Group #15767968 — subgroups

Subgroups — #15767968

Order 3328.

526 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 155 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	#13115643	C₂	1	2	yes	maximal
2	#13114996	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114995	C₂	1	2	yes	maximal, complemented
4	#13112663	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1957	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#102	—	2	2	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#384	C₂²	1	1	yes	characteristic
10	C₂×#88072	—	2	4	no	—
11	C₂×#88058	—	2	2	no	—
12	C₂×#88047	—	2	4	no	—
13	#11718596	C₂²	1	2	yes	—
14	C₂×C₄×#1055	C₂²	1	2	yes	—
15	C₂²×D₄×C₁₃	—	2	1	no	—
16	C₂²×D₄×C₁₃	—	2	2	no	—
17	C₂²×C₈×C₁₃	D₄	1	2	yes	—
18	C₁₃×#127	—	2	1	no	—
19	C₁₃×#127	—	2	2	no	—
20	C₁₃×#127	Q₈	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂²×#1055	D₄	1	4	yes	—
28	C₁₃×#102	—	4	4	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	#88072	—	4	8	no	—
31	#88058	—	4	4	no	—
32	#88047	—	4	8	no	—
33	C₄×#1055	—	2	2	no	—
34	C₄×#1055	—	2	2	no	—
35	#10673	—	13	1	no	maximal, Sylow
36	C₂⁴×C₁₃	—	4	1	no	—
37	C₂×D₄×C₁₃	—	2	4	no	—
38	C₂×D₄×C₁₃	—	2	2	no	—
39	C₂×D₄×C₁₃	—	4	2	no	—
40	C₂×D₄×C₁₃	—	4	1	no	—
41	C₂×D₄×C₁₃	—	4	4	no	—
42	C₂×D₄×C₁₃	—	4	2	no	—
43	C₂×D₄×C₁₃	—	4	4	no	—
44	C₂×D₄×C₁₃	—	4	2	no	—
45	C₂²×C₄×C₁₃	—	4	1	no	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₁₃	—	4	2	no	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	D₈	1	4	yes	—
52	C₁₃×C₄⋊C₄	—	4	2	no	—
53	C₁₃×C₄⋊C₄	—	4	4	no	—
54	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₁₃	#41	1	1	yes	characteristic
56	C₄²×C₁₃	#41	1	2	yes	—
57	C₄⋊Dic₁₃	—	4	4	no	—
58	C₄⋊Dic₁₃	—	4	4	no	—
59	C₄⋊Dic₁₃	—	4	4	no	—
60	C₂×#1055	—	2	4	no	—
61	C₂×#1055	—	2	4	no	—
62	C₂×#1055	D₈	1	8	yes	—
63	#1957	—	13	2	no	—
64	#1957	D₁₃	1	1	yes	characteristic
65	#1845	—	13	1	no	—
66	#1845	—	13	2	no	—
67	#1273	—	13	1	no	—
68	C₂³×C₁₃	—	4	2	no	—
69	C₂³×C₁₃	—	8	1	no	—
70	C₂³×C₁₃	—	8	2	no	—
71	C₂³×C₁₃	—	8	1	no	—
72	C₂³×C₁₃	—	8	2	no	—
73	C₂³×C₁₃	#127	1	1	yes	characteristic
74	D₄×C₁₃	—	4	4	no	—
75	D₄×C₁₃	—	4	8	no	—
76	D₄×C₁₃	—	8	4	no	—
77	D₄×C₁₃	—	8	2	no	—
78	C₂×C₄×C₁₃	—	8	1	no	—
79	C₂×C₄×C₁₃	—	8	2	no	—
80	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#123	1	2	yes	—
83	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
85	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₁₃	#137	1	4	yes	—
87	C₂×C₄×C₁₃	#137	1	2	yes	—
88	C₂×Dic₁₃	—	8	2	no	—
89	C₂×Dic₁₃	—	8	2	no	—
90	C₂×Dic₁₃	—	8	2	no	—
91	C₈×C₁₃	—	2	4	no	—
92	#1055	—	2	8	no	—
93	C₂×#127	C₂×D₁₃	1	1	yes	characteristic
94	C₂×#107	—	26	4	no	—
95	C₂×#107	—	26	2	no	—
96	C₂×#102	—	2	2	no	—
97	C₂×#102	—	26	4	no	—
98	C₂×C₄×C₈	—	13	2	no	—
99	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
100	#384	—	13	2	no	—
101	#384	C₂×D₁₃	1	1	yes	characteristic
102	C₂²×C₁₃	—	8	4	no	—
103	C₂²×C₁₃	—	8	2	no	—
104	C₂²×C₁₃	—	16	1	no	—
105	C₂²×C₁₃	—	16	1	no	—
106	C₂²×C₁₃	—	16	2	no	—
107	C₂²×C₁₃	#399	1	1	yes	characteristic
108	C₂²×C₁₃	#493	1	1	yes	characteristic
109	C₂²×C₁₃	#493	1	2	yes	—
110	C₂²×C₁₃	#497	1	1	yes	characteristic
111	C₂²×C₁₃	#497	1	2	yes	—
112	C₄×C₁₃	—	16	1	no	—
113	C₄×C₁₃	#474	1	2	yes	—
114	C₄×C₁₃	#474	1	4	yes	—
115	C₄×C₁₃	#480	1	4	yes	—
116	C₄×C₁₃	#480	1	2	yes	—
117	Dic₁₃	—	16	2	no	—
118	C₂²×D₄	—	2	2	no	—
119	C₂²×D₄	—	2	1	no	—
120	C₂²×C₈	—	13	4	no	—
121	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
122	#127	—	2	1	no	—
123	#127	—	2	2	no	—
124	#127	Dic₂₆	1	2	yes	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	26	2	no	—
128	C₂×C₄⋊C₄	—	26	2	no	—
129	C₂×C₄⋊C₄	—	26	2	no	—
130	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
131	#107	—	52	8	no	—
132	#107	—	52	4	no	—
133	#102	—	4	4	no	—
134	#102	—	52	8	no	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	—	26	2	no	—
137	C₄×C₈	—	26	2	no	—
138	C₂×C₁₃	—	16	2	no	—
139	C₂×C₁₃	#1712	1	1	yes	characteristic
140	C₂×C₁₃	#1722	1	1	yes	characteristic
141	C₂×C₁₃	#1722	1	2	yes	—
142	C₂×C₁₃	#2093	1	2	yes	—
143	C₂×C₁₃	#2093	1	1	yes	characteristic
144	C₂⁴	—	4	1	no	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	4	no	—
149	C₂×D₄	—	4	4	no	—
150	C₂×D₄	—	4	2	no	—
151	C₂×D₄	—	4	1	no	—
152	C₂×D₄	—	4	2	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	—	52	2	no	—
155	C₂²×C₄	D₄×D₁₃	1	2	yes	—
156	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	13	8	no	—
160	C₂×C₈	—	26	4	no	—
161	C₂×C₈	—	26	4	no	—
162	C₂×C₈	#6104	1	4	yes	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	52	4	no	—
166	C₄⋊C₄	—	52	4	no	—
167	C₄⋊C₄	—	52	4	no	—
168	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
169	C₄²	#6127	1	1	yes	characteristic
170	C₄²	#6129	1	2	yes	—
171	C₁₃	#10673	1	1	yes	characteristic, complemented, Sylow
172	C₂³	—	4	2	no	—
173	C₂³	—	8	1	no	—
174	C₂³	—	8	1	no	—
175	C₂³	—	8	2	no	—
176	C₂³	—	8	2	no	—
177	C₂³	#88194	1	1	yes	characteristic
178	D₄	—	4	4	no	—
179	D₄	—	4	8	no	—
180	D₄	—	8	4	no	—
181	D₄	—	8	2	no	—
182	C₂×C₄	—	8	1	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	104	2	no	—
185	C₂×C₄	—	104	2	no	—
186	C₂×C₄	—	104	2	no	—
187	C₂×C₄	#88132	1	2	yes	—
188	C₂×C₄	#88137	1	2	yes	—
189	C₂×C₄	D₈×D₁₃	1	4	yes	—
190	C₂×C₄	#88169	1	4	yes	—
191	C₂×C₄	#88178	1	1	yes	characteristic
192	C₂×C₄	C₂×#6104	1	2	yes	—
193	C₂×C₄	#88189	1	1	yes	characteristic
194	C₂×C₄	#88205	1	2	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	26	8	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	2	no	—
199	C₂²	—	16	1	no	—
200	C₂²	—	16	1	no	—
201	C₂²	—	16	2	no	—
202	C₂²	#11718610	1	1	yes	characteristic
203	C₂²	#11718721	1	1	yes	characteristic
204	C₂²	#11718734	1	1	yes	characteristic
205	C₂²	#11718812	1	2	yes	—
206	C₂²	#11718828	1	2	yes	—
207	C₄	—	16	1	no	—
208	C₄	—	208	2	no	—
209	C₄	#11718420	1	4	yes	—
210	C₄	#11718435	1	4	yes	—
211	C₄	#11718668	1	2	yes	—
212	C₄	#11718694	1	2	yes	—
213	C₂	—	16	2	no	—
214	C₂	#13114450	1	1	yes	characteristic
215	C₂	#13114505	1	1	yes	characteristic
216	C₂	#13114509	1	2	yes	—
217	C₂	#13116362	1	2	yes	—
218	C₂	#13116364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115643

C₂

yes

maximal

#13114996

C₂

yes

maximal, characteristic, complemented

#13114995

C₂

yes

maximal, complemented

#13112663

C₂

yes

maximal, characteristic, complemented

C₁₃×#1957

C₂

yes

maximal, characteristic

C₂×C₁₃×#127

C₂²

yes

characteristic

C₂×C₁₃×#102

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

—

—

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

—

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×#127

—

C₁₃×#127

—

C₁₃×#127

Q₈

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

D₄

yes

—

C₁₃×#102

—

C₄×C₈×C₁₃

—

—

—

—

—

—

—

C₂⁴×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₈

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×Q₈

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

D₈

yes

—

#1957

—

#1957

D₁₃

yes

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

#127

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

#123

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

#1055

—

C₂×#127

C₂×D₁₃

yes

—

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

100

#384

—

101

#384

C₂×D₁₃

yes

characteristic

102

C₂²×C₁₃

—

103

C₂²×C₁₃

—

104

C₂²×C₁₃

—

105

C₂²×C₁₃

—

106

C₂²×C₁₃

—

107

C₂²×C₁₃

#399

yes

characteristic

108

C₂²×C₁₃

#493

yes

characteristic

109

C₂²×C₁₃

#493

yes

—

110

C₂²×C₁₃

#497

yes

characteristic

111

C₂²×C₁₃

#497

yes

—

112

C₄×C₁₃

—

113

C₄×C₁₃

#474

yes

—

114

C₄×C₁₃

#474

yes

—

115

C₄×C₁₃

#480

yes

—

116

C₄×C₁₃

#480

yes

—

117

Dic₁₃

—

118

C₂²×D₄

—

119

C₂²×D₄

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂²⋊D₁₃

yes

—

122

#127

—

123

#127

—

124

#127

Dic₂₆

yes

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄²

C₂²×D₁₃

yes

characteristic

131

#107

—

132

#107

—

133

#102

—

134

#102

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₂×C₁₃

—

139

C₂×C₁₃

#1712

yes

characteristic

140

C₂×C₁₃

#1722

yes

characteristic

141

C₂×C₁₃

#1722

yes

—

142

C₂×C₁₃

#2093

yes

—

143

C₂×C₁₃

#2093

yes

characteristic

144

C₂⁴

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

D₄×D₁₃

yes

—

156

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

#6104

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄²

C₂×Dic₂₆

yes

characteristic

169

C₄²

#6127

yes

characteristic

170

C₄²

#6129

yes

—

171

C₁₃

#10673

yes

characteristic, complemented, Sylow

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

—

176

C₂³

—

177

C₂³

#88194

yes

characteristic

178

D₄

—

179

D₄

—

180

D₄

—

181

D₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

104

—

185

C₂×C₄

—

104

—

186

C₂×C₄

—

104

—

187

C₂×C₄

#88132

yes

—

188

C₂×C₄

#88137

yes

—

189

C₂×C₄

D₈×D₁₃

yes

—

190

C₂×C₄

#88169

yes

—

191

C₂×C₄

#88178

yes

characteristic

192

C₂×C₄

C₂×#6104

yes

—

193

C₂×C₄

#88189

yes

characteristic

194

C₂×C₄

#88205

yes

—

195

C₈

—

196

C₈

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

#11718610

yes

characteristic

203

C₂²

#11718721

yes

characteristic

204

C₂²

#11718734

yes

characteristic

205

C₂²

#11718812

yes

—

206

C₂²

#11718828

yes

—

207

C₄

—

208

C₄

—

208

—

209

C₄

#11718420

yes

—

210

C₄

#11718435

yes

—

211

C₄

#11718668

yes

—

212

C₄

#11718694

yes

—

213

C₂

—

214

C₂

#13114450

yes

characteristic

215

C₂

#13114505

yes

characteristic

216

C₂

#13114509

yes

—

217

C₂

#13116362

yes

—

218

C₂

#13116364

yes

characteristic

	subgroup	G/H
center	C₂³	#88194
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1957	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×C₁₃×#127	C₂²
hypercenter	#1957	D₁₃
hypocenter	C₁₃	#10673
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

subgroup

G/H

center

C₂³

#88194

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1957

C₂

socle

C₂³×C₁₃

#127

prime radical

C₂×C₁₃×#127

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

N	H ≅ G/N	type
2. #13114996	C₂	semidirect
3. #13114995	C₂	semidirect
4. #13112663	C₂	semidirect
171. C₁₃	#10673	semidirect

H ≅ G/N

type

2. #13114996

C₂

semidirect

3. #13114995

C₂

semidirect

4. #13112663

C₂

semidirect

171. C₁₃

#10673

semidirect

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

subgroup	index
#13115643	2
#13114996	2
#13114995	2
#13112663	2
C₁₃×#1957	2
#10673	13

subgroup

index

C₁₃×#1957

Subgroups — #15767968

Conjugacy classes of subgroups

Complements

Maximal subgroups