Group #30748779 — subgroups

Subgroups — #30748779

Order 17152.

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

normal subgroups only

This group has 362 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×Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22558970	C₂	1	2	yes	maximal
3	#22558553	C₂	1	1	yes	maximal, characteristic
4	#22558078	C₂	1	2	yes	maximal
5	C₆₇×#1467	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₆₇⋊C₈	—	2	2	no	—
7	Q₈×C₆₇⋊C₈	—	2	2	no	—
8	#16863453	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863148	—	2	2	no	—
14	#16863146	—	2	2	no	—
15	#16863122	—	2	2	no	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	#432×C₆₇	C₂²	1	2	yes	—
18	#324×C₆₇	—	2	2	no	—
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
26	#14523954	—	2	2	no	—
27	C₄×C₆₇⋊C₈	—	2	2	no	—
28	C₄×C₆₇⋊C₈	—	2	2	no	—
29	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
31	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
32	C₄×Q₈×C₆₇	—	2	2	no	—
33	C₄×Q₈×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₆₇	—	2	2	no	—
39	C₂²⋊C₈×C₆₇	—	2	2	no	—
40	#97×C₆₇	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	4	1	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
48	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Q₈×C₆₇	—	2	4	no	—
50	C₂×Q₈×C₆₇	—	2	2	no	—
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	4	2	no	—
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₄²×C₆₇	—	2	2	no	—
60	C₄²×C₆₇	—	2	2	no	—
61	C₄²×C₆₇	C₂×D₄	1	2	yes	—
62	C₄²×C₆₇	#41	1	2	yes	—
63	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	Q₈×C₆₇	—	2	8	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	1	no	—
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	4	1	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	#108	1	4	yes	—
74	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#121	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₆₇	—	8	2	no	—
79	C₆₇⋊C₈	—	2	1	no	—
80	C₆₇⋊C₈	—	2	2	no	—
81	C₆₇⋊C₈	—	4	2	no	—
82	C₆₇⋊C₈	—	8	2	no	—
83	C₆₇⋊C₈	#101	1	2	yes	—
84	C₂²×C₆₇	—	2	1	no	—
85	C₂²×C₆₇	—	2	1	no	—
86	C₂²×C₆₇	#380	1	1	yes	characteristic
87	C₂²×C₆₇	C₂×#108	1	2	yes	—
88	C₄×C₆₇	—	2	1	no	—
89	C₄×C₆₇	—	2	4	no	—
90	C₄×C₆₇	—	2	2	no	—
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	—	4	2	no	—
93	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
94	C₄×C₆₇	#413	1	1	yes	characteristic
95	#8621	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	—	2	2	no	—
97	C₂×C₆₇	#1801	1	1	yes	characteristic
98	C₂×C₆₇	#1810	1	1	yes	characteristic
99	C₂×C₆₇	#1812	1	1	yes	characteristic
100	C₂×C₈×Q₈	—	67	1	no	—
101	#1568	—	67	2	no	—
102	#1467	—	67	1	no	—
103	#1467	D₆₇	1	1	yes	characteristic
104	#1290	—	67	2	no	—
105	C₆₇	#8621	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
107	C₈×Q₈	—	134	2	no	—
108	C₈×Q₈	—	134	2	no	—
109	#432	—	67	2	no	—
110	#432	C₂×D₆₇	1	2	yes	—
111	C₂×C₄⋊C₈	—	67	2	no	—
112	C₂×C₄⋊C₈	—	67	1	no	—
113	C₂×C₄×C₈	—	67	1	no	—
114	C₂×C₄×C₈	—	67	2	no	—
115	#335	—	134	2	no	—
116	#334	—	134	2	no	—
117	#324	—	2	2	no	—
118	#324	—	134	2	no	—
119	C₂²×Q₈	Dic₁₃₄	1	1	yes	characteristic
120	C₂²×C₈	—	67	1	no	—
121	C₂²×C₈	—	67	2	no	—
122	C₂²×C₈	—	67	1	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	C₂×C₄⋊C₄	Dic₁₃₄	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
127	C₂×C₄²	C₄×D₆₇	1	2	yes	—
128	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	134	4	no	—
131	C₄⋊C₈	—	134	2	no	—
132	C₄⋊C₈	—	134	2	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₂²⋊C₈	—	134	2	no	—
135	#97	—	2	2	no	—
136	#97	—	134	2	no	—
137	C₄×C₈	—	67	2	no	—
138	C₄×C₈	—	67	2	no	—
139	C₄×C₈	—	134	2	no	—
140	C₄×C₈	—	134	2	no	—
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
144	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	268	2	no	—
150	C₂×C₈	—	268	2	no	—
151	C₂×C₈	—	268	1	no	—
152	C₂×C₈	—	67	1	no	—
153	C₂×C₈	—	67	1	no	—
154	C₂×C₈	—	67	2	no	—
155	C₂×C₈	—	134	2	no	—
156	C₂×C₈	—	134	2	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	D₄×D₆₇	1	2	yes	—
162	C₄²	#11761627	1	2	yes	—
163	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
164	Q₈	—	2	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	#14523978	1	4	yes	—
174	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
175	C₂×C₄	#14523999	1	1	yes	characteristic
176	C₂×C₄	#14524015	1	2	yes	—
177	C₂×C₄	#14524020	1	2	yes	—
178	C₈	—	8	2	no	—
179	C₈	—	268	2	no	—
180	C₈	—	536	2	no	—
181	C₈	—	67	2	no	—
182	C₈	—	134	1	no	—
183	C₈	—	134	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	#16863254	1	1	yes	characteristic
187	C₂²	C₂×#14523978	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	#101×D₆₇	1	1	yes	characteristic
194	C₄	#16863379	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	#22560203	1	1	yes	characteristic
197	C₂	#22560245	1	1	yes	characteristic
198	C₂	#22560253	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#22558970

C₂

yes

maximal

#22558553

C₂

yes

maximal, characteristic

#22558078

C₂

yes

maximal

C₆₇×#1467

C₂

yes

maximal, characteristic

Q₈×C₆₇⋊C₈

—

Q₈×C₆₇⋊C₈

—

#16863453

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₄

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

C₂×C₄×Q₈×C₆₇

C₂²

yes

characteristic

#432×C₆₇

C₂²

yes

—

#324×C₆₇

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂²×Q₈×C₆₇

Q₈

yes

characteristic

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

Q₈

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#108

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#101

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#380

yes

characteristic

C₂²×C₆₇

C₂×#108

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

100

C₂×C₈×Q₈

—

101

#1568

—

102

#1467

—

103

#1467

D₆₇

yes

characteristic

104

#1290

—

105

C₆₇

#8621

yes

characteristic, complemented, Sylow

106

C₂×C₄×Q₈

C₂×D₆₇

yes

characteristic

107

C₈×Q₈

—

134

—

108

C₈×Q₈

—

134

—

109

#432

—

110

#432

C₂×D₆₇

yes

—

111

C₂×C₄⋊C₈

—

112

C₂×C₄⋊C₈

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

—

115

#335

—

134

—

116

#334

—

134

—

117

#324

—

118

#324

—

134

—

119

C₂²×Q₈

Dic₁₃₄

yes

characteristic

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

C₂×C₄⋊C₄

Dic₁₃₄

yes

characteristic

126

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

127

C₂×C₄²

C₄×D₆₇

yes

—

128

C₂×C₄²

C₂²×D₆₇

yes

characteristic

129

C₄⋊C₈

—

130

C₄⋊C₈

—

134

—

131

C₄⋊C₈

—

134

—

132

C₄⋊C₈

—

134

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

134

—

135

#97

—

136

#97

—

134

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

134

—

140

C₄×C₈

—

134

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

Dic₆₇⋊C₄

yes

—

144

C₂²×C₄

Dic₆₇⋊C₄

yes

—

145

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

146

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

147

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

148

C₂×C₈

—

149

C₂×C₈

—

268

—

150

C₂×C₈

—

268

—

151

C₂×C₈

—

268

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

134

—

156

C₂×C₈

—

134

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

D₄×D₆₇

yes

—

162

C₄²

#11761627

yes

—

163

C₂³

C₂×Dic₆₇⋊C₄

yes

characteristic

164

Q₈

—

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

#14523978

yes

—

174

C₂×C₄

C₂²⋊C₄×D₆₇

yes

characteristic

175

C₂×C₄

#14523999

yes

characteristic

176

C₂×C₄

#14524015

yes

—

177

C₂×C₄

#14524020

yes

—

178

C₈

—

179

C₈

—

268

—

180

C₈

—

536

—

181

C₈

—

182

C₈

—

134

—

183

C₈

—

134

—

184

C₂²

—

185

C₂²

—

186

C₂²

#16863254

yes

characteristic

187

C₂²

C₂×#14523978

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#101×D₆₇

yes

characteristic

194

C₄

#16863379

yes

characteristic

195

C₂

—

196

C₂

#22560203

yes

characteristic

197

C₂

#22560245

yes

characteristic

198

C₂

#22560253

yes

characteristic

	subgroup	G/H
center	C₂²	#16863254
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1467	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1467	D₆₇
hypocenter	C₆₇	#8621
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

subgroup

G/H

center

C₂²

#16863254

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1467

C₂

socle

C₂²×C₆₇

#380

prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

N	H ≅ G/N	type
105. C₆₇	#8621	semidirect

H ≅ G/N

type

105. C₆₇

#8621

semidirect

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

subgroup	index
C₂×Q₈×C₆₇⋊C₈	2
#22558970	2
#22558553	2
#22558078	2
C₆₇×#1467	2
#8621	67

subgroup

index

C₂×Q₈×C₆₇⋊C₈

C₆₇×#1467

Subgroups — #30748779

Conjugacy classes of subgroups

Complements

Maximal subgroups