no	—
144	C₂×C₁₃	#1957	1	1	yes	characteristic
145	C₂×C₁₃	#2140	1	2	yes	—
146	C₂×C₁₃	#2244	1	2	yes	—
147	C₂×C₁₃	#2250	1	2	yes	—
148	D₁₃	—	16	4	no	—
149	C₂⁴	—	52	2	no	—
150	C₂×D₄	—	26	4	no	—
151	C₂×D₄	—	52	2	no	—
152	C₂×D₄	—	52	2	no	—
153	C₂×D₄	—	52	2	no	—
154	C₂×D₄	—	52	2	no	—
155	C₂×D₄	—	52	2	no	—
156	C₂×D₄	—	52	8	no	—
157	C₂×D₄	—	104	8	no	—
158	C₂²×C₄	—	52	2	no	—
159	C₂²×C₄	D₁₀₄	1	2	yes	—
160	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
161	D₈	—	26	8	no	—
162	D₈	—	52	4	no	—
163	D₈	—	52	8	no	—
164	C₂×C₈	#6095	1	2	yes	—
165	C₂×C₈	D₁₀₄	1	2	yes	—
166	C₂×C₈	#6103	1	2	yes	—
167	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
168	C₂×C₈	C₂×D₅₂	1	2	yes	—
169	C₂×C₈	#6127	1	2	yes	—
170	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
171	C₄⋊C₄	—	52	2	no	—
172	C₄⋊C₄	—	52	2	no	—
173	C₄²	#6095	1	2	yes	—
174	C₄²	#6103	1	2	yes	—
175	C₁₆	—	2	4	no	—
176	C₁₃	#13206	1	1	yes	characteristic, complemented, Sylow
177	C₂³	—	52	4	no	—
178	C₂³	—	104	8	no	—
179	C₂³	—	104	2	no	—
180	C₂³	—	104	2	no	—
181	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
182	D₄	—	52	4	no	—
183	D₄	—	52	4	no	—
184	D₄	—	104	16	no	—
185	D₄	—	104	2	no	—
186	D₄	—	104	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	104	4	no	—
189	C₂×C₄	—	104	2	no	—
190	C₂×C₄	#88061	1	4	yes	—
191	C₂×C₄	#88128	1	1	yes	characteristic
192	C₂×C₄	#88129	1	1	yes	characteristic
193	C₂×C₄	C₂×#6095	1	1	yes	characteristic
194	C₂×C₄	C₂×#6095	1	1	yes	characteristic
195	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
196	C₂×C₄	C₂×#6103	1	1	yes	characteristic
197	C₈	#88061	1	4	yes	—
198	C₈	C₄×D₅₂	1	2	yes	—
199	C₈	#88184	1	2	yes	—
200	C₂²	—	104	8	no	—
201	C₂²	—	104	4	no	—
202	C₂²	—	208	4	no	—
203	C₂²	—	208	4	no	—
204	C₂²	#11718355	1	1	yes	characteristic
205	C₂²	#11718356	1	1	yes	characteristic
206	C₂²	#11718569	1	4	yes	—
207	C₂²	C₂×#88061	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	208	2	no	—
210	C₄	#11718354	1	1	yes	characteristic
211	C₄	#11718355	1	1	yes	characteristic
212	C₄	#11718600	1	1	yes	characteristic
213	C₄	C₂×#88061	1	1	yes	characteristic
214	C₂	—	208	4	no	—
215	C₂	#13115642	1	1	yes	characteristic
216	C₂	#13116553	1	2	yes	—
217	C₂	#13117078	1	2	yes	—
218	C₂	#13117102	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718356

C₂

yes

maximal, characteristic

C₂×#11718101

C₂

yes

maximal, complemented

#13115000

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×#346

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₀₄

—

C₂²×D₁₀₄

C₂²

yes

—

C₂×#88128

—

C₂²×C₁₃×C₁₆

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88058

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₁₃×#346

—

#11718101

—

C₂²×D₅₂

—

C₂²×D₅₂

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

C₂×C₄

yes

—

#88128

—

#88128

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

D₄

yes

complemented

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

D₈

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

SD₁₆

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

C₁₃×C₁₆

—

C₂×#493

—

C₂×#357

—

C₂×#346

D₁₃

yes

characteristic, complemented

#1846

—

C₂³×C₁₃

C₂×D₈

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#102

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

#102

yes

—

C₈×C₁₃

C₄×D₄

yes

—

C₈×C₁₃

#121

yes

—

C₂²×D₈

—

C₂²×D₈

—

C₂×#127

—

C₂²×C₁₆

C₂×D₁₃

yes

—

#493

—

100

#493

—

101

C₂×#107

—

102

C₂×#106

—

103

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

104

#382

—

105

#357

—

106

#346

—

107

C₂²×C₁₃

C₂×#102

yes

characteristic

108

C₂²×C₁₃

#486

yes

characteristic

109

C₂²×C₁₃

#493

yes

characteristic

110

C₂²×C₁₃

#509

yes

—

111

C₂×D₁₃

—

112

C₂×D₁₃

—

113

C₂×D₁₃

—

114

C₂×D₁₃

—

115

C₄×C₁₃

—

116

C₄×C₁₃

#390

yes

characteristic

117

C₄×C₁₃

C₂×#102

yes

characteristic

118

C₄×C₁₃

#486

yes

characteristic

119

C₄×C₁₃

#492

yes

characteristic

120

Dic₁₃

—

121

C₂²×D₄

—

122

C₂²×D₄

—

123

C₂×D₈

—

124

C₂×D₈

—

125

C₂×D₈

—

126

C₂×D₈

—

127

C₂×D₈

—

128

C₂×D₈

—

129

C₂×D₈

—

130

C₂²×C₈

D₅₂

yes

characteristic

131

C₂²×C₈

C₂²×D₁₃

yes

characteristic

132

#127

—

133

#127

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄²

D₅₂

yes

characteristic

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

D₅₂

yes

complemented

139

#107

—

140

#106

—

141

C₄×C₈

C₄×D₁₃

yes

—

142

C₄×C₈

C₂²⋊D₁₃

yes

—

143

#95

—

144

C₂×C₁₃

#1957

yes

characteristic

145

C₂×C₁₃

#2140

yes

—

146

C₂×C₁₃

#2244

yes

—

147

C₂×C₁₃

#2250

yes

—

148

D₁₃

—

149

C₂⁴

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

104

—

158

C₂²×C₄

—

159

C₂²×C₄

D₁₀₄

yes

—

160

C₂²×C₄

C₂×D₅₂

yes

characteristic

161

D₈

—

162

D₈

—

163

D₈

—

164

C₂×C₈

#6095

yes

—

165

C₂×C₈

D₁₀₄

yes

—

166

C₂×C₈

#6103

yes

—

167

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

168

C₂×C₈

C₂×D₅₂

yes

—

169

C₂×C₈

#6127

yes

—

170

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄²

#6095

yes

—

174

C₄²

#6103

yes

—

175

C₁₆

—

176

C₁₃

#13206

yes

characteristic, complemented, Sylow

177

C₂³

—

178

C₂³

—

104

—

179

C₂³

—

104

—

180

C₂³

—

104

—

181

C₂³

C₂×D₁₀₄

yes

characteristic

182

D₄

—

183

D₄

—

184

D₄

—

104

—

185

D₄

—

104

—

186

D₄

—

104

—

187

C₂×C₄

—

188

C₂×C₄

—

104

—

189

C₂×C₄

—

104

—

190

C₂×C₄

#88061

yes

—

191

C₂×C₄

#88128

yes

characteristic

192

C₂×C₄

#88129

yes

characteristic

193

C₂×C₄

C₂×#6095

yes

characteristic

194

C₂×C₄

C₂×#6095

yes

characteristic

195

C₂×C₄

C₂×D₁₀₄

yes

characteristic

196

C₂×C₄

C₂×#6103

yes

characteristic

197

C₈

#88061

yes

—

198

C₈

C₄×D₅₂

yes

—

199

C₈

#88184

yes

—

200

C₂²

—

104

—

201

C₂²

—

104

—

202

C₂²

—

208

—

203

C₂²

—

208

—

204

C₂²

#11718355

yes

characteristic

205

C₂²

#11718356

yes

characteristic

206

C₂²

#11718569

yes

—

207

C₂²

C₂×#88061

yes

characteristic

208

C₄

—

209

C₄

—

208

—

210

C₄

#11718354

yes

characteristic

211

C₄

#11718355

yes

characteristic

212

C₄

#11718600

yes

characteristic

213

C₄

C₂×#88061

yes

characteristic

214

C₂

—

208

—

215

C₂

#13115642

yes

characteristic

216

C₂

#13116553

yes

—

217

C₂

#13117078

yes

—

218

C₂

#13117102

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂×#11718356	C₂
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#13206
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₁₀₄

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₂×C₁₃×#346

C₂

socle

C₂³×C₁₃

C₂×D₈

prime radical

C₂×#11718356

C₂

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. C₂×#11718101	C₂	semidirect
3. #13115000	C₂	semidirect
4. C₂×C₁₃×#346	C₂	semidirect
14. #11718356	C₄	semidirect
34. C₂×C₁₃×C₁₆	D₄	semidirect
70. C₂×#346	D₁₃	semidirect
138. C₂×C₁₆	D₅₂	semidirect
176. C₁₃	#13206	semidirect

H ≅ G/N

type

2. C₂×#11718101

C₂

semidirect

3. #13115000

C₂

semidirect

4. C₂×C₁₃×#346

C₂

semidirect

14. #11718356

C₄

semidirect

34. C₂×C₁₃×C₁₆

D₄

semidirect

70. C₂×#346

D₁₃

semidirect

138. C₂×C₁₆

D₅₂

semidirect

176. C₁₃

#13206

semidirect

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

subgroup	index
C₂×#11718356	2
C₂×#11718101	2
#13115000	2
C₂×C₁₃×#346	2
#13206	13

subgroup

index

C₂×C₁₃×#346

Subgroups — #15781151

Conjugacy classes of subgroups

Complements

Maximal subgroups