Group #15753932 — subgroups

Subgroups — #15753932

Order 3328.

436 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 143 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	#13113498	C₂	1	1	yes	maximal, characteristic
2	#13113428	C₂	1	1	yes	maximal, characteristic
3	#13113125	C₂	1	2	yes	maximal, complemented
4	#13112725	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₄	1	2	yes	—
7	C₁₃×#532	C₂²	1	1	yes	characteristic
8	C₁₃×#436	—	2	2	no	—
9	C₈×D₄×C₁₃	—	2	2	no	—
10	C₁₃×#423	C₂²	1	1	yes	characteristic
11	C₁₃×#408	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718586	C₂²	1	2	yes	—
14	C₄×#6099	C₂²	1	2	yes	—
15	#11718138	C₄	1	2	yes	—
16	#11718136	C₄	1	2	yes	—
17	#11718132	C₄	1	4	yes	—
18	#11718047	—	2	2	no	—
19	#11718046	—	2	2	no	—
20	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
21	C₁₃×#131	—	4	4	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
28	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
29	C₁₃×#124	C₂×C₄	1	2	yes	—
30	C₁₃×#121	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×#6099	—	2	2	no	—
34	C₁₃×C₄⋊C₈	—	2	2	no	—
35	C₁₃×C₂²⋊C₈	—	2	4	no	—
36	C₄×C₈×C₁₃	D₄	1	2	yes	—
37	C₄×C₈×C₁₃	D₄	1	2	yes	—
38	C₂²⋊#1055	—	2	2	no	—
39	C₄⋊#1055	C₂×C₄	1	4	yes	—
40	#88043	C₂×C₄	1	4	yes	—
41	C₄×#1055	—	2	2	no	—
42	C₄×#1055	—	2	2	no	—
43	#8243	—	13	1	no	maximal, Sylow
44	C₁₃×#41	—	4	2	no	—
45	C₁₃×#41	—	4	2	no	—
46	C₂×Q₈×C₁₃	C₄²	1	2	yes	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	—	2	2	no	—
49	C₂×D₄×C₁₃	C₄²	1	2	yes	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	#41	1	2	yes	—
53	C₁₃×M₁₆	—	4	4	no	—
54	C₁₃×M₁₆	—	4	2	no	—
55	C₂×C₈×C₁₃	—	2	2	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	—	4	1	no	—
58	C₂×C₈×C₁₃	—	4	4	no	—
59	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₂²⋊C₄	—	2	4	no	—
64	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂×D₄	1	2	yes	—
67	#6099	—	4	4	no	—
68	C₂×#1055	—	2	4	no	—
69	C₂×#1055	—	2	4	no	—
70	#2931	D₁₃	1	1	yes	characteristic
71	#1543	—	13	1	no	—
72	#1525	—	13	1	no	—
73	#1451	—	13	2	no	—
74	#1299	—	13	2	no	—
75	C₂³×C₁₃	—	2	2	no	—
76	C₂³×C₁₃	#117	1	1	yes	characteristic
77	Q₈×C₁₃	—	4	2	no	—
78	D₄×C₁₃	—	4	2	no	—
79	D₄×C₁₃	—	4	4	no	—
80	D₄×C₁₃	—	4	4	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	4	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	4	2	no	—
86	C₂×C₄×C₁₃	—	4	2	no	—
87	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
91	C₈×C₁₃	—	2	1	no	—
92	C₈×C₁₃	—	2	2	no	—
93	C₈×C₁₃	—	4	2	no	—
94	C₈×C₁₃	#100	1	2	yes	—
95	#1055	—	4	4	no	—
96	#1055	—	4	4	no	—
97	C₂×#131	Dic₁₃	1	2	yes	—
98	#532	C₂×D₁₃	1	1	yes	characteristic
99	#436	—	2	2	no	—
100	C₈×D₄	—	2	2	no	—
101	#431	—	13	2	no	—
102	#423	C₂×D₁₃	1	1	yes	characteristic
103	#408	Dic₁₃	1	2	yes	—
104	C₄×M₁₆	—	13	2	no	—
105	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
106	#333	—	13	2	no	—
107	#331	—	13	2	no	—
108	#329	—	13	4	no	—
109	#322	—	26	2	no	—
110	#322	—	26	2	no	—
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	—	4	4	no	—
115	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	2	no	—
118	C₄×C₁₃	—	4	2	no	—
119	C₄×C₁₃	C₂×#100	1	1	yes	characteristic
120	C₄×C₁₃	#413	1	1	yes	characteristic
121	C₄×C₁₃	#444	1	4	yes	—
122	C₂×#41	C₂×Dic₁₃	1	2	yes	—
123	#131	—	4	4	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂×M₁₆	—	26	2	no	—
126	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
129	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
130	C₄⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
131	#127	C₄×D₁₃	1	1	yes	characteristic
132	#124	C₄×D₁₃	1	2	yes	—
133	#121	—	2	2	no	—
134	C₄×D₄	—	2	2	no	—
135	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	13	4	no	—
138	C₂²⋊C₈	—	2	4	no	—
139	C₂²⋊C₈	—	26	2	no	—
140	#97	—	13	4	no	—
141	C₄×C₈	—	26	2	no	—
142	C₄×C₈	—	26	2	no	—
143	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
144	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
145	C₂×C₁₃	—	4	1	no	—
146	C₂×C₁₃	—	8	2	no	—
147	C₂×C₁₃	C₄×#100	1	1	yes	characteristic
148	C₂×C₁₃	#1765	1	2	yes	—
149	#41	—	4	2	no	—
150	#41	—	4	2	no	—
151	C₂×Q₈	C₄×Dic₁₃	1	2	yes	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	C₄×Dic₁₃	1	2	yes	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	#6127	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
158	M₁₆	—	4	4	no	—
159	M₁₆	—	4	2	no	—
160	M₁₆	—	52	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	4	no	—
165	C₂×C₈	—	26	4	no	—
166	C₂×C₈	—	26	4	no	—
167	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
168	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
173	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
174	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
175	C₁₃	#8243	1	1	yes	characteristic, complemented, Sylow
176	C₂³	—	2	2	no	—
177	C₂³	#88180	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
189	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
190	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
191	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	52	4	no	—
196	C₈	—	52	4	no	—
197	C₈	#88073	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	4	no	—
202	C₂²	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	4	2	no	—
206	C₄	#11718806	1	4	yes	—
207	C₄	C₂×#88073	1	1	yes	characteristic
208	C₄	#11718898	1	1	yes	characteristic
209	C₂	—	4	1	no	—
210	C₂	—	8	2	no	—
211	C₂	C₄×#88073	1	1	yes	characteristic
212	C₂	#13114678	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13113498

C₂

yes

maximal, characteristic

#13113428

C₂

yes

maximal, characteristic

#13113125

C₂

yes

maximal, complemented

#13112725

C₂

yes

maximal, complemented

C₁₃×#2931

C₂

yes

maximal, characteristic

C₂×C₁₃×#131

C₄

yes

—

C₁₃×#532

C₂²

yes

characteristic

C₁₃×#436

—

C₈×D₄×C₁₃

—

C₁₃×#423

C₂²

yes

characteristic

C₁₃×#408

C₄

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718586

C₂²

yes

—

C₄×#6099

C₂²

yes

—

#11718138

C₄

yes

—

#11718136

C₄

yes

—

#11718132

C₄

yes

—

#11718047

—

#11718046

—

C₂×C₁₃×#41

C₂×C₄

yes

—

C₁₃×#131

—

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₁₃×C₄⋊Q₈

C₂×C₄

yes

characteristic

C₁₃×#127

C₂×C₄

yes

characteristic

C₁₃×#124

C₂×C₄

yes

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₂²⋊#1055

—

C₄⋊#1055

C₂×C₄

yes

—

#88043

C₂×C₄

yes

—

—

—

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

C₄²

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₄²

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

—

—

—

D₁₃

yes

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

#117

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

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×C₄²

yes

characteristic

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#100

yes

—

#1055

—

#1055

—

C₂×#131

Dic₁₃

yes

—

#532

C₂×D₁₃

yes

characteristic

#436

—

100

C₈×D₄

—

101

#431

—

102

#423

C₂×D₁₃

yes

characteristic

103

#408

Dic₁₃

yes

—

104

C₄×M₁₆

—

105

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

106

#333

—

107

#331

—

108

#329

—

109

#322

—

110

#322

—

111

C₂²×C₁₃

—

112

C₂²×C₁₃

—

113

C₂²×C₁₃

—

114

C₂²×C₁₃

—

115

C₂²×C₁₃

C₄×C₂²⋊C₄

yes

characteristic

116

C₄×C₁₃

—

117

C₄×C₁₃

—

118

C₄×C₁₃

—

119

C₄×C₁₃

C₂×#100

yes

characteristic

120

C₄×C₁₃

#413

yes

characteristic

121

C₄×C₁₃

#444

yes

—

122

C₂×#41

C₂×Dic₁₃

yes

—

123

#131

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂×M₁₆

C₂×Dic₁₃

yes

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂×Dic₁₃

yes

characteristic

129

C₂²×C₈

C₂×Dic₁₃

yes

characteristic

130

C₄⋊Q₈

C₄×D₁₃

yes

characteristic

131

#127

C₄×D₁₃

yes

characteristic

132

#124

C₄×D₁₃

yes

—

133

#121

—

134

C₄×D₄

—

135

C₂×C₄²

C₂²×D₁₃

yes

characteristic

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₂²⋊C₈

—

139

C₂²⋊C₈

—

140

#97

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

C₂²⋊D₁₃

yes

—

144

C₄×C₈

C₂²⋊D₁₃

yes

—

145

C₂×C₁₃

—

146

C₂×C₁₃

—

147

C₂×C₁₃

C₄×#100

yes

characteristic

148

C₂×C₁₃

#1765

yes

—

149

#41

—

150

#41

—

151

C₂×Q₈

C₄×Dic₁₃

yes

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

C₄×Dic₁₃

yes

—

155

C₂²×C₄

—

156

C₂²×C₄

#6127

yes

—

157

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

158

M₁₆

—

159

M₁₆

—

160

M₁₆

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

C₂²⋊Dic₁₃

yes

—

168

C₂×C₈

C₂²⋊Dic₁₃

yes

characteristic

169

C₂×C₈

C₂²⋊Dic₁₃

yes

characteristic

170

C₄⋊C₄

—

171

C₂²⋊C₄

—

172

C₄²

C₂×C₄×D₁₃

yes

characteristic

173

C₄²

C₂×C₄×D₁₃

yes

characteristic

174

C₄²

C₂×C₂²⋊D₁₃

yes

—

175

C₁₃

#8243

yes

characteristic, complemented, Sylow

176

C₂³

—

177

C₂³

#88180

yes

characteristic

178

Q₈

—

179

D₄

—

180

D₄

—

181

D₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

C₂×C₄×Dic₁₃

yes

characteristic

189

C₂×C₄

C₄×C₂²⋊D₁₃

yes

—

190

C₂×C₄

C₄×C₂²⋊D₁₃

yes

—

191

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

#88073

yes

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

C₄×C₂²⋊Dic₁₃

yes

characteristic

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

#11718806

yes

—

207

C₄

C₂×#88073

yes

characteristic

208

C₄

#11718898

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

C₄×#88073

yes

characteristic

212

C₂

#13114678

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2931	C₂
socle	C₂²×C₁₃	C₄×C₂²⋊C₄
prime radical	C₁₃×#532	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2931	D₁₃
hypocenter	C₁₃	#8243
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₁₃

commutator

C₂×C₄×C₁₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#2931

C₂

socle

C₂²×C₁₃

C₄×C₂²⋊C₄

prime radical

C₁₃×#532

C₂²

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

N	H ≅ G/N	type
3. #13113125	C₂	semidirect
4. #13112725	C₂	semidirect
175. C₁₃	#8243	semidirect

H ≅ G/N

type

3. #13113125

C₂

semidirect

4. #13112725

C₂

semidirect

175. C₁₃

#8243

semidirect

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

subgroup	index
#13113498	2
#13113428	2
#13113125	2
#13112725	2
C₁₃×#2931	2
#8243	13

subgroup

index

C₁₃×#2931

Subgroups — #15753932

Conjugacy classes of subgroups

Complements

Maximal subgroups