Group #15756798 — subgroups

Subgroups — #15756798

Order 3328.

386 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 131 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	#13123921	C₂	1	1	yes	maximal, characteristic
2	#13115773	C₂	1	1	yes	maximal, characteristic
3	#13113588	C₂	1	2	yes	maximal
4	#13112674	C₂	1	2	yes	maximal
5	C₄×C₁₃×#101	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#101	C₄	1	2	yes	—
8	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
9	#11718685	—	2	2	no	—
10	#11718685	—	2	2	no	—
11	#11718587	—	2	2	no	—
12	#11718586	C₂²	1	2	yes	—
13	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	C₁₃×#343	C₄	1	1	yes	characteristic
16	C₁₃×#343	C₄	1	1	yes	characteristic
17	#11718116	—	2	2	no	—
18	#11718050	—	2	2	no	—
19	#11718049	—	2	2	no	—
20	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
23	C₄×Q₈×C₁₃	—	2	2	no	—
24	C₄×Q₈×C₁₃	—	2	2	no	—
25	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃×C₄⋊C₄	Q₈	1	2	yes	—
27	C₂×C₄²×C₁₃	D₄	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×#1055	—	2	1	no	—
30	C₂²×#1055	—	2	1	no	—
31	C₁₃×#101	—	2	4	no	—
32	C₁₃×#97	—	2	2	no	—
33	C₄×C₈×C₁₃	—	2	2	no	—
34	C₂²⋊#1055	—	2	2	no	—
35	C₄⋊#1055	—	2	2	no	—
36	C₄⋊#1055	—	2	2	no	—
37	C₄⋊#1055	—	2	2	no	—
38	C₄⋊#1055	D₄	1	2	yes	—
39	C₄⋊#1055	D₄	1	2	yes	—
40	#88043	—	2	4	no	—
41	C₄×#1055	—	2	2	no	—
42	C₄×#1055	—	2	2	no	—
43	#8838	—	13	1	no	maximal, Sylow
44	C₂×Q₈×C₁₃	—	2	2	no	—
45	C₂×Q₈×C₁₃	—	2	2	no	—
46	C₂×Q₈×C₁₃	—	2	1	no	—
47	C₂×Q₈×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₁₃×M₁₆	—	2	2	no	—
54	C₁₃×M₁₆	—	2	2	no	—
55	C₁₃×M₁₆	—	2	2	no	—
56	C₁₃×M₁₆	M₁₆	1	4	yes	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	—	2	4	no	—
61	C₄²×C₁₃	—	2	2	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	C₂×D₄	1	2	yes	—
64	C₄²×C₁₃	#41	1	2	yes	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	2	2	no	—
67	C₂×#1055	—	2	2	no	—
68	C₂×#1055	—	4	1	no	—
69	C₂×#1055	—	4	1	no	—
70	C₂×#1055	—	4	2	no	—
71	C₂×#1055	—	4	1	no	—
72	#2983	—	13	1	no	—
73	#1984	—	13	1	no	—
74	C₄×#101	D₁₃	1	1	yes	characteristic
75	#1565	—	13	2	no	—
76	#1278	—	13	2	no	—
77	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	Q₈×C₁₃	—	4	2	no	—
79	Q₈×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	4	1	no	—
87	C₂×C₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
91	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
95	C₈×C₁₃	—	2	4	no	—
96	C₈×C₁₃	—	4	2	no	—
97	#1055	—	4	2	no	—
98	#1055	—	4	2	no	—
99	#1055	—	4	2	no	—
100	#1055	—	8	1	no	—
101	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
102	#446	—	26	2	no	—
103	#446	—	26	2	no	—
104	#432	—	26	2	no	—
105	#431	—	13	2	no	—
106	C₂×C₄⋊C₈	—	13	1	no	—
107	C₂×#101	Dic₁₃	1	2	yes	—
108	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
109	C₂×C₄×C₈	—	13	1	no	—
110	#343	Dic₁₃	1	1	yes	characteristic
111	#343	Dic₁₃	1	1	yes	characteristic
112	#336	—	26	2	no	—
113	#335	—	26	2	no	—
114	#334	—	26	2	no	—
115	C₂²×C₁₃	—	2	1	no	—
116	C₂²×C₁₃	—	2	1	no	—
117	C₂²×C₁₃	#385	1	1	yes	characteristic
118	C₂²×C₁₃	#423	1	1	yes	characteristic
119	C₂²×C₁₃	#424	1	1	yes	characteristic
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	4	2	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	#435	1	2	yes	—
125	C₄×C₁₃	#436	1	2	yes	—
126	C₄×C₁₃	#454	1	1	yes	characteristic
127	C₄×C₁₃	#456	1	1	yes	characteristic
128	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
129	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
130	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
131	C₂²×C₈	—	26	1	no	—
132	C₂²×C₈	—	26	1	no	—
133	C₄×Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	Dic₂₆	1	2	yes	—
136	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
137	C₂×C₄²	D₅₂	1	2	yes	—
138	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
139	C₄⋊C₈	—	13	2	no	—
140	C₄⋊C₈	—	13	2	no	—
141	C₄⋊C₈	—	26	2	no	—
142	C₄⋊C₈	—	26	2	no	—
143	C₄⋊C₈	—	26	2	no	—
144	#101	—	2	4	no	—
145	C₂²⋊C₈	—	26	2	no	—
146	#97	—	2	2	no	—
147	#97	—	26	4	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	26	2	no	—
150	C₄×C₈	—	26	2	no	—
151	C₂×C₁₃	—	2	1	no	—
152	C₂×C₁₃	—	2	1	no	—
153	C₂×C₁₃	#1851	1	1	yes	characteristic
154	C₂×C₁₃	#1857	1	1	yes	characteristic
155	C₂×C₁₃	#1858	1	1	yes	characteristic
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	1	no	—
160	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
161	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
162	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
164	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
165	M₁₆	—	2	2	no	—
166	M₁₆	—	2	2	no	—
167	M₁₆	—	2	2	no	—
168	M₁₆	#6099	1	4	yes	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	26	2	no	—
172	C₂×C₈	—	26	2	no	—
173	C₂×C₈	—	26	2	no	—
174	C₂×C₈	—	52	1	no	—
175	C₂×C₈	—	52	1	no	—
176	C₂×C₈	—	52	2	no	—
177	C₂×C₈	—	52	1	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	D₄×D₁₃	1	2	yes	—
183	C₄²	#6129	1	2	yes	—
184	C₁₃	#8838	1	1	yes	characteristic, complemented, Sylow
185	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
186	Q₈	—	4	2	no	—
187	Q₈	—	4	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	#88132	1	2	yes	—
197	C₂×C₄	#88136	1	1	yes	characteristic
198	C₂×C₄	#88140	1	1	yes	characteristic
199	C₂×C₄	C₂×#6099	1	2	yes	—
200	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
201	C₂×C₄	#88202	1	1	yes	characteristic
202	C₂×C₄	#88202	1	1	yes	characteristic
203	C₈	—	2	4	no	—
204	C₈	—	4	2	no	—
205	C₈	—	52	2	no	—
206	C₈	—	52	2	no	—
207	C₈	—	52	2	no	—
208	C₈	—	104	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	#11718585	1	1	yes	characteristic
212	C₂²	#11718611	1	1	yes	characteristic
213	C₂²	#11718660	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	#11718413	1	1	yes	characteristic
219	C₄	#11718414	1	1	yes	characteristic
220	C₄	#11718672	1	2	yes	—
221	C₄	#11718673	1	2	yes	—
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#13115021	1	1	yes	characteristic
225	C₂	#13115053	1	1	yes	characteristic
226	C₂	#13115058	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13123921

C₂

yes

maximal, characteristic

#13115773

C₂

yes

maximal, characteristic

#13113588

C₂

yes

maximal

#13112674

C₂

yes

maximal

C₄×C₁₃×#101

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×#101

C₄

yes

—

C₄×C₁₃×M₁₆

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₄⋊#1055

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

C₁₃×#343

C₄

yes

characteristic

C₁₃×#343

C₄

yes

—

—

—

C₂²×Q₈×C₁₃

C₂×C₄

yes

characteristic

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₄×Q₈×C₁₃

—

C₄×Q₈×C₁₃

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

characteristic

C₂×C₁₃×C₄⋊C₄

Q₈

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₂²×#1055

—

C₁₃×#101

—

C₁₃×#97

—

C₄×C₈×C₁₃

—

C₂²⋊#1055

—

—

—

—

D₄

yes

—

C₄⋊#1055

D₄

yes

—

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

M₁₆

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

—

—

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic

#1565

—

#1278

—

C₂³×C₁₃

C₂×C₄⋊C₄

yes

characteristic

Q₈×C₁₃

—

Q₈×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

—

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

#120

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#123

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#1055

—

#1055

—

100

#1055

—

101

C₂×C₄×Q₈

C₂×D₁₃

yes

characteristic

102

#446

—

103

#446

—

104

#432

—

105

#431

—

106

C₂×C₄⋊C₈

—

107

C₂×#101

Dic₁₃

yes

—

108

C₄×M₁₆

C₂×D₁₃

yes

—

109

C₂×C₄×C₈

—

110

#343

Dic₁₃

yes

characteristic

111

#343

Dic₁₃

yes

characteristic

112

#336

—

113

#335

—

114

#334

—

115

C₂²×C₁₃

—

116

C₂²×C₁₃

—

117

C₂²×C₁₃

#385

yes

characteristic

118

C₂²×C₁₃

#423

yes

characteristic

119

C₂²×C₁₃

#424

yes

characteristic

120

C₄×C₁₃

—

121

C₄×C₁₃

—

122

C₄×C₁₃

—

123

C₄×C₁₃

—

124

C₄×C₁₃

#435

yes

—

125

C₄×C₁₃

#436

yes

—

126

C₄×C₁₃

#454

yes

characteristic

127

C₄×C₁₃

#456

yes

characteristic

128

C₂²×Q₈

C₂×Dic₁₃

yes

characteristic

129

C₂×M₁₆

C₂×Dic₁₃

yes

—

130

C₂×M₁₆

C₂×Dic₁₃

yes

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

C₄×Q₈

—

134

C₄×Q₈

—

135

C₂×C₄⋊C₄

Dic₂₆

yes

—

136

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

characteristic

137

C₂×C₄²

D₅₂

yes

—

138

C₂×C₄²

C₂²×D₁₃

yes

characteristic

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

#101

—

145

C₂²⋊C₈

—

146

#97

—

147

#97

—

148

C₄×C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₂×C₁₃

—

152

C₂×C₁₃

—

153

C₂×C₁₃

#1851

yes

characteristic

154

C₂×C₁₃

#1857

yes

characteristic

155

C₂×C₁₃

#1858

yes

characteristic

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂²×C₄

C₄⋊Dic₁₃

yes

—

161

C₂²×C₄

C₄⋊Dic₁₃

yes

—

162

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

163

C₂²×C₄

C₂×D₅₂

yes

characteristic

164

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

165

M₁₆

—

166

M₁₆

—

167

M₁₆

—

168

M₁₆

#6099

yes

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄²

—

181

C₄²

—

182

C₄²

D₄×D₁₃

yes

—

183

C₄²

#6129

yes

—

184

C₁₃

#8838

yes

characteristic, complemented, Sylow

185

C₂³

C₂×C₄⋊Dic₁₃

yes

characteristic

186

Q₈

—

187

Q₈

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

#88132

yes

—

197

C₂×C₄

#88136

yes

characteristic

198

C₂×C₄

#88140

yes

characteristic

199

C₂×C₄

C₂×#6099

yes

—

200

C₂×C₄

D₄×Dic₁₃

yes

—

201

C₂×C₄

#88202

yes

characteristic

202

C₂×C₄

#88202

yes

characteristic

203

C₈

—

204

C₈

—

205

C₈

—

206

C₈

—

207

C₈

—

208

C₈

—

104

—

209

C₂²

—

210

C₂²

—

211

C₂²

#11718585

yes

characteristic

212

C₂²

#11718611

yes

characteristic

213

C₂²

#11718660

yes

characteristic

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

#11718413

yes

characteristic

219

C₄

#11718414

yes

characteristic

220

C₄

#11718672

yes

—

221

C₄

#11718673

yes

—

222

C₂

—

223

C₂

—

224

C₂

#13115021

yes

characteristic

225

C₂

#13115053

yes

characteristic

226

C₂

#13115058

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88136
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#101	C₂
socle	C₂²×C₁₃	#385
prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#101	D₁₃
hypocenter	C₁₃	#8838
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

subgroup

G/H

center

C₂×C₄

#88136

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₄×C₁₃×#101

C₂

socle

C₂²×C₁₃

#385

prime radical

C₂³×C₁₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

N	H ≅ G/N	type
184. C₁₃	#8838	semidirect

H ≅ G/N

type

184. C₁₃

#8838

semidirect

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

subgroup	index
#13123921	2
#13115773	2
#13113588	2
#13112674	2
C₄×C₁₃×#101	2
#8838	13

subgroup

index

C₄×C₁₃×#101

Subgroups — #15756798

Conjugacy classes of subgroups

Complements

Maximal subgroups