Group #31011332 — subgroups

Subgroups — #31011332

Order 17152.

482 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 187 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	#22570721	C₂	1	2	yes	maximal
2	#22562494	C₂	1	4	yes	maximal, complemented
3	#22562493	C₂	1	4	yes	maximal, complemented
4	#22561993	C₂	1	1	yes	maximal, characteristic
5	#22561991	C₂	1	2	yes	maximal
6	C₂×#16863171	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic
8	#16863949	—	2	2	no	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	#16863800	C₂²	1	4	yes	—
11	#16863797	C₂²	1	4	yes	—
12	#16863796	—	2	2	no	—
13	#16863793	—	2	2	no	—
14	#16863434	C₂²	1	2	yes	—
15	#16863432	—	2	2	no	—
16	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
17	#16863224	C₂²	1	2	yes	—
18	#16863223	C₂²	1	4	yes	—
19	#16863222	C₂²	1	2	yes	—
20	#16863221	C₂²	1	4	yes	—
21	#16863171	C₂²	1	2	yes	—
22	#16863171	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	#427×C₆₇	C₂²	1	2	yes	—
25	#367×C₆₇	C₂²	1	2	yes	—
26	#366×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	#14524042	—	2	4	no	—
29	#14524034	—	2	4	no	—
30	#14524020	—	4	4	no	—
31	C₄×Dic₁₃₄	—	4	4	no	—
32	#14524002	—	2	2	no	—
33	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
34	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
35	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
36	#14523977	C₂³	1	4	yes	—
37	#14523976	—	2	4	no	—
38	#14523975	—	2	2	no	—
39	#14523974	—	2	4	no	—
40	C₈×Dic₆₇	D₄	1	2	yes	—
41	C₈×Dic₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	#117×C₆₇	—	2	2	no	—
44	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
45	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
47	#107×C₆₇	C₂³	1	4	yes	—
48	#106×C₆₇	—	2	2	no	—
49	C₂×Dic₁₃₄	—	4	4	no	—
50	C₂²×Dic₆₇	D₈	1	2	yes	—
51	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	—
52	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
53	C₄⋊Dic₆₇	—	2	4	no	—
54	C₄⋊Dic₆₇	—	4	4	no	—
55	Dic₆₇⋊C₄	—	4	4	no	—
56	Dic₆₇⋊C₄	—	4	4	no	—
57	C₂²⋊Dic₆₇	—	4	2	no	—
58	C₄×Dic₆₇	—	4	2	no	—
59	C₄×Dic₆₇	D₈	1	2	yes	—
60	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
61	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
65	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
66	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₂²⋊C₄×C₆₇	—	4	2	no	—
69	C₄²×C₆₇	—	4	2	no	—
70	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
71	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
72	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	—	4	4	no	—
74	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₆₇	—	4	2	no	—
79	C₂×Dic₆₇	—	4	4	no	—
80	C₂×Dic₆₇	C₂×D₈	1	4	yes	—
81	Dic₁₃₄	—	8	8	no	—
82	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
83	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
85	C₆₇⋊C₈	C₂×Q₁₆	1	4	yes	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	#500	1	2	yes	—
88	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
89	C₄×C₆₇	—	8	4	no	—
90	C₄×C₆₇	#500	1	2	yes	—
91	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
92	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
93	Dic₆₇	—	2	4	no	—
94	Dic₆₇	—	8	4	no	—
95	#33728	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	—	2	2	no	—
97	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
98	C₂×C₆₇	#3414	1	2	yes	—
99	#3160	—	67	2	no	—
100	#2255	—	67	4	no	—
101	#2255	—	67	4	no	—
102	#2158	—	67	1	no	—
103	#2158	—	67	2	no	—
104	#2158	D₆₇	1	1	yes	characteristic
105	C₂×#346	—	67	1	no	—
106	C₆₇	#33728	1	1	yes	characteristic, complemented, Sylow
107	#533	—	134	2	no	—
108	C₂²×C₁₆	—	67	1	no	—
109	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
110	#500	—	67	4	no	—
111	#499	—	67	4	no	—
112	#499	—	134	2	no	—
113	#498	—	134	2	no	—
114	#427	—	67	2	no	—
115	#427	—	134	2	no	—
116	#427	C₂×D₆₇	1	2	yes	—
117	C₂×C₄×C₈	—	67	1	no	—
118	#367	—	67	2	no	—
119	#367	—	67	4	no	—
120	#367	C₂×D₆₇	1	2	yes	—
121	#366	—	67	2	no	—
122	#366	—	67	4	no	—
123	#366	C₂×D₆₇	1	2	yes	—
124	#346	—	67	2	no	—
125	#346	—	67	2	no	—
126	C₂²×C₈	—	67	1	no	—
127	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
128	C₄⋊Q₈	—	134	4	no	—
129	#125	—	134	4	no	—
130	C₂²⋊Q₈	—	268	4	no	—
131	C₄×Q₈	—	268	4	no	—
132	#117	—	2	2	no	—
133	#117	—	134	2	no	—
134	C₂×C₄²	—	67	1	no	—
135	C₂×C₁₆	—	67	4	no	—
136	C₂×C₁₆	—	67	2	no	—
137	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
138	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
139	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
140	#107	—	67	4	no	—
141	#107	—	134	4	no	—
142	#107	C₂²×D₆₇	1	4	yes	—
143	#106	—	2	2	no	—
144	#106	—	134	4	no	—
145	#106	—	134	2	no	—
146	C₄×C₈	—	67	2	no	—
147	C₄×C₈	—	67	2	no	—
148	C₂×Q₈	—	268	4	no	—
149	C₂²×C₄	—	67	2	no	—
150	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
151	C₂×C₈	—	67	4	no	—
152	C₂×C₈	—	67	2	no	—
153	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
154	C₂×C₈	Q₈×D₆₇	1	2	yes	—
155	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
156	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	268	4	no	—
159	C₄⋊C₄	—	268	4	no	—
160	C₄⋊C₄	—	268	4	no	—
161	C₄⋊C₄	—	134	4	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	268	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	268	2	no	—
166	C₄²	—	67	2	no	—
167	C₄²	—	67	2	no	—
168	C₁₆	—	67	4	no	—
169	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
170	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
171	C₂³	D₈×D₆₇	1	1	yes	characteristic
172	Q₈	—	536	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	268	4	no	—
175	C₂×C₄	—	268	2	no	—
176	C₂×C₄	—	67	4	no	—
177	C₂×C₄	#14524042	1	2	yes	—
178	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
180	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
181	C₈	—	67	4	no	—
182	C₈	#14524042	1	2	yes	—
183	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
184	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
185	C₂²	—	2	2	no	—
186	C₂²	#16863800	1	2	yes	—
187	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	536	4	no	—
190	C₄	—	134	4	no	—
191	C₄	#16863800	1	2	yes	—
192	C₄	C₂×#14524042	1	1	yes	characteristic
193	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₂×#16863800	1	1	yes	characteristic
196	C₂	#22573342	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22570721

C₂

yes

maximal

#22562494

C₂

yes

maximal, complemented

#22562493

C₂

yes

maximal, complemented

#22561993

C₂

yes

maximal, characteristic

#22561991

C₂

yes

maximal

C₂×#16863171

C₂

yes

maximal, characteristic

C₆₇×#2158

C₂

yes

maximal, characteristic

#16863949

—

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

#16863800

C₂²

yes

—

#16863797

C₂²

yes

—

#16863796

—

#16863793

—

#16863434

C₂²

yes

—

#16863432

—

C₂×C₈×Dic₆₇

C₂²

yes

characteristic

#16863224

C₂²

yes

—

#16863223

C₂²

yes

—

#16863222

C₂²

yes

—

#16863221

C₂²

yes

—

#16863171

C₂²

yes

—

#16863171

C₂²

yes

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

—

#367×C₆₇

C₂²

yes

—

#366×C₆₇

C₂²

yes

—

C₂²×C₆₇⋊C₈

D₄

yes

—

—

—

C₄×Dic₁₃₄

—

#14524002

—

C₂×C₄×Dic₆₇

D₄

yes

characteristic

C₂×C₆₇⋊C₁₆

Q₈

yes

complemented

C₂×C₆₇⋊C₁₆

C₂³

yes

—

#14523977

C₂³

yes

—

#14523976

—

#14523975

—

#14523974

—

C₈×Dic₆₇

D₄

yes

—

C₈×Dic₆₇

C₂³

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

—

C₂×C₁₆×C₆₇

Q₈

yes

complemented

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

#107×C₆₇

C₂³

yes

—

#106×C₆₇

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

D₈

yes

—

C₂×C₆₇⋊C₈

Q₁₆

yes

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₈

yes

—

C₄×Dic₆₇

C₂×D₄

yes

—

C₆₇⋊C₁₆

C₂×Q₈

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₁₆×C₆₇

C₂×Q₈

yes

—

C₁₆×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₈

yes

—

Dic₁₃₄

—

C₈×C₆₇

C₄⋊Q₈

yes

—

C₈×C₆₇

C₂²×Q₈

yes

characteristic

C₈×C₆₇

C₂²×Q₈

yes

characteristic

C₆₇⋊C₈

C₂×Q₁₆

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

#500

yes

—

C₂²×C₆₇

C₂²×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#500

yes

—

C₄×C₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₆₇

C₂²×Q₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

—

#33728

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

C₂×#500

yes

characteristic

C₂×C₆₇

#3414

yes

—

#3160

—

100

#2255

—

101

#2255

—

102

#2158

—

103

#2158

—

104

#2158

D₆₇

yes

characteristic

105

C₂×#346

—

106

C₆₇

#33728

yes

characteristic, complemented, Sylow

107

#533

—

134

—

108

C₂²×C₁₆

—

109

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

110

#500

—

111

#499

—

112

#499

—

134

—

113

#498

—

134

—

114

#427

—

115

#427

—

134

—

116

#427

C₂×D₆₇

yes

—

117

C₂×C₄×C₈

—

118

#367

—

119

#367

—

120

#367

C₂×D₆₇

yes

—

121

#366

—

122

#366

—

123

#366

C₂×D₆₇

yes

—

124

#346

—

125

#346

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂²×D₆₇

yes

characteristic

128

C₄⋊Q₈

—

134

—

129

#125

—

134

—

130

C₂²⋊Q₈

—

268

—

131

C₄×Q₈

—

268

—

132

#117

—

133

#117

—

134

—

134

C₂×C₄²

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

Dic₁₃₄

yes

complemented

138

C₂×C₁₆

C₂²×D₆₇

yes

characteristic

139

C₂×C₁₆

C₂²×D₆₇

yes

characteristic

140

#107

—

141

#107

—

134

—

142

#107

C₂²×D₆₇

yes

—

143

#106

—

144

#106

—

134

—

145

#106

—

134

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₂×Q₈

—

268

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₆₇

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

D₄×D₆₇

yes

characteristic

154

C₂×C₈

Q₈×D₆₇

yes

—

155

C₂×C₈

C₂×Dic₁₃₄

yes

—

156

C₂×C₈

C₂³×D₆₇

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

268

—

159

C₄⋊C₄

—

268

—

160

C₄⋊C₄

—

268

—

161

C₄⋊C₄

—

134

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

268

—

164

C₄²

—

165

C₄²

—

268

—

166

C₄²

—

167

C₄²

—

168

C₁₆

—

169

C₁₆

C₂×Dic₁₃₄

yes

—

170

C₁₆

C₂×Dic₁₃₄

yes

—

171

C₂³

D₈×D₆₇

yes

characteristic

172

Q₈

—

536

—

173

C₂×C₄

—

174

C₂×C₄

—

268

—

175

C₂×C₄

—

268

—

176

C₂×C₄

—

177

C₂×C₄

#14524042

yes

—

178

C₂×C₄

D₈×D₆₇

yes

characteristic

179

C₂×C₄

Q₁₆×D₆₇

yes

—

180

C₂×C₄

C₂×D₄×D₆₇

yes

characteristic

181

C₈

—

182

C₈

#14524042

yes

—

183

C₈

C₂×Q₈×D₆₇

yes

characteristic

184

C₈

C₂²×Dic₁₃₄

yes

characteristic

185

C₂²

—

186

C₂²

#16863800

yes

—

187

C₂²

C₂×D₈×D₆₇

yes

characteristic

188

C₄

—

189

C₄

—

536

—

190

C₄

—

134

—

191

C₄

#16863800

yes

—

192

C₄

C₂×#14524042

yes

characteristic

193

C₄

C₂×Q₁₆×D₆₇

yes

characteristic

194

C₂

—

195

C₂

C₂×#16863800

yes

characteristic

196

C₂

#22573342

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2158	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#2158	D₆₇
hypocenter	C₆₇	#33728
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂×C₄

D₈×D₆₇

commutator, cosocle

C₂×C₈×C₆₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₆₇

Fitting

C₆₇×#2158

C₂

socle

C₂²×C₆₇

C₂²×D₈

prime radical

C₂³×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
2. #22562494	C₂	semidirect
3. #22562493	C₂	semidirect
34. C₂×C₆₇⋊C₁₆	Q₈	semidirect
44. C₂×C₁₆×C₆₇	Q₈	semidirect
106. C₆₇	#33728	semidirect
137. C₂×C₁₆	Dic₁₃₄	semidirect

H ≅ G/N

type

2. #22562494

C₂

semidirect

3. #22562493

C₂

semidirect

34. C₂×C₆₇⋊C₁₆

Q₈

semidirect

44. C₂×C₁₆×C₆₇

Q₈

semidirect

106. C₆₇

#33728

semidirect

137. C₂×C₁₆

Dic₁₃₄

semidirect

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

subgroup	index
#22570721	2
#22562494	2
#22562493	2
#22561993	2
#22561991	2
C₂×#16863171	2
C₆₇×#2158	2
#33728	67

subgroup

index

C₆₇×#2158

Subgroups — #31011332

Conjugacy classes of subgroups

Complements

Maximal subgroups