Group #15767902 — subgroups

Subgroups — #15767902

Order 3328.

372 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13115011	C₂	1	1	yes	maximal, characteristic, complemented
2	#13113606	C₂	1	2	yes	maximal, complemented
3	#13113261	C₂	1	2	yes	maximal
4	#13112667	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1955	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₂²⋊C₈	—	2	2	no	—
9	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
10	C₂×#88059	—	2	2	no	—
11	#11718586	C₂²	1	2	yes	—
12	#11718585	C₂²	1	2	yes	—
13	D₄⋊#1055	—	2	4	no	—
14	#11718072	—	2	4	no	—
15	C₂²×D₄×C₁₃	Q₈	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	—	2	1	no	—
17	C₂²×C₈×C₁₃	—	2	1	no	—
18	C₂²×C₈×C₁₃	D₄	1	2	yes	—
19	C₄×D₄×C₁₃	—	2	2	no	—
20	C₄×D₄×C₁₃	—	2	2	no	—
21	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
22	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×#6099	—	2	2	no	—
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	C₁₃×C₂²⋊C₈	—	4	4	no	—
27	C₁₃×#97	—	2	2	no	—
28	C₂²⋊#1055	—	2	2	no	—
29	#88059	—	4	4	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	—	2	2	no	—
33	C₄×#1055	D₄	1	4	yes	—
34	#10661	—	13	1	no	maximal, Sylow
35	C₂⁴×C₁₃	—	2	1	no	—
36	C₂×D₄×C₁₃	—	2	2	no	—
37	C₂×D₄×C₁₃	—	4	2	no	—
38	C₂²×C₄×C₁₃	—	2	1	no	—
39	C₂²×C₄×C₁₃	—	4	1	no	—
40	C₂²×C₄×C₁₃	—	4	1	no	—
41	C₂²×C₄×C₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	—	4	2	no	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
46	C₂×C₈×C₁₃	—	2	2	no	—
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	—	4	1	no	—
49	C₂×C₈×C₁₃	—	4	2	no	—
50	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
51	C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₁₃×C₂²⋊C₄	—	4	2	no	—
53	C₄²×C₁₃	C₂×D₄	1	2	yes	—
54	C₄²×C₁₃	#41	1	2	yes	—
55	#6099	—	4	4	no	—
56	C₂×#1055	—	2	4	no	—
57	C₂×#1055	—	4	2	no	—
58	#1955	D₁₃	1	1	yes	characteristic
59	#1849	—	13	1	no	—
60	#1570	—	13	2	no	—
61	#1486	—	13	2	no	—
62	#1275	—	13	1	no	—
63	C₂³×C₁₃	—	2	2	no	—
64	C₂³×C₁₃	—	4	1	no	—
65	C₂³×C₁₃	—	4	2	no	—
66	C₂³×C₁₃	—	4	1	no	—
67	C₂³×C₁₃	—	4	2	no	—
68	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
69	D₄×C₁₃	—	2	4	no	—
70	D₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	4	1	no	—
72	C₂×C₄×C₁₃	—	4	1	no	—
73	C₂×C₄×C₁₃	—	4	1	no	—
74	C₂×C₄×C₁₃	—	4	2	no	—
75	C₂×C₄×C₁₃	—	8	2	no	—
76	C₂×C₄×C₁₃	—	8	1	no	—
77	C₂×C₄×C₁₃	—	8	1	no	—
78	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄×C₁₃	#123	1	2	yes	—
80	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
83	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
84	C₈×C₁₃	—	2	4	no	—
85	C₈×C₁₃	—	8	1	no	—
86	#1055	—	4	4	no	—
87	#1055	—	8	2	no	—
88	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
89	#431	—	13	2	no	—
90	C₂×#108	—	26	2	no	—
91	#423	—	13	2	no	—
92	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
93	C₂×C₂²⋊C₈	—	2	2	no	—
94	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
95	#334	—	26	4	no	—
96	D₄⋊C₈	—	26	4	no	—
97	C₂²×C₁₃	—	2	1	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	—	4	4	no	—
100	C₂²×C₁₃	—	4	2	no	—
101	C₂²×C₁₃	—	8	2	no	—
102	C₂²×C₁₃	—	8	1	no	—
103	C₂²×C₁₃	—	8	1	no	—
104	C₂²×C₁₃	#399	1	1	yes	characteristic
105	C₂²×C₁₃	#476	1	2	yes	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	—	8	1	no	—
109	C₄×C₁₃	#481	1	2	yes	—
110	C₄×C₁₃	#492	1	1	yes	characteristic
111	C₄×C₁₃	#496	1	1	yes	characteristic
112	C₂²×D₄	Dic₂₆	1	1	yes	characteristic
113	C₂³×C₄	—	2	1	no	—
114	C₂×M₁₆	—	26	2	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
117	C₄×D₄	—	2	2	no	—
118	C₄×D₄	—	2	2	no	—
119	C₂×C₄⋊C₄	Dic₂₆	1	1	yes	characteristic
120	C₂×C₂²⋊C₄	—	2	1	no	—
121	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
122	#108	—	52	4	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	26	2	no	—
125	C₄⋊C₈	—	26	2	no	—
126	C₄⋊C₈	—	26	2	no	—
127	C₂²⋊C₈	—	4	4	no	—
128	C₂²⋊C₈	—	26	2	no	—
129	#97	—	2	2	no	—
130	C₄×C₈	—	13	4	no	—
131	C₂×C₁₃	—	2	2	no	—
132	C₂×C₁₃	—	8	2	no	—
133	C₂×C₁₃	#2091	1	1	yes	characteristic
134	C₂×C₁₃	#2094	1	1	yes	characteristic
135	C₂×C₁₃	#2098	1	1	yes	characteristic
136	C₂⁴	—	2	1	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	4	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
145	C₂²×C₄	#6127	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
147	M₁₆	—	52	4	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	26	4	no	—
153	C₂×C₈	—	52	2	no	—
154	C₂×C₈	#6105	1	4	yes	—
155	C₄⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	—	4	2	no	—
157	C₄²	D₄×D₁₃	1	2	yes	—
158	C₄²	#6129	1	2	yes	—
159	C₁₃	#10661	1	1	yes	characteristic, complemented, Sylow
160	C₂³	—	2	2	no	—
161	C₂³	—	4	1	no	—
162	C₂³	—	4	2	no	—
163	C₂³	—	4	2	no	—
164	C₂³	—	4	1	no	—
165	C₂³	#88178	1	1	yes	characteristic
166	D₄	—	2	4	no	—
167	D₄	—	4	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	#88132	1	2	yes	—
176	C₂×C₄	#88137	1	2	yes	—
177	C₂×C₄	C₂×#6105	1	2	yes	—
178	C₂×C₄	#88189	1	1	yes	characteristic
179	C₂×C₄	#88194	1	1	yes	characteristic
180	C₂×C₄	#88203	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	8	1	no	—
183	C₈	—	52	4	no	—
184	C₈	—	104	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	4	4	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	1	no	—
191	C₂²	—	8	1	no	—
192	C₂²	#11718610	1	1	yes	characteristic
193	C₂²	#11718669	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	8	1	no	—
197	C₄	#11718695	1	2	yes	—
198	C₄	#11718719	1	1	yes	characteristic
199	C₄	#11718737	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	—	8	2	no	—
202	C₂	#13116352	1	1	yes	characteristic
203	C₂	#13116369	1	1	yes	characteristic
204	C₂	#13116389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115011

C₂

yes

maximal, characteristic, complemented

#13113606

C₂

yes

maximal, complemented

#13113261

C₂

yes

maximal

#13112667

C₂

yes

maximal, characteristic, complemented

C₁₃×#1955

C₂

yes

maximal, characteristic

C₂×C₄×D₄×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₁₃×C₂²⋊C₈

—

C₂×C₁₃×#97

C₂²

yes

characteristic

C₂×#88059

—

#11718586

C₂²

yes

—

#11718585

C₂²

yes

—

D₄⋊#1055

—

#11718072

—

C₂²×D₄×C₁₃

Q₈

yes

characteristic

C₂³×C₄×C₁₃

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₄×D₄×C₁₃

—

C₄×D₄×C₁₃

—

C₂×C₁₃×C₄⋊C₄

Q₈

yes

characteristic

C₂×C₁₃×C₂²⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₁₃×#97

—

C₂²⋊#1055

—

—

—

—

—

D₄

yes

—

#10661

—

maximal, Sylow

C₂⁴×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

—

—

—

—

D₁₃

yes

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊Q₈

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#1055

—

C₂×C₄×D₄

C₂×D₁₃

yes

—

—

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×#97

C₂×D₁₃

yes

characteristic

#334

—

D₄⋊C₈

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

100

C₂²×C₁₃

—

101

C₂²×C₁₃

—

102

C₂²×C₁₃

—

103

C₂²×C₁₃

—

104

C₂²×C₁₃

#399

yes

characteristic

105

C₂²×C₁₃

#476

yes

—

106

C₄×C₁₃

—

107

C₄×C₁₃

—

108

C₄×C₁₃

—

109

C₄×C₁₃

#481

yes

—

110

C₄×C₁₃

#492

yes

characteristic

111

C₄×C₁₃

#496

yes

characteristic

112

C₂²×D₄

Dic₂₆

yes

characteristic

113

C₂³×C₄

—

114

C₂×M₁₆

—

115

C₂²×C₈

—

116

C₂²×C₈

C₂²⋊D₁₃

yes

—

117

C₄×D₄

—

118

C₄×D₄

—

119

C₂×C₄⋊C₄

Dic₂₆

yes

characteristic

120

C₂×C₂²⋊C₄

—

121

C₂×C₄²

C₂²×D₁₃

yes

characteristic

122

#108

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₂²⋊C₈

—

128

C₂²⋊C₈

—

129

#97

—

130

C₄×C₈

—

131

C₂×C₁₃

—

132

C₂×C₁₃

—

133

C₂×C₁₃

#2091

yes

characteristic

134

C₂×C₁₃

#2094

yes

characteristic

135

C₂×C₁₃

#2098

yes

characteristic

136

C₂⁴

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

145

C₂²×C₄

#6127

yes

characteristic

146

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

#6105

yes

—

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

157

C₄²

D₄×D₁₃

yes

—

158

C₄²

#6129

yes

—

159

C₁₃

#10661

yes

characteristic, complemented, Sylow

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

#88178

yes

characteristic

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#88132

yes

—

176

C₂×C₄

#88137

yes

—

177

C₂×C₄

C₂×#6105

yes

—

178

C₂×C₄

#88189

yes

characteristic

179

C₂×C₄

#88194

yes

characteristic

180

C₂×C₄

#88203

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

104

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

#11718610

yes

characteristic

193

C₂²

#11718669

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#11718695

yes

—

198

C₄

#11718719

yes

characteristic

199

C₄

#11718737

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

#13116352

yes

characteristic

203

C₂

#13116369

yes

characteristic

204

C₂

#13116389

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88194
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1955	C₂
socle	C₂²×C₁₃	#399
prime radical	C₂²×D₄×C₁₃	Q₈
hypercenter	#1955	D₁₃
hypocenter	C₁₃	#10661
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1955

subgroup

G/H

center

C₂×C₄

#88194

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1955

C₂

socle

C₂²×C₁₃

#399

prime radical

C₂²×D₄×C₁₃

Q₈

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1955

N	H ≅ G/N	type
1. #13115011	C₂	semidirect
2. #13113606	C₂	semidirect
4. #13112667	C₂	semidirect
159. C₁₃	#10661	semidirect

H ≅ G/N

type

1. #13115011

C₂

semidirect

2. #13113606

C₂

semidirect

4. #13112667

C₂

semidirect

159. C₁₃

#10661

semidirect

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

subgroup	index
#13115011	2
#13113606	2
#13113261	2
#13112667	2
C₁₃×#1955	2
#10661	13

subgroup

index

C₁₃×#1955

Subgroups — #15767902

Conjugacy classes of subgroups

Complements

Maximal subgroups