Group #13115270 — subgroups

Subgroups — #13115270

Order 1664.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₁₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6099	C₂	1	1	yes	maximal, characteristic
3	C₂×#88075	C₂	1	2	yes	maximal, complemented
4	#11718873	C₂	1	1	yes	maximal, characteristic
5	#11718121	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×C₂²⋊Q₈	—	2	2	no	—
9	C₁₃×C₂²⋊Q₈	C₄	1	4	yes	complemented
10	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
13	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
14	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂×#6099	—	2	2	no	—
16	C₂×#6099	—	2	4	no	—
17	C₂²×#1055	C₂²	1	2	yes	—
18	#88075	—	2	4	no	—
19	#88071	—	2	1	no	—
20	C₂⁴×C₁₃	D₄	1	1	yes	characteristic
21	C₂×Q₈×C₁₃	—	2	1	no	—
22	C₂×Q₈×C₁₃	—	2	2	no	—
23	C₂×Q₈×C₁₃	—	2	2	no	—
24	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₃	—	2	2	no	—
26	C₂²×C₄×C₁₃	—	2	2	no	—
27	C₂²×C₄×C₁₃	—	2	1	no	—
28	C₂²×C₄×C₁₃	—	2	1	no	—
29	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	D₄	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₃	—	2	2	no	—
34	C₁₃×C₄⋊C₄	—	2	4	no	—
35	C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₁₃×C₂²⋊C₄	—	2	4	no	—
38	C₂²⋊Dic₁₃	—	4	2	no	—
39	C₄⋊Dic₁₃	—	2	2	no	—
40	C₄⋊Dic₁₃	—	2	2	no	—
41	#6099	—	2	8	no	—
42	C₂×#1055	—	2	2	no	—
43	C₂×#1055	—	2	2	no	—
44	C₂×#1055	D₄	1	4	yes	—
45	#1894	—	13	1	no	maximal, Sylow
46	C₂³×C₁₃	—	2	2	no	—
47	C₂³×C₁₃	—	2	1	no	—
48	C₂³×C₁₃	—	2	1	no	—
49	C₂³×C₁₃	—	2	2	no	—
50	C₂³×C₁₃	C₂²⋊C₄	1	2	yes	—
51	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
52	Q₈×C₁₃	—	2	4	no	—
53	Q₈×C₁₃	—	4	1	no	—
54	Q₈×C₁₃	—	4	1	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	4	no	—
57	C₂×C₄×C₁₃	—	2	2	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	—	4	2	no	—
62	C₂×C₄×C₁₃	—	4	1	no	—
63	C₂×C₄×C₁₃	—	4	1	no	—
64	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#41	1	2	yes	—
69	C₂×Dic₁₃	—	4	2	no	—
70	C₂×Dic₁₃	—	4	2	no	—
71	C₂×Dic₁₃	—	4	2	no	—
72	#1055	—	2	4	no	—
73	C₂²×M₁₆	—	13	1	no	—
74	C₂×C₂²⋊Q₈	D₁₃	1	1	yes	characteristic
75	C₂×#103	—	13	2	no	—
76	#380	—	13	1	no	—
77	#340	—	13	2	no	—
78	C₂²×C₁₃	—	2	2	no	—
79	C₂²×C₁₃	—	2	2	no	—
80	C₂²×C₁₃	—	2	2	no	—
81	C₂²×C₁₃	—	4	2	no	—
82	C₂²×C₁₃	—	4	2	no	—
83	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₁₃	#121	1	2	yes	—
85	C₂²×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂²×C₁₃	#137	1	2	yes	—
87	C₂²×C₁₃	#137	1	1	yes	characteristic
88	C₄×C₁₃	—	2	2	no	—
89	C₄×C₁₃	—	4	2	no	—
90	C₄×C₁₃	—	4	2	no	—
91	C₄×C₁₃	C₄×D₄	1	2	yes	—
92	C₄×C₁₃	#120	1	1	yes	characteristic
93	C₄×C₁₃	#123	1	1	yes	characteristic
94	Dic₁₃	—	8	2	no	—
95	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
96	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
97	C₂×M₁₆	—	26	4	no	—
98	C₂×M₁₆	—	26	2	no	—
99	C₂²×C₈	—	13	2	no	—
100	C₂²⋊Q₈	—	2	2	no	—
101	C₂²⋊Q₈	Dic₁₃	1	4	yes	complemented
102	C₂×C₄⋊C₄	—	2	1	no	—
103	C₂×C₄⋊C₄	—	13	2	no	—
104	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
105	C₂×C₂²⋊C₄	—	2	1	no	—
106	C₂×C₂²⋊C₄	—	26	1	no	—
107	#103	—	26	4	no	—
108	#95	—	26	1	no	—
109	C₂×C₁₃	—	4	2	no	—
110	C₂×C₁₃	#386	1	1	yes	characteristic
111	C₂×C₁₃	#417	1	1	yes	characteristic
112	C₂×C₁₃	#419	1	1	yes	characteristic
113	C₂×C₁₃	#468	1	2	yes	—
114	C₂×C₁₃	#470	1	2	yes	—
115	C₂⁴	C₂²⋊D₁₃	1	1	yes	characteristic
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	26	2	no	—
125	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
126	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
128	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
129	M₁₆	—	26	8	no	—
130	C₂×C₈	—	13	4	no	—
131	C₂×C₈	—	26	2	no	—
132	C₂×C₈	—	26	2	no	—
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	26	2	no	—
136	C₄⋊C₄	—	26	2	no	—
137	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
138	C₂²⋊C₄	—	2	4	no	—
139	C₂²⋊C₄	—	52	2	no	—
140	C₁₃	#1894	1	1	yes	characteristic, complemented, Sylow
141	C₂³	—	2	2	no	—
142	C₂³	—	2	2	no	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	C₂²⋊Dic₁₃	1	2	yes	—
146	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
147	Q₈	—	2	4	no	—
148	Q₈	—	4	1	no	—
149	Q₈	—	4	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	52	2	no	—
160	C₂×C₄	—	52	2	no	—
161	C₂×C₄	—	52	2	no	—
162	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
163	C₂×C₄	D₄×D₁₃	1	2	yes	—
164	C₂×C₄	#6129	1	2	yes	—
165	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
166	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
167	C₈	—	26	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	#88193	1	2	yes	—
174	C₂²	#88196	1	2	yes	—
175	C₂²	#88203	1	1	yes	characteristic
176	C₂²	#88205	1	1	yes	characteristic
177	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	104	2	no	—
182	C₄	D₄×Dic₁₃	1	2	yes	—
183	C₄	#88189	1	1	yes	characteristic
184	C₄	#88191	1	1	yes	characteristic
185	C₂	—	4	2	no	—
186	C₂	#11718710	1	2	yes	—
187	C₂	#11718713	1	2	yes	—
188	C₂	#11718881	1	1	yes	characteristic
189	C₂	#11718886	1	1	yes	characteristic
190	C₂	#11718893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₂²⋊Q₈

C₂

yes

maximal, characteristic

C₂²×#6099

C₂

yes

maximal, characteristic

C₂×#88075

C₂

yes

maximal, complemented

#11718873

C₂

yes

maximal, characteristic

#11718121

C₂

yes

maximal, complemented

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂³×C₄×C₁₃

C₂²

yes

characteristic

C₁₃×C₂²⋊Q₈

—

C₁₃×C₂²⋊Q₈

C₄

yes

complemented

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₃×C₂²⋊C₄

—

C₂×C₂²⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6099

—

C₂×#6099

—

C₂²×#1055

C₂²

yes

—

#88075

—

#88071

—

C₂⁴×C₁₃

D₄

yes

characteristic

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₂²⋊C₄

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

D₄

yes

—

#1894

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₄

yes

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

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

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

—

C₂²×M₁₆

—

C₂×C₂²⋊Q₈

D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

#121

yes

—

C₂²×C₁₃

AGL(1,ℤ/8)

yes

characteristic

C₂²×C₁₃

#137

yes

—

C₂²×C₁₃

#137

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂³×C₄

C₂×D₁₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

100

C₂²⋊Q₈

—

101

C₂²⋊Q₈

Dic₁₃

yes

complemented

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

105

C₂×C₂²⋊C₄

—

106

C₂×C₂²⋊C₄

—

107

#103

—

108

#95

—

109

C₂×C₁₃

—

110

C₂×C₁₃

#386

yes

characteristic

111

C₂×C₁₃

#417

yes

characteristic

112

C₂×C₁₃

#419

yes

characteristic

113

C₂×C₁₃

#468

yes

—

114

C₂×C₁₃

#470

yes

—

115

C₂⁴

C₂²⋊D₁₃

yes

characteristic

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

C₂×Dic₁₃

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×Dic₁₃

yes

—

126

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

127

C₂²×C₄

C₂²⋊D₁₃

yes

—

128

C₂²×C₄

C₂²×D₁₃

yes

characteristic

129

M₁₆

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

C₂×Dic₁₃

yes

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₁₃

#1894

yes

characteristic, complemented, Sylow

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂²⋊Dic₁₃

yes

—

146

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂²⋊Dic₁₃

yes

—

163

C₂×C₄

D₄×D₁₃

yes

—

164

C₂×C₄

#6129

yes

—

165

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

166

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

167

C₈

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

#88193

yes

—

174

C₂²

#88196

yes

—

175

C₂²

#88203

yes

characteristic

176

C₂²

#88205

yes

characteristic

177

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

104

—

182

C₄

D₄×Dic₁₃

yes

—

183

C₄

#88189

yes

characteristic

184

C₄

#88191

yes

characteristic

185

C₂

—

186

C₂

#11718710

yes

—

187

C₂

#11718713

yes

—

188

C₂

#11718881

yes

characteristic

189

C₂

#11718886

yes

characteristic

190

C₂

#11718893

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×C₂²⋊Q₈	C₂
socle	C₂³×C₁₃	C₂×D₄
prime radical	C₂⁴×C₁₃	D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₃
hypocenter	C₁₃	#1894
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₁₃×C₂²⋊Q₈

C₂

socle

C₂³×C₁₃

C₂×D₄

prime radical

C₂⁴×C₁₃

D₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₂×C₂²⋊Q₈

D₁₃

hypocenter

C₁₃

#1894

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

N	H ≅ G/N	type
3. C₂×#88075	C₂	semidirect
5. #11718121	C₂	semidirect
9. C₁₃×C₂²⋊Q₈	C₄	semidirect
101. C₂²⋊Q₈	Dic₁₃	semidirect
140. C₁₃	#1894	semidirect

H ≅ G/N

type

3. C₂×#88075

C₂

semidirect

5. #11718121

C₂

semidirect

9. C₁₃×C₂²⋊Q₈

C₄

semidirect

101. C₂²⋊Q₈

Dic₁₃

semidirect

140. C₁₃

#1894

semidirect

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

subgroup	index
C₂×C₁₃×C₂²⋊Q₈	2
C₂²×#6099	2
C₂×#88075	2
#11718873	2
#11718121	2
#1894	13

subgroup

index

C₂×C₁₃×C₂²⋊Q₈

C₂²×#6099

Subgroups — #13115270

Conjugacy classes of subgroups

Complements

Maximal subgroups