Group #30760581 — subgroups

Subgroups — #30760581

Order 17152.

350 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22561145	C₂	1	1	yes	maximal, characteristic
2	#22560211	C₂	1	2	yes	maximal
3	#22558092	C₂	1	2	yes	maximal
4	#22558044	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×#14523966	—	2	2	no	—
10	#16863158	—	2	2	no	—
11	#16863150	—	2	2	no	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#384×C₆₇	C₂²	1	2	yes	—
14	#340×C₆₇	—	2	2	no	—
15	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	—	2	1	no	—
18	C₄×Dic₁₃₄	—	2	2	no	—
19	C₄×Dic₁₃₄	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
21	C₂×Dic₆₇⋊C₄	—	2	1	no	—
22	C₂×C₄×Dic₆₇	—	2	1	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	#14523966	—	4	4	no	—
27	C₂²×C₈×C₆₇	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₆₇	—	2	2	no	—
33	C₂×Dic₁₃₄	—	2	2	no	—
34	C₂×Dic₁₃₄	—	2	2	no	—
35	C₂×Dic₁₃₄	—	2	1	no	—
36	C₂×Dic₁₃₄	Q₁₆	1	2	yes	—
37	C₂²×Dic₆₇	—	2	1	no	—
38	C₂²×Dic₆₇	—	2	1	no	—
39	C₂×C₆₇⋊C₈	—	4	2	no	—
40	C₂×C₆₇⋊C₈	—	4	1	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₄⋊Dic₆₇	—	2	2	no	—
45	Dic₆₇⋊C₄	—	4	2	no	—
46	C₄×Dic₆₇	—	4	1	no	—
47	C₄×Dic₆₇	—	4	1	no	—
48	C₂²×C₄×C₆₇	—	4	2	no	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₄⋊C₄×C₆₇	—	4	2	no	—
56	C₄⋊C₄×C₆₇	—	4	2	no	—
57	C₄⋊C₄×C₆₇	—	4	2	no	—
58	C₄⋊C₄×C₆₇	—	4	2	no	—
59	C₄⋊C₄×C₆₇	—	8	2	no	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₄²×C₆₇	#41	1	2	yes	—
62	C₂³×C₆₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	8	2	no	—
65	C₂×C₄×C₆₇	—	8	2	no	—
66	C₂×C₄×C₆₇	—	8	2	no	—
67	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#123	1	2	yes	—
69	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#126	1	2	yes	—
71	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#135	1	2	yes	—
73	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
74	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₆₇	#137	1	2	yes	—
76	C₂×Dic₆₇	—	2	2	no	—
77	C₂×Dic₆₇	—	4	1	no	—
78	C₂×Dic₆₇	—	4	1	no	—
79	C₂×Dic₆₇	—	4	1	no	—
80	C₂×Dic₆₇	—	4	2	no	—
81	Dic₁₃₄	—	2	4	no	—
82	Dic₁₃₄	—	4	2	no	—
83	C₈×C₆₇	—	4	2	no	—
84	C₆₇⋊C₈	—	8	2	no	—
85	C₆₇⋊C₈	—	8	1	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	—	16	2	no	—
93	C₄×C₆₇	#482	1	2	yes	—
94	C₄×C₆₇	#484	1	2	yes	—
95	C₄×C₆₇	#485	1	2	yes	—
96	C₄×C₆₇	#488	1	1	yes	characteristic
97	C₄×C₆₇	#491	1	1	yes	characteristic
98	Dic₆₇	—	4	2	no	—
99	Dic₆₇	—	8	1	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	—	67	1	no	—
107	#1834	D₆₇	1	1	yes	characteristic
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	—	67	1	no	—
113	C₂×C₄⋊C₈	—	67	2	no	—
114	C₂×C₄⋊C₈	—	67	1	no	—
115	C₂×#103	—	134	2	no	—
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	#384	C₂×D₆₇	1	2	yes	—
118	#340	—	2	2	no	—
119	#340	—	134	2	no	—
120	#336	—	134	2	no	—
121	C₂²×Q₈	—	67	1	no	—
122	C₂²×C₈	—	134	2	no	—
123	C₂²×C₈	—	134	1	no	—
124	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
125	C₄×Q₈	—	134	2	no	—
126	C₄×Q₈	—	134	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	67	1	no	—
131	C₂×C₄⋊C₄	—	134	1	no	—
132	C₂×C₄²	—	134	1	no	—
133	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
134	C₄⋊C₈	—	134	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	—	67	2	no	—
140	C₂×Q₈	—	134	2	no	—
141	C₂×Q₈	—	134	2	no	—
142	C₂×Q₈	—	134	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	134	1	no	—
145	C₂²×C₄	—	134	1	no	—
146	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂²×C₄	#11761627	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	—	268	2	no	—
153	C₂×C₈	—	268	1	no	—
154	C₂×C₈	—	268	2	no	—
155	C₂×C₈	—	268	2	no	—
156	C₂×C₈	—	268	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	8	2	no	—
162	C₄⋊C₄	—	268	2	no	—
163	C₄⋊C₄	—	134	2	no	—
164	C₄²	—	268	1	no	—
165	C₄²	—	268	1	no	—
166	C₄²	#11761628	1	2	yes	—
167	C₄²	#11761629	1	2	yes	—
168	C₂³	#14524014	1	1	yes	characteristic
169	Q₈	—	268	2	no	—
170	Q₈	—	134	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	268	1	no	—
176	C₂×C₄	—	268	1	no	—
177	C₂×C₄	—	268	1	no	—
178	C₂×C₄	—	268	2	no	—
179	C₂×C₄	—	134	2	no	—
180	C₂×C₄	#14524026	1	1	yes	characteristic
181	C₂×C₄	#14524028	1	2	yes	—
182	C₂×C₄	#14524032	1	1	yes	characteristic
183	C₂×C₄	#14524038	1	2	yes	—
184	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
185	C₂×C₄	#14524063	1	2	yes	—
186	C₂×C₄	#14524066	1	1	yes	characteristic
187	C₂×C₄	#14524067	1	2	yes	—
188	C₂×C₄	#14524073	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	536	1	no	—
191	C₈	—	536	2	no	—
192	C₂²	#16863322	1	1	yes	characteristic
193	C₂²	#16863615	1	2	yes	—
194	C₂²	#16863634	1	1	yes	characteristic
195	C₂²	#16863646	1	1	yes	characteristic
196	C₂²	#16863656	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	268	2	no	—
199	C₄	—	16	2	no	—
200	C₄	—	536	1	no	—
201	C₄	#16863723	1	2	yes	—
202	C₄	#16863733	1	2	yes	—
203	C₄	#16863737	1	2	yes	—
204	C₄	#16863753	1	1	yes	characteristic
205	C₄	#16863766	1	1	yes	characteristic
206	C₂	#22559664	1	2	yes	—
207	C₂	#22559774	1	2	yes	—
208	C₂	#22561628	1	1	yes	characteristic
209	C₂	#22561638	1	1	yes	characteristic
210	C₂	#22561655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561145

C₂

yes

maximal, characteristic

#22560211

C₂

yes

maximal

#22558092

C₂

yes

maximal

#22558044

C₂

yes

maximal, characteristic

C₆₇×#1834

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₃₄

C₂²

yes

characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

—

#340×C₆₇

—

C₂²×Dic₁₃₄

D₄

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

—

C₂×C₄⋊Dic₆₇

D₄

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523966

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

Q₁₆

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

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

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#121

yes

characteristic

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

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

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

#482

yes

—

C₄×C₆₇

#484

yes

—

C₄×C₆₇

#485

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

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

—

107

#1834

D₆₇

yes

characteristic

108

#1803

—

109

#1296

—

110

#1277

—

111

C₆₇

#10624

yes

characteristic, complemented, Sylow

112

C₂×C₄×Q₈

—

113

C₂×C₄⋊C₈

—

114

C₂×C₄⋊C₈

—

115

C₂×#103

—

134

—

116

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

117

#384

C₂×D₆₇

yes

—

118

#340

—

119

#340

—

134

—

120

#336

—

134

—

121

C₂²×Q₈

—

122

C₂²×C₈

—

134

—

123

C₂²×C₈

—

134

—

124

C₂²×C₈

C₂²⋊D₆₇

yes

—

125

C₄×Q₈

—

134

—

126

C₄×Q₈

—

134

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

134

—

132

C₂×C₄²

—

134

—

133

C₂×C₄²

C₂²×D₆₇

yes

characteristic

134

C₄⋊C₈

—

134

—

135

C₄⋊C₈

—

134

—

136

C₄⋊C₈

—

134

—

137

#103

—

268

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

134

—

141

C₂×Q₈

—

134

—

142

C₂×Q₈

—

134

—

143

C₂²×C₄

—

144

C₂²×C₄

—

134

—

145

C₂²×C₄

—

134

—

146

C₂²×C₄

D₄×D₆₇

yes

characteristic

147

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

148

C₂²×C₄

#11761627

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

268

—

153

C₂×C₈

—

268

—

154

C₂×C₈

—

268

—

155

C₂×C₈

—

268

—

156

C₂×C₈

—

268

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

268

—

163

C₄⋊C₄

—

134

—

164

C₄²

—

268

—

165

C₄²

—

268

—

166

C₄²

#11761628

yes

—

167

C₄²

#11761629

yes

—

168

C₂³

#14524014

yes

characteristic

169

Q₈

—

268

—

170

Q₈

—

134

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

268

—

176

C₂×C₄

—

268

—

177

C₂×C₄

—

268

—

178

C₂×C₄

—

268

—

179

C₂×C₄

—

134

—

180

C₂×C₄

#14524026

yes

characteristic

181

C₂×C₄

#14524028

yes

—

182

C₂×C₄

#14524032

yes

characteristic

183

C₂×C₄

#14524038

yes

—

184

C₂×C₄

Q₁₆×D₆₇

yes

characteristic

185

C₂×C₄

#14524063

yes

—

186

C₂×C₄

#14524066

yes

characteristic

187

C₂×C₄

#14524067

yes

—

188

C₂×C₄

#14524073

yes

—

189

C₈

—

190

C₈

—

536

—

191

C₈

—

536

—

192

C₂²

#16863322

yes

characteristic

193

C₂²

#16863615

yes

—

194

C₂²

#16863634

yes

characteristic

195

C₂²

#16863646

yes

characteristic

196

C₂²

#16863656

yes

—

197

C₄

—

198

C₄

—

268

—

199

C₄

—

200

C₄

—

536

—

201

C₄

#16863723

yes

—

202

C₄

#16863733

yes

—

203

C₄

#16863737

yes

—

204

C₄

#16863753

yes

characteristic

205

C₄

#16863766

yes

characteristic

206

C₂

#22559664

yes

—

207

C₂

#22559774

yes

—

208

C₂

#22561628

yes

characteristic

209

C₂

#22561638

yes

characteristic

210

C₂

#22561655

yes

characteristic

	subgroup	G/H
center	C₂³	#14524014
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1834	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	#1834	D₆₇
hypocenter	C₆₇	#10624
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#14524014

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1834

C₂

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

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
#22561145	2
#22560211	2
#22558092	2
#22558044	2
C₆₇×#1834	2
#10624	67

subgroup

index

C₆₇×#1834

Subgroups — #30760581

Conjugacy classes of subgroups

Complements

Maximal subgroups