Group #30760580 — subgroups

Subgroups — #30760580

Order 17152.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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	#22560337	C₂	1	1	yes	maximal, characteristic
2	#22560214	C₂	1	2	yes	maximal
3	#22558093	C₂	1	2	yes	maximal
4	#22558044	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14523967	—	2	2	no	—
8	#16863266	C₂²	1	2	yes	—
9	#16863159	—	2	2	no	—
10	#16863158	—	2	2	no	—
11	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#336×C₆₇	—	2	2	no	—
15	C₂²×C₆₇⋊C₈	—	2	2	no	—
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	#14523967	—	4	4	no	—
22	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	—	2	1	no	—
24	C₂²×C₈×C₆₇	D₄	1	2	yes	—
25	C₄×Q₈×C₆₇	—	2	2	no	—
26	C₄×Q₈×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	—	2	1	no	—
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₇	—	2	2	no	—
32	C₄×C₈×C₆₇	—	2	2	no	—
33	C₂²×Dic₆₇	—	4	2	no	—
34	C₂×C₆₇⋊C₈	—	4	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	4	2	no	—
37	C₄⋊Dic₆₇	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₄⋊Dic₆₇	—	4	2	no	—
41	C₄⋊Dic₆₇	—	8	2	no	—
42	C₂×Q₈×C₆₇	—	2	1	no	—
43	C₂×Q₈×C₆₇	—	2	2	no	—
44	C₂×Q₈×C₆₇	—	2	2	no	—
45	C₂×Q₈×C₆₇	Q₁₆	1	2	yes	—
46	C₂²×C₄×C₆₇	—	2	1	no	—
47	C₂²×C₄×C₆₇	—	2	1	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	4	1	no	—
55	C₂×C₈×C₆₇	—	4	2	no	—
56	C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₄⋊C₄×C₆₇	—	4	2	no	—
58	C₄²×C₆₇	—	4	1	no	—
59	C₄²×C₆₇	—	4	1	no	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₄²×C₆₇	#41	1	2	yes	—
62	C₂³×C₆₇	#121	1	1	yes	characteristic
63	Q₈×C₆₇	—	2	4	no	—
64	Q₈×C₆₇	—	4	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	4	1	no	—
68	C₂×C₄×C₆₇	—	4	1	no	—
69	C₂×C₄×C₆₇	—	4	1	no	—
70	C₂×C₄×C₆₇	—	4	2	no	—
71	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#123	1	2	yes	—
73	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#126	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#135	1	2	yes	—
77	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
78	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₆₇	#137	1	2	yes	—
80	C₂×Dic₆₇	—	8	2	no	—
81	C₂×Dic₆₇	—	8	2	no	—
82	C₂×Dic₆₇	—	8	2	no	—
83	C₈×C₆₇	—	4	2	no	—
84	C₈×C₆₇	—	8	1	no	—
85	C₆₇⋊C₈	—	8	2	no	—
86	C₂²×C₆₇	#397	1	1	yes	characteristic
87	C₂²×C₆₇	#464	1	2	yes	—
88	C₂²×C₆₇	#467	1	1	yes	characteristic
89	C₂²×C₆₇	#469	1	1	yes	characteristic
90	C₂²×C₆₇	#471	1	2	yes	—
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	—	4	2	no	—
93	C₄×C₆₇	—	8	1	no	—
94	C₄×C₆₇	#482	1	2	yes	—
95	C₄×C₆₇	#484	1	2	yes	—
96	C₄×C₆₇	#485	1	2	yes	—
97	C₄×C₆₇	#488	1	1	yes	characteristic
98	C₄×C₆₇	#491	1	1	yes	characteristic
99	Dic₆₇	—	16	2	no	—
100	#10624	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	#1684	1	2	yes	—
102	C₂×C₆₇	#1702	1	2	yes	—
103	C₂×C₆₇	#2074	1	1	yes	characteristic
104	C₂×C₆₇	#2076	1	1	yes	characteristic
105	C₂×C₆₇	#2079	1	1	yes	characteristic
106	#1984	D₆₇	1	1	yes	characteristic
107	#1834	—	67	1	no	—
108	#1803	—	67	2	no	—
109	#1296	—	67	2	no	—
110	#1277	—	67	1	no	—
111	C₆₇	#10624	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
113	C₂×C₄⋊C₈	—	67	2	no	—
114	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
115	C₂×#103	—	134	2	no	—
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	#384	—	67	2	no	—
118	#340	—	134	2	no	—
119	#340	—	134	2	no	—
120	#336	—	2	2	no	—
121	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	134	2	no	—
124	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
125	C₄×Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	134	2	no	—
129	C₂×C₄⋊C₄	—	134	2	no	—
130	C₂×C₄⋊C₄	—	134	2	no	—
131	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	134	2	no	—
136	C₄⋊C₈	—	134	2	no	—
137	#103	—	268	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	Dic₂₆₈	1	2	yes	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	268	2	no	—
146	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
148	C₂²×C₄	#11761628	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	268	2	no	—
155	C₂×C₈	—	268	2	no	—
156	C₂×C₈	—	268	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	268	2	no	—
160	C₄⋊C₄	—	268	2	no	—
161	C₄⋊C₄	—	268	2	no	—
162	C₄⋊C₄	—	268	2	no	—
163	C₄⋊C₄	—	536	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	#11761627	1	2	yes	—
167	C₄²	#11761627	1	2	yes	—
168	C₂³	#14524017	1	1	yes	characteristic
169	Q₈	—	2	4	no	—
170	Q₈	—	4	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	536	2	no	—
178	C₂×C₄	—	536	2	no	—
179	C₂×C₄	—	536	2	no	—
180	C₂×C₄	#14524024	1	1	yes	characteristic
181	C₂×C₄	#14524027	1	2	yes	—
182	C₂×C₄	#14524030	1	1	yes	characteristic
183	C₂×C₄	#14524036	1	2	yes	—
184	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
185	C₂×C₄	#14524063	1	2	yes	—
186	C₂×C₄	#14524065	1	1	yes	characteristic
187	C₂×C₄	#14524071	1	2	yes	—
188	C₂×C₄	#14524074	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	1	no	—
191	C₈	—	536	2	no	—
192	C₂²	#16863321	1	1	yes	characteristic
193	C₂²	#16863618	1	2	yes	—
194	C₂²	#16863637	1	1	yes	characteristic
195	C₂²	#16863649	1	1	yes	characteristic
196	C₂²	#16863659	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	1072	2	no	—
201	C₄	#16863721	1	2	yes	—
202	C₄	#16863732	1	2	yes	—
203	C₄	#16863734	1	2	yes	—
204	C₄	#16863750	1	1	yes	characteristic
205	C₄	#16863764	1	1	yes	characteristic
206	C₂	#22559667	1	2	yes	—
207	C₂	#22559777	1	2	yes	—
208	C₂	#22561627	1	1	yes	characteristic
209	C₂	#22561637	1	1	yes	characteristic
210	C₂	#22561654	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560337

C₂

yes

maximal, characteristic

#22560214

C₂

yes

maximal

#22558093

C₂

yes

maximal

#22558044

C₂

yes

maximal, characteristic

C₆₇×#1984

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×#14523967

—

#16863266

C₂²

yes

—

#16863159

—

#16863158

—

C₂×C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#336×C₆₇

—

C₂²×C₆₇⋊C₈

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523967

—

C₂²×Q₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

Q₁₆

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#121

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#123

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

#126

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#397

yes

characteristic

C₂²×C₆₇

#464

yes

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#482

yes

—

C₄×C₆₇

#484

yes

—

C₄×C₆₇

#485

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

100

#10624

—

maximal, Sylow

101

C₂×C₆₇

#1684

yes

—

102

C₂×C₆₇

#1702

yes

—

103

C₂×C₆₇

#2074

yes

characteristic

104

C₂×C₆₇

#2076

yes

characteristic

105

C₂×C₆₇

#2079

yes

characteristic

106

#1984

D₆₇

yes

characteristic

107

#1834

—

108

#1803

—

109

#1296

—

110

#1277

—

111

C₆₇

#10624

yes

characteristic, complemented, Sylow

112

C₂×C₄×Q₈

C₂×D₆₇

yes

characteristic

113

C₂×C₄⋊C₈

—

114

C₂×C₄⋊C₈

C₂×D₆₇

yes

characteristic

115

C₂×#103

—

134

—

116

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

117

#384

—

118

#340

—

134

—

119

#340

—

134

—

120

#336

—

121

C₂²×Q₈

D₂₆₈

yes

characteristic

122

C₂²×C₈

—

123

C₂²×C₈

—

134

—

124

C₂²×C₈

C₂²⋊D₆₇

yes

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

134

—

129

C₂×C₄⋊C₄

—

134

—

130

C₂×C₄⋊C₄

—

134

—

131

C₂×C₄⋊C₄

D₂₆₈

yes

characteristic

132

C₂×C₄²

—

133

C₂×C₄²

C₂²×D₆₇

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

134

—

136

C₄⋊C₈

—

134

—

137

#103

—

268

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

Dic₂₆₈

yes

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

268

—

146

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

147

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

148

C₂²×C₄

#11761628

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

268

—

155

C₂×C₈

—

268

—

156

C₂×C₈

—

268

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

268

—

160

C₄⋊C₄

—

268

—

161

C₄⋊C₄

—

268

—

162

C₄⋊C₄

—

268

—

163

C₄⋊C₄

—

536

—

164

C₄²

—

165

C₄²

—

166

C₄²

#11761627

yes

—

167

C₄²

#11761627

yes

—

168

C₂³

#14524017

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

536

—

178

C₂×C₄

—

536

—

179

C₂×C₄

—

536

—

180

C₂×C₄

#14524024

yes

characteristic

181

C₂×C₄

#14524027

yes

—

182

C₂×C₄

#14524030

yes

characteristic

183

C₂×C₄

#14524036

yes

—

184

C₂×C₄

C₂×Dic₂₆₈

yes

characteristic

185

C₂×C₄

#14524063

yes

—

186

C₂×C₄

#14524065

yes

characteristic

187

C₂×C₄

#14524071

yes

—

188

C₂×C₄

#14524074

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

536

—

192

C₂²

#16863321

yes

characteristic

193

C₂²

#16863618

yes

—

194

C₂²

#16863637

yes

characteristic

195

C₂²

#16863649

yes

characteristic

196

C₂²

#16863659

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

1072

—

201

C₄

#16863721

yes

—

202

C₄

#16863732

yes

—

203

C₄

#16863734

yes

—

204

C₄

#16863750

yes

characteristic

205

C₄

#16863764

yes

characteristic

206

C₂

#22559667

yes

—

207

C₂

#22559777

yes

—

208

C₂

#22561627

yes

characteristic

209

C₂

#22561637

yes

characteristic

210

C₂

#22561654

yes

characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1984	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	#1984	D₆₇
hypocenter	C₆₇	#10624
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

subgroup

G/H

center

C₂³

#14524017

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1984

C₂

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

N	H ≅ G/N	type
111. C₆₇	#10624	semidirect

H ≅ G/N

type

111. C₆₇

#10624

semidirect

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

subgroup	index
#22560337	2
#22560214	2
#22558093	2
#22558044	2
C₆₇×#1984	2
#10624	67

subgroup

index

C₆₇×#1984

Subgroups — #30760580

Conjugacy classes of subgroups

Complements

Maximal subgroups