Group #15756739 — subgroups

Subgroups — #15756739

Order 3328.

322 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 139 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	#13115007	C₂	1	1	yes	maximal, characteristic
2	#13114973	C₂	1	1	yes	maximal, characteristic
3	#13113595	C₂	1	1	yes	maximal, characteristic
4	#13113594	C₂	1	1	yes	maximal, characteristic
5	#13113591	C₂	1	1	yes	maximal, characteristic
6	C₂×#11718047	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1566	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#423	C₂²	1	1	yes	characteristic
9	C₁₃×#423	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88062	—	2	2	no	—
12	#11718585	C₂²	1	1	yes	characteristic
13	#11718585	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₁₃×#334	—	2	2	no	—
17	#11718150	—	2	2	no	—
18	#11718072	—	2	2	no	—
19	#11718071	—	2	2	no	—
20	#11718050	—	2	2	no	—
21	#11718047	C₄	1	2	yes	—
22	#11718047	C₄	1	2	yes	—
23	C₂×C₁₃×M₁₆	—	2	1	no	—
24	C₂×C₁₃×M₁₆	—	2	1	no	—
25	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×#6099	—	2	1	no	—
28	C₂×#6099	—	2	1	no	—
29	C₂²×#1055	D₄	1	2	yes	—
30	C₂²×#1055	Q₈	1	2	yes	—
31	C₁₃×C₄⋊C₈	—	2	1	no	—
32	C₁₃×C₄⋊C₈	—	2	1	no	—
33	C₁₃×C₄⋊C₈	—	2	1	no	—
34	C₁₃×C₄⋊C₈	—	2	1	no	—
35	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃	D₄	1	2	yes	—
38	#88062	—	4	4	no	—
39	C₄⋊#1055	—	2	1	no	—
40	C₄⋊#1055	—	2	1	no	—
41	C₄⋊#1055	—	2	1	no	—
42	C₄⋊#1055	—	2	1	no	—
43	C₄×#1055	C₂×C₄	1	2	yes	—
44	C₄×#1055	C₂×C₄	1	2	yes	—
45	C₄×#1055	D₄	1	2	yes	—
46	C₄×#1055	Q₈	1	2	yes	—
47	#8828	—	13	1	no	maximal, Sylow
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₁₃×M₁₆	—	4	2	no	—
52	C₁₃×M₁₆	—	4	2	no	—
53	C₂×C₈×C₁₃	—	2	4	no	—
54	C₂×C₈×C₁₃	—	4	1	no	—
55	C₂×C₈×C₁₃	—	4	1	no	—
56	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
58	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	#6099	—	4	2	no	—
63	#6099	—	4	2	no	—
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	—	4	1	no	—
68	C₂×#1055	—	4	1	no	—
69	C₂×#1055	C₄⋊C₄	1	4	yes	—
70	C₂×#1055	C₄⋊C₄	1	4	yes	—
71	#1848	—	13	1	no	—
72	#1837	—	13	1	no	—
73	#1567	—	13	1	no	—
74	#1567	—	13	1	no	—
75	#1566	—	13	1	no	—
76	#1566	D₁₃	1	1	yes	characteristic
77	C₂×#322	—	13	1	no	—
78	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×C₄×C₁₃	#137	1	2	yes	—
89	C₈×C₁₃	—	2	2	no	—
90	C₈×C₁₃	—	8	1	no	—
91	C₈×C₁₃	—	8	1	no	—
92	C₈×C₁₃	#108	1	4	yes	complemented
93	#1055	—	2	4	no	—
94	#1055	—	2	4	no	—
95	#1055	—	8	1	no	—
96	#1055	—	8	1	no	—
97	C₂×#108	—	26	2	no	—
98	#423	—	13	1	no	—
99	#423	—	13	1	no	—
100	#423	C₂×D₁₃	1	1	yes	characteristic
101	#423	C₂×D₁₃	1	1	yes	characteristic
102	C₂×C₄×C₈	—	13	1	no	—
103	C₂×C₄×C₈	—	13	1	no	—
104	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
105	#341	—	26	2	no	—
106	#335	—	26	2	no	—
107	#335	—	26	2	no	—
108	#334	—	2	2	no	—
109	#334	—	26	2	no	—
110	#322	—	13	2	no	—
111	#322	—	13	2	no	—
112	C₂²×C₁₃	—	2	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	#384	1	1	yes	characteristic
115	C₂²×C₁₃	#501	1	2	yes	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	#428	1	2	yes	—
119	C₄×C₁₃	C₂×#108	1	2	yes	—
120	C₄×C₁₃	#430	1	2	yes	—
121	C₄×C₁₃	#496	1	1	yes	characteristic
122	C₄×C₁₃	#497	1	1	yes	characteristic
123	C₂×M₁₆	—	2	1	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂×M₁₆	—	26	1	no	—
126	C₂×M₁₆	—	26	1	no	—
127	C₂²×C₈	—	13	2	no	—
128	C₂²×C₈	—	13	2	no	—
129	C₂²×C₈	Dic₂₆	1	2	yes	—
130	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
131	#108	—	52	4	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	—	2	1	no	—
135	C₄⋊C₈	—	2	1	no	—
136	C₄⋊C₈	—	26	1	no	—
137	C₄⋊C₈	—	26	1	no	—
138	C₄⋊C₈	—	26	1	no	—
139	C₄⋊C₈	—	26	1	no	—
140	C₄×C₈	—	13	2	no	—
141	C₄×C₈	—	13	2	no	—
142	C₄×C₈	—	13	2	no	—
143	C₄×C₈	—	13	2	no	—
144	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
145	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
147	C₂×C₁₃	—	2	2	no	—
148	C₂×C₁₃	#1847	1	1	yes	characteristic
149	C₂×C₁₃	#1848	1	1	yes	characteristic
150	C₂×C₁₃	#1849	1	1	yes	characteristic
151	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
152	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
153	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
154	M₁₆	—	4	2	no	—
155	M₁₆	—	4	2	no	—
156	M₁₆	—	52	2	no	—
157	M₁₆	—	52	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	13	4	no	—
162	C₂×C₈	—	13	4	no	—
163	C₂×C₈	—	26	2	no	—
164	C₂×C₈	—	26	2	no	—
165	C₂×C₈	—	52	2	no	—
166	C₂×C₈	—	52	1	no	—
167	C₂×C₈	—	52	1	no	—
168	C₂×C₈	#6101	1	2	yes	—
169	C₂×C₈	#6101	1	2	yes	—
170	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
171	C₄²	D₄×D₁₃	1	1	yes	characteristic
172	C₄²	Q₈×D₁₃	1	1	yes	characteristic
173	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
174	C₁₃	#8828	1	1	yes	characteristic, complemented, Sylow
175	C₂³	#88142	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	#88142	1	1	yes	characteristic
179	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
180	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
181	C₂×C₄	#88168	1	2	yes	—
182	C₂×C₄	#88169	1	2	yes	—
183	C₂×C₄	C₂×#6101	1	1	yes	characteristic
184	C₂×C₄	#88194	1	1	yes	characteristic
185	C₂×C₄	#88198	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	8	1	no	—
188	C₈	—	8	1	no	—
189	C₈	—	26	4	no	—
190	C₈	—	26	4	no	—
191	C₈	—	104	1	no	—
192	C₈	—	104	1	no	—
193	C₈	#88062	1	4	yes	complemented
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#11718517	1	2	yes	—
197	C₂²	#11718637	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	#11718522	1	2	yes	—
201	C₄	#11718554	1	2	yes	—
202	C₄	C₂×#88062	1	2	yes	—
203	C₄	#11718828	1	1	yes	characteristic
204	C₄	#11718837	1	1	yes	characteristic
205	C₂	—	2	2	no	—
206	C₂	#13115001	1	1	yes	characteristic
207	C₂	#13115007	1	1	yes	characteristic
208	C₂	#13115009	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115007

C₂

yes

maximal, characteristic

#13114973

C₂

yes

maximal, characteristic

#13113595

C₂

yes

maximal, characteristic

#13113594

C₂

yes

maximal, characteristic

#13113591

C₂

yes

maximal, characteristic

C₂×#11718047

C₂

yes

maximal, characteristic

C₁₃×#1566

C₂

yes

maximal, characteristic

C₁₃×#423

C₂²

yes

characteristic

C₁₃×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88062

—

#11718585

C₂²

yes

characteristic

#11718585

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

C₁₃×#334

—

—

—

—

—

C₄

yes

—

#11718047

C₄

yes

—

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₂×#6099

—

C₂²×#1055

D₄

yes

—

C₂²×#1055

Q₈

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

—

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

D₄

yes

—

C₄×#1055

Q₈

yes

—

#8828

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#1055

C₄⋊C₄

yes

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂×#322

—

C₂³×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

yes

—

—

—

—

—

—

—

100

#423

C₂×D₁₃

yes

characteristic

101

#423

C₂×D₁₃

yes

characteristic

102

C₂×C₄×C₈

—

103

C₂×C₄×C₈

—

104

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

105

#341

—

106

#335

—

107

#335

—

108

#334

—

109

#334

—

110

#322

—

111

#322

—

112

C₂²×C₁₃

—

113

C₂²×C₁₃

—

114

C₂²×C₁₃

#384

yes

characteristic

115

C₂²×C₁₃

#501

yes

—

116

C₄×C₁₃

—

117

C₄×C₁₃

—

118

C₄×C₁₃

#428

yes

—

119

C₄×C₁₃

C₂×#108

yes

—

120

C₄×C₁₃

#430

yes

—

121

C₄×C₁₃

#496

yes

characteristic

122

C₄×C₁₃

#497

yes

characteristic

123

C₂×M₁₆

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

Dic₂₆

yes

—

130

C₂×C₄²

C₂²×D₁₃

yes

characteristic

131

#108

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

C₄×D₁₃

yes

characteristic

145

C₄×C₈

C₄×D₁₃

yes

characteristic

146

C₄×C₈

C₂²⋊D₁₃

yes

—

147

C₂×C₁₃

—

148

C₂×C₁₃

#1847

yes

characteristic

149

C₂×C₁₃

#1848

yes

characteristic

150

C₂×C₁₃

#1849

yes

characteristic

151

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

152

C₂²×C₄

D₄×D₁₃

yes

characteristic

153

C₂²×C₄

Q₈×D₁₃

yes

characteristic

154

M₁₆

—

155

M₁₆

—

156

M₁₆

—

157

M₁₆

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

#6101

yes

—

169

C₂×C₈

#6101

yes

—

170

C₄²

C₂×C₄×D₁₃

yes

characteristic

171

C₄²

D₄×D₁₃

yes

characteristic

172

C₄²

Q₈×D₁₃

yes

characteristic

173

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

174

C₁₃

#8828

yes

characteristic, complemented, Sylow

175

C₂³

#88142

yes

characteristic

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

#88142

yes

characteristic

179

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

180

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

181

C₂×C₄

#88168

yes

—

182

C₂×C₄

#88169

yes

—

183

C₂×C₄

C₂×#6101

yes

characteristic

184

C₂×C₄

#88194

yes

characteristic

185

C₂×C₄

#88198

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₈

—

104

—

192

C₈

—

104

—

193

C₈

#88062

yes

complemented

194

C₂²

—

195

C₂²

—

196

C₂²

#11718517

yes

—

197

C₂²

#11718637

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

#11718522

yes

—

201

C₄

#11718554

yes

—

202

C₄

C₂×#88062

yes

—

203

C₄

#11718828

yes

characteristic

204

C₄

#11718837

yes

characteristic

205

C₂

—

206

C₂

#13115001

yes

characteristic

207

C₂

#13115007

yes

characteristic

208

C₂

#13115009

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1566	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1566	D₁₃
hypocenter	C₁₃	#8828
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1566

C₂

socle

C₂²×C₁₃

#384

prime radical

C₂³×C₁₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
92. C₈×C₁₃	#108	semidirect
174. C₁₃	#8828	semidirect
193. C₈	#88062	semidirect

H ≅ G/N

type

92. C₈×C₁₃

#108

semidirect

174. C₁₃

#8828

semidirect

193. C₈

#88062

semidirect

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

subgroup	index
#13115007	2
#13114973	2
#13113595	2
#13113594	2
#13113591	2
C₂×#11718047	2
C₁₃×#1566	2
#8828	13

subgroup

index

C₁₃×#1566

Subgroups — #15756739

Conjugacy classes of subgroups

Complements

Maximal subgroups