Group #30772313 — subgroups

Subgroups — #30772313

Order 17152.

370 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22573312	C₂	1	1	yes	maximal, characteristic
2	#22561975	C₂	1	2	yes	maximal, complemented
3	#22561823	C₂	1	1	yes	maximal, characteristic
4	#22558298	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1757	C₂	1	1	yes	maximal, characteristic
6	#16863813	—	2	4	no	—
7	C₂×#14523983	—	2	2	no	—
8	C₂×#14523983	C₂²	1	1	yes	characteristic
9	C₂×#14523983	C₂²	1	2	yes	—
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	#16863217	—	2	4	no	—
12	C₂²⋊(C₆₇⋊C₁₆)	—	2	2	no	—
13	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
14	#413×C₆₇	C₄	1	1	yes	characteristic
15	#410×C₆₇	C₄	1	1	yes	characteristic
16	#405×C₆₇	C₂²	1	2	yes	—
17	#344×C₆₇	C₄	1	2	yes	—
18	#14523983	—	2	4	no	—
19	#14523983	—	2	2	no	—
20	#14523983	—	2	2	no	—
21	#14523983	D₄	1	4	yes	—
22	C₂×C₆₇⋊C₁₆	—	2	4	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	1	no	—
25	C₂×C₆₇⋊C₁₆	—	2	1	no	—
26	C₂×C₆₇⋊C₁₆	—	2	2	no	—
27	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
28	#131×C₆₇	—	2	2	no	—
29	#131×C₆₇	—	2	2	no	—
30	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
33	C₂²×C₈×C₆₇	D₄	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	#117×C₆₇	C₂×C₄	1	2	yes	—
36	#101×C₆₇	C₈	1	2	yes	—
37	#100×C₆₇	C₈	1	2	yes	—
38	#99×C₆₇	—	2	2	no	—
39	#97×C₆₇	—	2	2	no	—
40	C₄×C₈×C₆₇	—	2	2	no	—
41	C₆₇⋊C₁₆	—	2	4	no	—
42	C₆₇⋊C₁₆	—	4	1	no	—
43	C₆₇⋊C₁₆	—	4	1	no	—
44	#41×C₆₇	—	2	2	no	—
45	#41×C₆₇	—	2	2	no	—
46	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
47	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
48	C₂×D₄×C₆₇	M₁₆	1	2	yes	—
49	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	M₁₆×C₆₇	—	2	2	no	—
52	M₁₆×C₆₇	—	2	2	no	—
53	M₁₆×C₆₇	—	2	4	no	—
54	M₁₆×C₆₇	C₂×C₈	1	4	yes	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	2	yes	—
63	C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂²⋊C₄×C₆₇	—	2	2	no	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₂³×C₆₇	C₄⋊C₈	1	2	yes	—
67	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
68	Q₈×C₆₇	—	2	2	no	—
69	D₄×C₆₇	—	2	4	no	—
70	D₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	4	2	no	—
74	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#131	1	2	yes	—
79	C₈×C₆₇	—	2	2	no	—
80	C₈×C₆₇	—	2	4	no	—
81	C₈×C₆₇	#120	1	1	yes	characteristic
82	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
83	C₈×C₆₇	#123	1	1	yes	characteristic
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	—	4	1	no	—
86	C₂²×C₆₇	—	4	2	no	—
87	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₆₇	#435	1	2	yes	—
89	C₄×C₆₇	—	2	2	no	—
90	C₄×C₆₇	—	8	2	no	—
91	C₄×C₆₇	#385	1	1	yes	characteristic
92	C₄×C₆₇	#424	1	1	yes	characteristic
93	C₄×C₆₇	C₈×D₄	1	2	yes	—
94	#12916	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	—	2	2	no	—
96	C₂×C₆₇	—	2	1	no	—
97	C₂×C₆₇	—	4	2	no	—
98	C₂×C₆₇	#1850	1	1	yes	characteristic
99	#3407	—	67	1	no	—
100	#2153	—	67	2	no	—
101	#2113	—	67	1	no	—
102	#1757	D₆₇	1	1	yes	characteristic
103	#1379	—	67	2	no	—
104	C₆₇	#12916	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
106	#504	—	134	4	no	—
107	C₂×M₃₂	—	67	2	no	—
108	C₂×M₃₂	—	67	1	no	—
109	C₂×M₃₂	—	134	2	no	—
110	C₂²×C₁₆	—	67	1	no	—
111	#413	Dic₆₇	1	1	yes	characteristic
112	#410	Dic₆₇	1	1	yes	characteristic
113	#405	C₂×D₆₇	1	2	yes	—
114	#364	—	134	4	no	—
115	C₂²⋊C₁₆	—	134	2	no	—
116	#344	Dic₆₇	1	2	yes	—
117	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
121	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
122	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
123	C₂²×C₈	D₂₆₈	1	2	yes	—
124	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
125	#117	C₂×Dic₆₇	1	2	yes	—
126	M₃₂	—	67	4	no	—
127	M₃₂	—	134	2	no	—
128	M₃₂	—	134	4	no	—
129	M₃₂	—	134	2	no	—
130	C₂×C₁₆	—	134	1	no	—
131	C₂×C₁₆	—	134	1	no	—
132	C₂×C₁₆	—	134	2	no	—
133	C₂×C₁₆	—	134	4	no	—
134	C₂×C₁₆	—	134	2	no	—
135	#101	C₆₇⋊C₈	1	2	yes	—
136	#100	C₆₇⋊C₈	1	2	yes	—
137	#99	—	2	2	no	—
138	#97	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
143	C₂×D₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
144	C₂×D₄	#11761612	1	2	yes	—
145	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	2	no	—
149	M₁₆	—	2	4	no	—
150	M₁₆	C₂×C₆₇⋊C₈	1	4	yes	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
155	C₂×C₈	D₄×D₆₇	1	2	yes	—
156	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
157	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
158	C₂×C₈	#11761627	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₁₆	—	268	1	no	—
163	C₁₆	—	268	1	no	—
164	C₁₆	—	134	4	no	—
165	C₂³	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
166	C₂³	C₂×#11761612	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
176	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
177	C₂×C₄	#14524050	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	2	4	no	—
180	C₈	#14524013	1	1	yes	characteristic
181	C₈	#14524020	1	2	yes	—
182	C₈	#14524027	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
187	C₂²	#16863468	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	8	2	no	—
190	C₄	#16863271	1	1	yes	characteristic
191	C₄	#16863423	1	1	yes	characteristic
192	C₄	D₄×C₆₇⋊C₈	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#22560389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22573312

C₂

yes

maximal, characteristic

#22561975

C₂

yes

maximal, complemented

#22561823

C₂

yes

maximal, characteristic

#22558298

C₂

yes

maximal, complemented

C₆₇×#1757

C₂

yes

maximal, characteristic

#16863813

—

C₂×#14523983

—

C₂×#14523983

C₂²

yes

characteristic

C₂×#14523983

C₂²

yes

—

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

#16863217

—

C₂²⋊(C₆₇⋊C₁₆)

—

C₂×#131×C₆₇

C₂²

yes

characteristic

#413×C₆₇

C₄

yes

characteristic

#410×C₆₇

C₄

yes

characteristic

#405×C₆₇

C₂²

yes

—

#344×C₆₇

C₄

yes

—

—

—

—

D₄

yes

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×#41×C₆₇

C₂×C₄

yes

characteristic

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₇

Q₈

yes

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

C₂×C₄

yes

—

#101×C₆₇

C₈

yes

—

#100×C₆₇

C₈

yes

—

#99×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₈

yes

characteristic

C₂×D₄×C₆₇

C₂×C₈

yes

characteristic

C₂×D₄×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₄⋊C₈

yes

—

C₂³×C₆₇

C₂×M₁₆

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#120

yes

characteristic

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

#123

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₇

#435

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₈×D₄

yes

—

#12916

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1850

yes

characteristic

#3407

—

100

#2153

—

101

#2113

—

102

#1757

D₆₇

yes

characteristic

103

#1379

—

104

C₆₇

#12916

yes

characteristic, complemented, Sylow

105

C₂×#131

C₂×D₆₇

yes

characteristic

106

#504

—

134

—

107

C₂×M₃₂

—

108

C₂×M₃₂

—

109

C₂×M₃₂

—

134

—

110

C₂²×C₁₆

—

111

#413

Dic₆₇

yes

characteristic

112

#410

Dic₆₇

yes

characteristic

113

#405

C₂×D₆₇

yes

—

114

#364

—

134

—

115

C₂²⋊C₁₆

—

134

—

116

#344

Dic₆₇

yes

—

117

C₂×#41

C₂×Dic₆₇

yes

characteristic

118

#131

—

119

#131

—

120

C₂×M₁₆

Dic₁₃₄

yes

—

121

C₂×M₁₆

C₂×Dic₆₇

yes

—

122

C₂×M₁₆

C₂×Dic₆₇

yes

characteristic

123

C₂²×C₈

D₂₆₈

yes

—

124

C₂²×C₈

C₂²×D₆₇

yes

characteristic

125

#117

C₂×Dic₆₇

yes

—

126

M₃₂

—

127

M₃₂

—

134

—

128

M₃₂

—

134

—

129

M₃₂

—

134

—

130

C₂×C₁₆

—

134

—

131

C₂×C₁₆

—

134

—

132

C₂×C₁₆

—

134

—

133

C₂×C₁₆

—

134

—

134

C₂×C₁₆

—

134

—

135

#101

C₆₇⋊C₈

yes

—

136

#100

C₆₇⋊C₈

yes

—

137

#99

—

138

#97

—

139

C₄×C₈

—

140

#41

—

141

#41

—

142

C₂×Q₈

C₂×C₆₇⋊C₈

yes

characteristic

143

C₂×D₄

C₂×C₆₇⋊C₈

yes

characteristic

144

C₂×D₄

#11761612

yes

—

145

C₂²×C₄

C₄⋊Dic₆₇

yes

—

146

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

C₂×C₆₇⋊C₈

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

C₄⋊Dic₆₇

yes

—

155

C₂×C₈

D₄×D₆₇

yes

—

156

C₂×C₈

C₂×D₂₆₈

yes

characteristic

157

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

158

C₂×C₈

#11761627

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₁₆

—

268

—

163

C₁₆

—

268

—

164

C₁₆

—

134

—

165

C₂³

C₄⋊(C₆₇⋊C₈)

yes

—

166

C₂³

C₂×#11761612

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

C₄⋊(C₆₇⋊C₈)

yes

—

174

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

175

C₂×C₄

D₄×Dic₆₇

yes

—

176

C₂×C₄

C₂²×C₆₇⋊C₈

yes

characteristic

177

C₂×C₄

#14524050

yes

—

178

C₈

—

179

C₈

—

180

C₈

#14524013

yes

characteristic

181

C₈

#14524020

yes

—

182

C₈

#14524027

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×C₄⋊(C₆₇⋊C₈)

yes

characteristic

187

C₂²

#16863468

yes

—

188

C₄

—

189

C₄

—

190

C₄

#16863271

yes

characteristic

191

C₄

#16863423

yes

characteristic

192

C₄

D₄×C₆₇⋊C₈

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#22560389

yes

characteristic

	subgroup	G/H
center	C₈	#14524013
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1757	C₂
socle	C₂×C₆₇	#1850
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1757	D₆₇
hypocenter	C₆₇	#12916
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1757

subgroup

G/H

center

C₈

#14524013

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1757

C₂

socle

C₂×C₆₇

#1850

prime radical

C₂×#41×C₆₇

C₂×C₄

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1757

N	H ≅ G/N	type
2. #22561975	C₂	semidirect
4. #22558298	C₂	semidirect
104. C₆₇	#12916	semidirect

H ≅ G/N

type

2. #22561975

C₂

semidirect

4. #22558298

C₂

semidirect

104. C₆₇

#12916

semidirect

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

subgroup	index
#22573312	2
#22561975	2
#22561823	2
#22558298	2
C₆₇×#1757	2
#12916	67

subgroup

index

C₆₇×#1757

Subgroups — #30772313

Conjugacy classes of subgroups

Complements

Maximal subgroups