Group #30754911 — subgroups

Subgroups — #30754911

Order 17152.

538 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 167 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	#22568607	C₂	1	1	yes	maximal, characteristic
2	#22560062	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863144	C₂	1	4	yes	maximal
4	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524043	C₄	1	4	yes	—
6	C₂×#14524043	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14523954	—	2	1	no	—
9	#16863266	—	2	1	no	—
10	#16863266	—	2	1	no	—
11	#16863266	—	2	2	no	—
12	#16863151	—	2	1	no	—
13	#16863144	—	2	8	no	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	#336×C₆₇	—	2	2	no	—
17	C₂²×Dic₁₃₄	C₂×C₄	1	4	yes	—
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	#14524043	—	2	4	no	—
20	#14524043	—	2	4	no	—
21	#14524043	—	4	8	no	—
22	C₂×C₄⋊Dic₆₇	—	2	4	no	—
23	C₂×C₄⋊Dic₆₇	—	2	4	no	—
24	C₂×C₄⋊Dic₆₇	—	2	2	no	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
28	#14523954	—	4	2	no	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₇	—	2	4	no	—
36	C₂×Dic₁₃₄	—	2	8	no	—
37	C₂×Dic₁₃₄	—	4	8	no	—
38	C₂²×Dic₆₇	—	2	4	no	—
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂×C₆₇⋊C₈	—	4	2	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₄⋊Dic₆₇	—	4	8	no	—
43	C₄⋊Dic₆₇	—	4	8	no	—
44	C₄⋊Dic₆₇	—	4	4	no	—
45	C₄⋊Dic₆₇	—	4	2	no	—
46	C₄⋊Dic₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	—	4	2	no	—
53	C₂×C₈×C₆₇	—	4	4	no	—
54	C₄²×C₆₇	—	2	4	no	—
55	C₄²×C₆₇	—	2	4	no	—
56	C₄²×C₆₇	—	2	2	no	—
57	C₄²×C₆₇	SD₁₆	1	2	yes	—
58	C₄²×C₆₇	Q₁₆	1	2	yes	—
59	C₄²×C₆₇	C₂×D₄	1	2	yes	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	#100	1	4	yes	—
68	C₂×C₄×C₆₇	#103	1	4	yes	—
69	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₆₇	#121	1	2	yes	—
71	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#137	1	4	yes	—
76	C₂×C₄×C₆₇	#137	1	2	yes	—
77	C₂×Dic₆₇	—	4	4	no	—
78	C₂×Dic₆₇	—	4	8	no	—
79	Dic₁₃₄	—	8	8	no	—
80	C₈×C₆₇	—	8	2	no	—
81	C₆₇⋊C₈	—	4	4	no	—
82	C₂²×C₆₇	#390	1	1	yes	characteristic
83	C₂²×C₆₇	C₂×#100	1	2	yes	—
84	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
85	C₂²×C₆₇	#419	1	2	yes	—
86	C₂²×C₆₇	#419	1	1	yes	characteristic
87	C₄×C₆₇	—	2	4	no	—
88	C₄×C₆₇	—	2	4	no	—
89	C₄×C₆₇	—	2	2	no	—
90	C₄×C₆₇	#461	1	2	yes	—
91	C₄×C₆₇	#462	1	2	yes	—
92	C₄×C₆₇	#491	1	2	yes	—
93	C₄×C₆₇	#497	1	2	yes	—
94	Dic₆₇	—	8	4	no	—
95	#9702	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	#1548	1	2	yes	—
97	C₂×C₆₇	#1558	1	2	yes	—
98	C₂×C₆₇	#1960	1	1	yes	characteristic
99	C₂×C₆₇	#1965	1	1	yes	characteristic
100	C₂×C₆₇	#1981	1	1	yes	characteristic
101	#2868	—	67	1	no	—
102	#1831	D₆₇	1	1	yes	characteristic
103	#1767	—	67	1	no	—
104	C₂×#333	—	67	4	no	—
105	C₆₇	#9702	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₄⋊Q₈	—	67	4	no	—
107	C₂×C₄⋊Q₈	—	67	2	no	—
108	C₂×C₄⋊C₈	—	67	2	no	—
109	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
110	C₂×#97	—	134	1	no	—
111	#384	—	134	1	no	—
112	#384	—	134	1	no	—
113	#384	—	134	2	no	—
114	C₄³	C₂×D₆₇	1	1	yes	characteristic
115	#336	—	2	2	no	—
116	#336	—	134	1	no	—
117	#333	—	134	8	no	—
118	C₂²×Q₈	—	67	4	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	134	2	no	—
121	C₄⋊Q₈	—	268	8	no	—
122	C₄⋊Q₈	—	134	4	no	—
123	C₄⋊Q₈	—	134	4	no	—
124	C₂×C₄⋊C₄	—	134	4	no	—
125	C₂×C₄⋊C₄	—	134	4	no	—
126	C₂×C₄⋊C₄	—	134	2	no	—
127	C₂×C₄⋊C₄	—	134	2	no	—
128	C₂×C₄²	C₄×D₆₇	1	2	yes	—
129	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
130	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
131	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
132	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	67	4	no	—
135	C₄⋊C₈	—	134	2	no	—
136	#97	—	268	2	no	—
137	C₂×Q₈	—	268	8	no	—
138	C₂×Q₈	—	134	8	no	—
139	C₂²×C₄	—	134	4	no	—
140	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
141	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
142	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	268	2	no	—
148	C₂×C₈	—	268	2	no	—
149	C₂×C₈	—	134	4	no	—
150	C₄⋊C₄	—	268	8	no	—
151	C₄⋊C₄	—	268	8	no	—
152	C₄⋊C₄	—	268	4	no	—
153	C₄⋊C₄	—	268	2	no	—
154	C₄⋊C₄	—	268	2	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	#11761617	1	2	yes	—
159	C₄²	Dic₂₆₈	1	2	yes	—
160	C₄²	D₄×D₆₇	1	2	yes	—
161	C₄²	#11761629	1	2	yes	—
162	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
163	Q₈	—	536	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	268	4	no	—
170	C₂×C₄	—	268	8	no	—
171	C₂×C₄	#14523959	1	4	yes	—
172	C₂×C₄	#14523968	1	4	yes	—
173	C₂×C₄	#14524005	1	2	yes	—
174	C₂×C₄	#14524018	1	2	yes	—
175	C₂×C₄	#14524032	1	1	yes	characteristic
176	C₂×C₄	#14524039	1	1	yes	characteristic
177	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
178	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
179	C₂×C₄	#14524073	1	4	yes	—
180	C₂×C₄	#14524076	1	2	yes	—
181	C₈	—	8	2	no	—
182	C₈	—	268	4	no	—
183	C₂²	#16863297	1	1	yes	characteristic
184	C₂²	C₂×#14523959	1	2	yes	—
185	C₂²	C₂×#14523968	1	1	yes	characteristic
186	C₂²	#16863404	1	2	yes	—
187	C₂²	#16863406	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	536	4	no	—
192	C₄	#16863602	1	2	yes	—
193	C₄	#16863607	1	2	yes	—
194	C₄	#16863766	1	2	yes	—
195	C₄	#16863791	1	2	yes	—
196	C₂	#22558889	1	2	yes	—
197	C₂	#22558934	1	2	yes	—
198	C₂	#22561028	1	1	yes	characteristic
199	C₂	#22561048	1	1	yes	characteristic
200	C₂	#22561131	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22568607

C₂

yes

maximal, characteristic

#22560062

C₂

yes

maximal, characteristic

C₂×#16863144

C₂

yes

maximal

C₆₇×#1831

C₂

yes

maximal, characteristic

C₂×#14524043

C₄

yes

—

C₂×#14524043

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

—

C₂²×Dic₁₃₄

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

—

#14524043

—

#14524043

—

#14524043

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

#14523954

—

C₂²×C₈×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

Q₁₆

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#100

yes

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

#137

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

C₂×#100

yes

—

C₂²×C₆₇

C₂×#103

yes

characteristic

C₂²×C₆₇

#419

yes

—

C₂²×C₆₇

#419

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#461

yes

—

C₄×C₆₇

#462

yes

—

C₄×C₆₇

#491

yes

—

C₄×C₆₇

#497

yes

—

Dic₆₇

—

#9702

—

maximal, Sylow

C₂×C₆₇

#1548

yes

—

C₂×C₆₇

#1558

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

100

C₂×C₆₇

#1981

yes

characteristic

101

#2868

—

102

#1831

D₆₇

yes

characteristic

103

#1767

—

104

C₂×#333

—

105

C₆₇

#9702

yes

characteristic, complemented, Sylow

106

C₂×C₄⋊Q₈

—

107

C₂×C₄⋊Q₈

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

110

C₂×#97

—

134

—

111

#384

—

134

—

112

#384

—

134

—

113

#384

—

134

—

114

C₄³

C₂×D₆₇

yes

characteristic

115

#336

—

116

#336

—

134

—

117

#333

—

134

—

118

C₂²×Q₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

134

—

121

C₄⋊Q₈

—

268

—

122

C₄⋊Q₈

—

134

—

123

C₄⋊Q₈

—

134

—

124

C₂×C₄⋊C₄

—

134

—

125

C₂×C₄⋊C₄

—

134

—

126

C₂×C₄⋊C₄

—

134

—

127

C₂×C₄⋊C₄

—

134

—

128

C₂×C₄²

C₄×D₆₇

yes

—

129

C₂×C₄²

D₂₆₈

yes

characteristic

130

C₂×C₄²

D₂₆₈

yes

characteristic

131

C₂×C₄²

C₂²⋊D₆₇

yes

—

132

C₂×C₄²

C₂²×D₆₇

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

134

—

136

#97

—

268

—

137

C₂×Q₈

—

268

—

138

C₂×Q₈

—

134

—

139

C₂²×C₄

—

134

—

140

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

141

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

142

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

143

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

144

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

268

—

148

C₂×C₈

—

268

—

149

C₂×C₈

—

134

—

150

C₄⋊C₄

—

268

—

151

C₄⋊C₄

—

268

—

152

C₄⋊C₄

—

268

—

153

C₄⋊C₄

—

268

—

154

C₄⋊C₄

—

268

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#11761617

yes

—

159

C₄²

Dic₂₆₈

yes

—

160

C₄²

D₄×D₆₇

yes

—

161

C₄²

#11761629

yes

—

162

C₂³

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

163

Q₈

—

536

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

268

—

170

C₂×C₄

—

268

—

171

C₂×C₄

#14523959

yes

—

172

C₂×C₄

#14523968

yes

—

173

C₂×C₄

#14524005

yes

—

174

C₂×C₄

#14524018

yes

—

175

C₂×C₄

#14524032

yes

characteristic

176

C₂×C₄

#14524039

yes

characteristic

177

C₂×C₄

C₂×#11761617

yes

characteristic

178

C₂×C₄

C₂×Dic₂₆₈

yes

characteristic

179

C₂×C₄

#14524073

yes

—

180

C₂×C₄

#14524076

yes

—

181

C₈

—

182

C₈

—

268

—

183

C₂²

#16863297

yes

characteristic

184

C₂²

C₂×#14523959

yes

—

185

C₂²

C₂×#14523968

yes

characteristic

186

C₂²

#16863404

yes

—

187

C₂²

#16863406

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

536

—

192

C₄

#16863602

yes

—

193

C₄

#16863607

yes

—

194

C₄

#16863766

yes

—

195

C₄

#16863791

yes

—

196

C₂

#22558889

yes

—

197

C₂

#22558934

yes

—

198

C₂

#22561028

yes

characteristic

199

C₂

#22561048

yes

characteristic

200

C₂

#22561131

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1831	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1831	D₆₇
hypocenter	C₆₇	#9702
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×(C₂×D₆₇)⋊C₄

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1831

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
105. C₆₇	#9702	semidirect

H ≅ G/N

type

105. C₆₇

#9702

semidirect

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

subgroup	index
#22568607	2
#22560062	2
C₂×#16863144	2
C₆₇×#1831	2
#9702	67

subgroup

index

C₆₇×#1831

Subgroups — #30754911

Conjugacy classes of subgroups

Complements

Maximal subgroups