Group #15765756 — subgroups

Subgroups — #15765756

Order 3328.

586 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 161 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×#11718740	C₂	1	2	yes	maximal
2	#13115639	C₂	1	2	yes	maximal
3	#13115638	C₂	1	1	yes	maximal, characteristic
4	#13112663	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#88123	C₂²	1	1	yes	characteristic
8	C₁₃×#498	—	2	4	no	—
9	C₂×C₁₃×#106	—	2	2	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88075	—	2	4	no	—
12	C₂×#88056	—	2	2	no	—
13	#11718740	—	2	4	no	—
14	#11718740	—	2	4	no	—
15	C₂×#88048	—	2	4	no	—
16	#11718596	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₂²	1	2	yes	—
18	C₂²×Q₈×C₁₃	—	2	2	no	—
19	C₂²×Dic₂₆	—	2	1	no	—
20	C₂²×C₈×C₁₃	D₄	1	2	yes	—
21	C₁₃×C₄⋊Q₈	—	2	2	no	—
22	C₁₃×C₄⋊Q₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
25	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₃	—	2	2	no	—
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×C₄⋊Dic₁₃	—	2	2	no	—
31	C₂²×#1055	D₄	1	4	yes	—
32	#88123	—	2	2	no	—
33	#88123	—	2	1	no	—
34	#88123	D₄	1	2	yes	—
35	C₁₃×#106	—	2	8	no	—
36	C₄×C₈×C₁₃	—	2	2	no	—
37	#88075	—	4	8	no	—
38	#88056	—	4	4	no	—
39	#88048	—	2	16	no	—
40	C₄×#1055	—	2	2	no	—
41	C₄×#1055	—	2	2	no	—
42	#10316	—	13	1	no	maximal, Sylow
43	C₂×Q₈×C₁₃	—	2	4	no	—
44	C₂×Q₈×C₁₃	—	4	2	no	—
45	C₂×Q₈×C₁₃	—	4	4	no	—
46	C₂×Q₈×C₁₃	—	4	4	no	—
47	C₂²×C₄×C₁₃	—	4	2	no	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	—	4	1	no	—
51	C₂²×Dic₁₃	—	4	1	no	—
52	C₂×Dic₂₆	—	2	2	no	—
53	C₂×Dic₂₆	—	4	2	no	—
54	C₂×Dic₂₆	—	4	2	no	—
55	C₂×Dic₂₆	—	4	1	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
59	C₁₃×C₄⋊C₄	—	2	4	no	—
60	C₁₃×C₄⋊C₄	—	2	4	no	—
61	C₁₃×C₄⋊C₄	—	4	2	no	—
62	C₁₃×C₄⋊C₄	—	4	2	no	—
63	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₃	C₂×D₄	1	2	yes	—
65	C₄²×C₁₃	#41	1	1	yes	characteristic
66	C₄⋊Dic₁₃	—	2	4	no	—
67	C₄⋊Dic₁₃	—	4	2	no	—
68	C₄⋊Dic₁₃	—	4	4	no	—
69	C₄⋊Dic₁₃	—	4	2	no	—
70	C₂×#1055	—	2	4	no	—
71	C₂×#1055	—	2	4	no	—
72	C₂×#1055	SD₁₆	1	8	yes	—
73	C₂×#498	—	13	2	no	—
74	C₂×#498	D₁₃	1	1	yes	characteristic
75	#1956	—	13	1	no	—
76	#1956	—	13	2	no	—
77	#1273	—	13	1	no	—
78	C₂³×C₁₃	#127	1	1	yes	characteristic
79	Q₈×C₁₃	—	4	8	no	—
80	Q₈×C₁₃	—	8	4	no	—
81	C₂×C₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	—	8	2	no	—
83	C₂×C₄×C₁₃	—	8	2	no	—
84	C₂×C₄×C₁₃	#120	1	2	yes	—
85	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#121	1	2	yes	—
88	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
89	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₁₃	#137	1	4	yes	—
91	C₂×C₄×C₁₃	#137	1	2	yes	—
92	C₂×Dic₁₃	—	4	2	no	—
93	C₂×Dic₁₃	—	8	1	no	—
94	C₂×Dic₁₃	—	8	1	no	—
95	C₂×Dic₁₃	—	8	1	no	—
96	C₂×Dic₁₃	—	8	2	no	—
97	Dic₂₆	—	4	4	no	—
98	Dic₂₆	—	8	2	no	—
99	C₈×C₁₃	—	2	4	no	—
100	#1055	—	2	8	no	—
101	C₂×C₄⋊Q₈	—	13	1	no	—
102	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
103	#498	—	2	4	no	—
104	#498	—	26	4	no	—
105	#498	—	26	4	no	—
106	C₂×#106	—	2	2	no	—
107	C₂×#106	—	26	4	no	—
108	C₂×#103	—	26	4	no	—
109	C₂×#103	—	26	2	no	—
110	C₂×C₄×C₈	—	13	2	no	—
111	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
112	#384	—	13	1	no	—
113	C₂²×C₁₃	#392	1	1	yes	characteristic
114	C₂²×C₁₃	#492	1	1	yes	characteristic
115	C₂²×C₁₃	#492	1	2	yes	—
116	C₂²×C₁₃	#497	1	1	yes	characteristic
117	C₂²×C₁₃	#497	1	2	yes	—
118	C₄×C₁₃	—	8	4	no	—
119	C₄×C₁₃	#448	1	4	yes	—
120	C₄×C₁₃	#448	1	2	yes	—
121	C₄×C₁₃	#461	1	2	yes	—
122	C₄×C₁₃	#461	1	4	yes	—
123	Dic₁₃	—	8	2	no	—
124	Dic₁₃	—	16	1	no	—
125	C₂²×Q₈	—	2	2	no	—
126	C₂²×Q₈	—	26	1	no	—
127	C₂²×C₈	—	13	4	no	—
128	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	13	2	no	—
133	C₄⋊Q₈	—	26	1	no	—
134	C₄⋊Q₈	—	26	2	no	—
135	C₄⋊Q₈	C₂²⋊D₁₃	1	4	yes	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	26	2	no	—
139	C₂×C₄⋊C₄	—	26	2	no	—
140	C₂×C₄⋊C₄	—	26	1	no	—
141	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
142	#106	—	2	8	no	—
143	#106	—	26	16	no	—
144	#103	—	52	8	no	—
145	#103	—	52	4	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	26	2	no	—
148	C₄×C₈	—	26	2	no	—
149	C₂×C₁₃	#1705	1	1	yes	characteristic
150	C₂×C₁₃	#1716	1	2	yes	—
151	C₂×C₁₃	#1716	1	1	yes	characteristic
152	C₂×C₁₃	#2013	1	2	yes	—
153	C₂×C₁₃	#2013	1	1	yes	characteristic
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	26	2	no	—
159	C₂×Q₈	—	52	1	no	—
160	C₂×Q₈	—	52	2	no	—
161	C₂×Q₈	—	52	2	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	52	1	no	—
164	C₂²×C₄	—	52	1	no	—
165	C₂²×C₄	D₄×D₁₃	1	2	yes	—
166	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	13	8	no	—
170	C₂×C₈	—	26	4	no	—
171	C₂×C₈	—	26	4	no	—
172	C₂×C₈	#6106	1	4	yes	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	26	4	no	—
178	C₄⋊C₄	—	52	2	no	—
179	C₄⋊C₄	—	52	4	no	—
180	C₄⋊C₄	—	52	2	no	—
181	C₄²	D₄×D₁₃	1	1	yes	characteristic
182	C₄²	#6129	1	1	yes	characteristic
183	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
184	C₁₃	#10316	1	1	yes	characteristic, complemented, Sylow
185	C₂³	#88194	1	1	yes	characteristic
186	Q₈	—	4	8	no	—
187	Q₈	—	8	4	no	—
188	Q₈	—	52	4	no	—
189	Q₈	—	104	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	52	2	no	—
194	C₂×C₄	—	104	1	no	—
195	C₂×C₄	—	104	1	no	—
196	C₂×C₄	—	104	1	no	—
197	C₂×C₄	—	104	2	no	—
198	C₂×C₄	SD₁₆×D₁₃	1	4	yes	—
199	C₂×C₄	#88169	1	4	yes	—
200	C₂×C₄	#88191	1	2	yes	—
201	C₂×C₄	#88192	1	1	yes	characteristic
202	C₂×C₄	#88193	1	2	yes	—
203	C₂×C₄	C₂×#6106	1	2	yes	—
204	C₂×C₄	#88196	1	2	yes	—
205	C₂×C₄	#88207	1	1	yes	characteristic
206	C₈	—	2	4	no	—
207	C₈	—	26	8	no	—
208	C₂²	#11718746	1	1	yes	characteristic
209	C₂²	#11718747	1	1	yes	characteristic
210	C₂²	#11718828	1	2	yes	—
211	C₂²	#11718836	1	2	yes	—
212	C₂²	#11718884	1	1	yes	characteristic
213	C₄	—	8	4	no	—
214	C₄	—	104	2	no	—
215	C₄	—	208	1	no	—
216	C₄	#11718744	1	2	yes	—
217	C₄	#11718823	1	4	yes	—
218	C₄	#11718831	1	4	yes	—
219	C₄	#11718888	1	2	yes	—
220	C₂	#13114424	1	1	yes	characteristic
221	C₂	#13114468	1	2	yes	—
222	C₂	#13114470	1	1	yes	characteristic
223	C₂	#13115917	1	2	yes	—
224	C₂	#13115919	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718740

C₂

yes

maximal

#13115639

C₂

yes

maximal

#13115638

C₂

yes

maximal, characteristic

#13112663

C₂

yes

maximal, characteristic

C₂×C₁₃×#498

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

—

C₂×#88123

C₂²

yes

characteristic

C₁₃×#498

—

C₂×C₁₃×#106

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

—

C₂²×Q₈×C₁₃

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

D₄

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

—

#88123

—

#88123

—

#88123

D₄

yes

—

C₁₃×#106

—

C₄×C₈×C₁₃

—

—

—

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

SD₁₆

yes

—

C₂×#498

—

C₂×#498

D₁₃

yes

—

—

—

C₂³×C₁₃

#127

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#120

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

100

#1055

—

101

C₂×C₄⋊Q₈

—

102

C₂×C₄⋊Q₈

C₂×D₁₃

yes

—

103

#498

—

104

#498

—

105

#498

—

106

C₂×#106

—

107

C₂×#106

—

108

C₂×#103

—

109

C₂×#103

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

112

#384

—

113

C₂²×C₁₃

#392

yes

characteristic

114

C₂²×C₁₃

#492

yes

characteristic

115

C₂²×C₁₃

#492

yes

—

116

C₂²×C₁₃

#497

yes

characteristic

117

C₂²×C₁₃

#497

yes

—

118

C₄×C₁₃

—

119

C₄×C₁₃

#448

yes

—

120

C₄×C₁₃

#448

yes

—

121

C₄×C₁₃

#461

yes

—

122

C₄×C₁₃

#461

yes

—

123

Dic₁₃

—

124

Dic₁₃

—

125

C₂²×Q₈

—

126

C₂²×Q₈

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²⋊D₁₃

yes

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

—

134

C₄⋊Q₈

—

135

C₄⋊Q₈

C₂²⋊D₁₃

yes

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄²

C₂²×D₁₃

yes

characteristic

142

#106

—

143

#106

—

144

#103

—

145

#103

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₂×C₁₃

#1705

yes

characteristic

150

C₂×C₁₃

#1716

yes

—

151

C₂×C₁₃

#1716

yes

characteristic

152

C₂×C₁₃

#2013

yes

—

153

C₂×C₁₃

#2013

yes

characteristic

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

D₄×D₁₃

yes

—

166

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

#6106

yes

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄²

D₄×D₁₃

yes

characteristic

182

C₄²

#6129

yes

characteristic

183

C₄²

C₂×C₂²⋊D₁₃

yes

—

184

C₁₃

#10316

yes

characteristic, complemented, Sylow

185

C₂³

#88194

yes

characteristic

186

Q₈

—

187

Q₈

—

188

Q₈

—

189

Q₈

—

104

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

104

—

195

C₂×C₄

—

104

—

196

C₂×C₄

—

104

—

197

C₂×C₄

—

104

—

198

C₂×C₄

SD₁₆×D₁₃

yes

—

199

C₂×C₄

#88169

yes

—

200

C₂×C₄

#88191

yes

—

201

C₂×C₄

#88192

yes

characteristic

202

C₂×C₄

#88193

yes

—

203

C₂×C₄

C₂×#6106

yes

—

204

C₂×C₄

#88196

yes

—

205

C₂×C₄

#88207

yes

characteristic

206

C₈

—

207

C₈

—

208

C₂²

#11718746

yes

characteristic

209

C₂²

#11718747

yes

characteristic

210

C₂²

#11718828

yes

—

211

C₂²

#11718836

yes

—

212

C₂²

#11718884

yes

characteristic

213

C₄

—

214

C₄

—

104

—

215

C₄

—

208

—

216

C₄

#11718744

yes

—

217

C₄

#11718823

yes

—

218

C₄

#11718831

yes

—

219

C₄

#11718888

yes

—

220

C₂

#13114424

yes

characteristic

221

C₂

#13114468

yes

—

222

C₂

#13114470

yes

characteristic

223

C₂

#13115917

yes

—

224

C₂

#13115919

yes

characteristic

	subgroup	G/H
center	C₂³	#88194
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#498	C₂
socle, prime radical	C₂³×C₁₃	#127
hypercenter	C₂×#498	D₁₃
hypocenter	C₁₃	#10316
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#88194

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₁₃×#498

C₂

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
184. C₁₃	#10316	semidirect

H ≅ G/N

type

184. C₁₃

#10316

semidirect

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

subgroup	index
C₂×#11718740	2
#13115639	2
#13115638	2
#13112663	2
C₂×C₁₃×#498	2
#10316	13

subgroup

index

C₂×C₁₃×#498

Subgroups — #15765756

Conjugacy classes of subgroups

Complements

Maximal subgroups