Group #13120046 — subgroups

Subgroups — #13120046

Order 1664.

500 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 185 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×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#88142	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#391	C₂	1	1	yes	maximal, characteristic
4	#11718891	C₂	1	1	yes	maximal, characteristic
5	#11718637	C₂	1	1	yes	maximal, characteristic
6	#11718596	C₂	1	1	yes	maximal, characteristic
7	C₄×#6101	C₂	1	1	yes	maximal, characteristic
8	#11718317	C₂	1	4	yes	maximal
9	#11718308	C₂	1	4	yes	maximal
10	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×Dic₂₆	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	Q₈×Dic₁₃	—	2	4	no	—
15	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
19	C₂×#6101	C₂²	1	2	yes	—
20	C₂×#6101	C₂²	1	4	yes	—
21	C₂×#6101	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
25	#88142	—	2	4	no	—
26	C₁₃×#95	C₂²	1	4	yes	—
27	#88071	C₂²	1	4	yes	—
28	#88071	C₂²	1	2	yes	—
29	C₂×Q₈×C₁₃	—	2	2	no	—
30	C₂×Q₈×C₁₃	—	2	2	no	—
31	C₂×Q₈×C₁₃	Q₈	1	4	yes	complemented
32	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₃	C₂³	1	2	yes	—
37	C₂²×Dic₁₃	C₂³	1	2	yes	—
38	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₃	C₂³	1	2	yes	—
40	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
41	C₂×Dic₂₆	—	2	4	no	—
42	C₂×Dic₂₆	—	4	4	no	—
43	C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₄²×C₁₃	—	2	2	no	—
45	C₄⋊Dic₁₃	—	2	2	no	—
46	C₄⋊Dic₁₃	—	2	2	no	—
47	C₄⋊Dic₁₃	—	2	4	no	—
48	C₄⋊Dic₁₃	—	4	4	no	—
49	#6101	—	2	4	no	—
50	#6101	—	4	4	no	—
51	#6101	Q₈	1	8	yes	complemented
52	C₄×Dic₁₃	—	2	2	no	—
53	C₄×Dic₁₃	—	2	4	no	—
54	C₄×Dic₁₃	D₄	1	4	yes	—
55	#2567	—	13	1	no	maximal, Sylow
56	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
57	Q₈×C₁₃	—	2	8	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	4	no	—
64	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
67	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₁₃	#41	1	2	yes	—
69	C₂×Dic₁₃	—	2	2	no	—
70	C₂×Dic₁₃	—	2	2	no	—
71	C₂×Dic₁₃	—	2	2	no	—
72	C₂×Dic₁₃	—	2	2	no	—
73	C₂×Dic₁₃	—	2	2	no	—
74	C₂×Dic₁₃	—	2	4	no	—
75	C₂×Dic₁₃	—	2	4	no	—
76	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
77	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
78	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
79	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
80	Dic₂₆	—	4	8	no	—
81	C₂×C₄⋊Q₈	—	13	1	no	—
82	C₂×C₄×Q₈	—	13	1	no	—
83	#395	—	13	4	no	—
84	#391	—	13	1	no	—
85	#391	D₁₃	1	1	yes	characteristic
86	#389	—	13	4	no	—
87	#384	—	13	1	no	—
88	#384	—	13	1	no	—
89	C₄×C₄⋊C₄	—	13	1	no	—
90	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
92	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
93	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
94	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
95	C₄×C₁₃	—	2	4	no	—
96	C₄×C₁₃	—	2	4	no	—
97	C₄×C₁₃	—	4	2	no	—
98	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
99	Dic₁₃	—	2	2	no	—
100	Dic₁₃	—	2	4	no	—
101	Dic₁₃	—	4	2	no	—
102	Dic₁₃	—	4	2	no	—
103	Dic₁₃	C₄⋊Q₈	1	4	yes	—
104	C₂²×Q₈	—	13	2	no	—
105	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
106	C₄⋊Q₈	—	26	4	no	—
107	C₄×Q₈	—	26	4	no	—
108	C₂×C₄⋊C₄	—	13	2	no	—
109	C₂×C₄⋊C₄	—	13	1	no	—
110	C₂×C₄⋊C₄	—	13	2	no	—
111	C₂×C₄⋊C₄	—	13	2	no	—
112	C₂×C₄⋊C₄	—	13	1	no	—
113	C₂×C₄⋊C₄	—	13	4	no	—
114	C₂×C₄⋊C₄	—	13	4	no	—
115	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
116	C₂×C₄²	—	13	1	no	—
117	C₂×C₄²	—	13	2	no	—
118	C₂×C₄²	—	13	1	no	—
119	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
120	#95	—	13	2	no	—
121	#95	—	13	4	no	—
122	#95	C₂×D₁₃	1	4	yes	—
123	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₁₃	#535	1	1	yes	characteristic
126	C₂×C₁₃	#554	1	2	yes	—
127	C₂×C₁₃	Q₈²	1	2	yes	—
128	C₂×Q₈	—	2	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	26	4	no	—
131	C₂×Q₈	—	52	4	no	—
132	C₂×Q₈	Dic₂₆	1	4	yes	complemented
133	C₂²×C₄	—	13	1	no	—
134	C₂²×C₄	—	13	2	no	—
135	C₂²×C₄	—	13	1	no	—
136	C₂²×C₄	—	13	2	no	—
137	C₂²×C₄	—	13	2	no	—
138	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
139	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
140	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
141	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	13	8	no	—
144	C₄⋊C₄	—	26	4	no	—
145	C₄⋊C₄	—	26	4	no	—
146	C₄⋊C₄	—	26	2	no	—
147	C₄⋊C₄	—	26	2	no	—
148	C₄⋊C₄	—	52	4	no	—
149	C₄⋊C₄	—	52	4	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	13	4	no	—
152	C₄²	—	26	4	no	—
153	C₄²	—	26	2	no	—
154	C₁₃	#2567	1	1	yes	characteristic, complemented, Sylow
155	C₂³	C₂³×D₁₃	1	1	yes	characteristic
156	Q₈	—	2	8	no	—
157	Q₈	—	52	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	13	2	no	—
165	C₂×C₄	—	13	4	no	—
166	C₂×C₄	—	13	2	no	—
167	C₂×C₄	—	13	4	no	—
168	C₂×C₄	—	26	2	no	—
169	C₂×C₄	—	26	2	no	—
170	C₂×C₄	—	26	2	no	—
171	C₂×C₄	—	26	2	no	—
172	C₂×C₄	—	26	2	no	—
173	C₂×C₄	—	26	4	no	—
174	C₂×C₄	—	26	4	no	—
175	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
176	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
177	C₂×C₄	D₄×D₁₃	1	2	yes	—
178	C₂×C₄	#6129	1	2	yes	—
179	C₂×C₄	Q₈×D₁₃	1	4	yes	—
180	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
181	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
182	C₂²	C₂×#6129	1	1	yes	characteristic
183	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
184	C₂²	#88256	1	2	yes	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	2	no	—
188	C₄	—	13	4	no	—
189	C₄	—	26	4	no	—
190	C₄	—	26	2	no	—
191	C₄	—	52	2	no	—
192	C₄	—	52	2	no	—
193	C₄	#88132	1	4	yes	—
194	C₂	C₂×#88132	1	1	yes	characteristic
195	C₂	Q₈×Dic₂₆	1	2	yes	—
196	C₂	#11719340	1	2	yes	—
197	C₂	#11719379	1	1	yes	characteristic
198	C₂	D₁₃×C₄⋊Q₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₁₃

C₂

yes

maximal, characteristic

C₂×#88142

C₂

yes

maximal, characteristic

C₁₃×#391

C₂

yes

maximal, characteristic

#11718891

C₂

yes

maximal, characteristic

#11718637

C₂

yes

maximal, characteristic

#11718596

C₂

yes

maximal, characteristic

C₄×#6101

C₂

yes

maximal, characteristic

#11718317

C₂

yes

maximal

#11718308

C₂

yes

maximal

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

Q₈×Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

#88142

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

Q₈

yes

complemented

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

#6101

Q₈

yes

complemented

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

—

#2567

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

C₂×Q₈

yes

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

#395

—

#391

—

#391

D₁₃

yes

—

—

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

Dic₁₃

—

100

Dic₁₃

—

101

Dic₁₃

—

102

Dic₁₃

—

103

Dic₁₃

C₄⋊Q₈

yes

—

104

C₂²×Q₈

—

105

C₂²×Q₈

C₂×D₁₃

yes

characteristic

106

C₄⋊Q₈

—

107

C₄×Q₈

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

—

119

C₂×C₄²

C₂×D₁₃

yes

characteristic

120

#95

—

121

#95

—

122

#95

C₂×D₁₃

yes

—

123

C₂×C₁₃

C₂×C₂²⋊Q₈

yes

characteristic

124

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

125

C₂×C₁₃

#535

yes

characteristic

126

C₂×C₁₃

#554

yes

—

127

C₂×C₁₃

Q₈²

yes

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

Dic₂₆

yes

complemented

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂²×D₁₃

yes

—

139

C₂²×C₄

C₂²×D₁₃

yes

—

140

C₂²×C₄

C₂²×D₁₃

yes

—

141

C₂²×C₄

C₂²×D₁₃

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₁₃

#2567

yes

characteristic, complemented, Sylow

155

C₂³

C₂³×D₁₃

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

C₂×Dic₂₆

yes

—

176

C₂×C₄

C₂×Dic₂₆

yes

—

177

C₂×C₄

D₄×D₁₃

yes

—

178

C₂×C₄

#6129

yes

—

179

C₂×C₄

Q₈×D₁₃

yes

—

180

C₂²

C₂²×Dic₂₆

yes

characteristic

181

C₂²

C₂×D₄×D₁₃

yes

characteristic

182

C₂²

C₂×#6129

yes

characteristic

183

C₂²

C₂×Q₈×D₁₃

yes

—

184

C₂²

#88256

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#88132

yes

—

194

C₂

C₂×#88132

yes

characteristic

195

C₂

Q₈×Dic₂₆

yes

—

196

C₂

#11719340

yes

—

197

C₂

#11719379

yes

characteristic

198

C₂

D₁₃×C₄⋊Q₈

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#391	C₂
hypercenter	#391	D₁₃
hypocenter	C₁₃	#2567
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#391

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
31. C₂×Q₈×C₁₃	Q₈	semidirect
51. #6101	Q₈	semidirect
132. C₂×Q₈	Dic₂₆	semidirect
154. C₁₃	#2567	semidirect

H ≅ G/N

type

31. C₂×Q₈×C₁₃

Q₈

semidirect

51. #6101

Q₈

semidirect

132. C₂×Q₈

Dic₂₆

semidirect

154. C₁₃

#2567

semidirect

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

subgroup	index
C₂×Q₈×Dic₁₃	2
C₂×#88142	2
C₁₃×#391	2
#11718891	2
#11718637	2
#11718596	2
C₄×#6101	2
#11718317	2
#11718308	2
#2567	13

subgroup

index

C₂×Q₈×Dic₁₃

C₁₃×#391

Subgroups — #13120046

Conjugacy classes of subgroups

Complements

Maximal subgroups