Group #30775661 — subgroups

Subgroups — #30775661

Order 17152.

302 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 95 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×#16863212	C₂	1	2	yes	maximal
2	#22560384	C₂	1	1	yes	maximal, characteristic
3	#22558307	C₂	1	2	yes	maximal
4	#22558293	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1971	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523979	—	2	1	no	—
8	C₂×#14523979	C₂²	1	2	yes	—
9	#16863420	—	2	1	no	—
10	#16863212	—	2	4	no	—
11	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#103×C₆₇	—	2	1	no	—
14	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
15	#391×C₆₇	—	2	1	no	—
16	C₂×#11761612	—	2	2	no	—
17	#14523983	—	2	2	no	—
18	#14523983	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	#14523979	—	2	2	no	—
21	#14523979	—	2	2	no	—
22	#14523979	—	4	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
24	C₂²×Q₈×C₆₇	—	2	1	no	—
25	C₂×Q₁₆×C₆₇	—	2	2	no	—
26	C₂×Q₁₆×C₆₇	—	2	2	no	—
27	C₂×Q₁₆×C₆₇	—	2	1	no	—
28	C₂×Q₁₆×C₆₇	D₄	1	2	yes	—
29	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	#106×C₆₇	—	2	2	no	—
34	#103×C₆₇	—	4	2	no	—
35	#97×C₆₇	—	2	2	no	—
36	#95×C₆₇	—	4	2	no	—
37	#11761612	—	4	2	no	—
38	#11761612	—	4	2	no	—
39	C₂×C₆₇⋊C₈	—	4	2	no	—
40	C₆₇⋊C₁₆	—	4	2	no	—
41	C₂×Q₈×C₆₇	—	2	2	no	—
42	C₂×Q₈×C₆₇	—	4	1	no	—
43	C₂×Q₈×C₆₇	—	4	1	no	—
44	C₂×Q₈×C₆₇	—	4	1	no	—
45	C₂×Q₈×C₆₇	—	4	1	no	—
46	C₂×Q₈×C₆₇	—	4	1	no	—
47	C₂²×C₄×C₆₇	—	4	1	no	—
48	C₂²×C₄×C₆₇	—	4	1	no	—
49	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	Q₁₆×C₆₇	—	2	4	no	—
52	Q₁₆×C₆₇	—	4	2	no	—
53	C₂×C₈×C₆₇	—	2	1	no	—
54	C₂×C₈×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	D₈	1	2	yes	—
56	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	#41	1	2	yes	—
58	C₂×C₈×C₆₇	#41	1	2	yes	—
59	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₆₇	—	4	1	no	—
61	C₄⋊C₄×C₆₇	—	4	1	no	—
62	C₄²×C₆₇	—	2	1	no	—
63	C₄²×C₆₇	—	2	1	no	—
64	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
65	Q₈×C₆₇	—	4	2	no	—
66	Q₈×C₆₇	—	4	2	no	—
67	Q₈×C₆₇	—	8	1	no	—
68	Q₈×C₆₇	—	8	1	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	4	2	no	—
71	C₂×C₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	—	8	1	no	—
73	C₂×C₄×C₆₇	—	8	1	no	—
74	C₂×C₄×C₆₇	—	8	2	no	—
75	C₂×C₄×C₆₇	—	8	1	no	—
76	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
82	C₈×C₆₇	—	2	2	no	—
83	C₈×C₆₇	#123	1	2	yes	—
84	C₈×C₆₇	#126	1	2	yes	—
85	C₆₇⋊C₈	—	8	2	no	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	#461	1	1	yes	characteristic
88	C₂²×C₆₇	#481	1	1	yes	characteristic
89	C₂²×C₆₇	#486	1	1	yes	characteristic
90	C₄×C₆₇	—	4	2	no	—
91	C₄×C₆₇	—	8	2	no	—
92	C₄×C₆₇	—	16	1	no	—
93	C₄×C₆₇	#397	1	1	yes	characteristic
94	C₄×C₆₇	#459	1	1	yes	characteristic
95	C₄×C₆₇	#480	1	1	yes	characteristic
96	C₄×C₆₇	#489	1	1	yes	characteristic
97	#13488	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	—	2	1	no	—
99	C₂×C₆₇	—	2	1	no	—
100	C₂×C₆₇	#2071	1	1	yes	characteristic
101	C₂×C₆₇	#2239	1	2	yes	—
102	C₂×#362	—	67	2	no	—
103	#1971	D₆₇	1	1	yes	characteristic
104	#1849	—	67	1	no	—
105	#1383	—	67	2	no	—
106	#1377	—	67	1	no	—
107	C₆₇	#13488	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
109	C₂×M₃₂	—	67	2	no	—
110	C₂×#108	—	67	2	no	—
111	C₂×#108	—	134	1	no	—
112	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
113	#423	—	134	1	no	—
114	C₂×#103	—	2	1	no	—
115	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
116	#391	—	2	1	no	—
117	#362	—	134	4	no	—
118	C₂²×Q₈	—	2	1	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×Q₁₆	D₂₆₈	1	2	yes	—
123	C₂×M₁₆	—	134	2	no	—
124	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
128	M₃₂	—	134	2	no	—
129	M₃₂	—	134	2	no	—
130	C₂×C₁₆	—	134	2	no	—
131	#108	—	268	2	no	—
132	#108	—	134	2	no	—
133	#108	—	134	2	no	—
134	#106	—	2	2	no	—
135	C₄⋊C₈	—	268	2	no	—
136	#103	—	4	2	no	—
137	#97	—	2	2	no	—
138	#95	—	4	2	no	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	4	1	no	—
141	C₂×Q₈	—	4	1	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	#11761616	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
149	Q₁₆	—	2	4	no	—
150	Q₁₆	—	4	2	no	—
151	M₁₆	—	268	2	no	—
152	M₁₆	—	268	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	268	2	no	—
156	C₂×C₈	#11761613	1	2	yes	—
157	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
158	C₂×C₈	#11761627	1	2	yes	—
159	C₂×C₈	#11761627	1	2	yes	—
160	C₂×C₈	#11761628	1	1	yes	characteristic
161	C₄⋊C₄	—	4	1	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	268	2	no	—
166	C₂³	C₂×#11761616	1	1	yes	characteristic
167	Q₈	—	4	2	no	—
168	Q₈	—	4	2	no	—
169	Q₈	—	8	1	no	—
170	Q₈	—	8	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	#14524017	1	1	yes	characteristic
179	C₂×C₄	#14524024	1	1	yes	characteristic
180	C₂×C₄	#14524030	1	1	yes	characteristic
181	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
182	C₂×C₄	#14524068	1	1	yes	characteristic
183	C₂×C₄	#14524073	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	536	2	no	—
186	C₈	#14524027	1	2	yes	—
187	C₈	#14524036	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#16863601	1	1	yes	characteristic
190	C₂²	#16863712	1	1	yes	characteristic
191	C₂²	#16863742	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	8	2	no	—
194	C₄	—	16	1	no	—
195	C₄	#16863321	1	1	yes	characteristic
196	C₄	#16863591	1	1	yes	characteristic
197	C₄	#16863707	1	1	yes	characteristic
198	C₄	#16863755	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#22561612	1	1	yes	characteristic
202	C₂	#22562426	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863212

C₂

yes

maximal

#22560384

C₂

yes

maximal, characteristic

#22558307

C₂

yes

maximal

#22558293

C₂

yes

maximal, characteristic

C₆₇×#1971

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

C₂×#14523979

—

C₂×#14523979

C₂²

yes

—

#16863420

—

#16863212

—

C₂²×Q₁₆×C₆₇

C₂²

yes

characteristic

C₂×#106×C₆₇

C₂²

yes

characteristic

C₂×#103×C₆₇

—

C₂×#97×C₆₇

C₂²

yes

characteristic

#391×C₆₇

—

C₂×#11761612

—

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

#14523979

—

#14523979

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×Q₈×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

#106×C₆₇

—

#103×C₆₇

—

#97×C₆₇

—

#95×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

SD₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

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

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×D₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₈×C₆₇

—

C₈×C₆₇

#123

yes

—

C₈×C₆₇

#126

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

100

C₂×C₆₇

#2071

yes

characteristic

101

C₂×C₆₇

#2239

yes

—

102

C₂×#362

—

103

#1971

D₆₇

yes

characteristic

104

#1849

—

105

#1383

—

106

#1377

—

107

C₆₇

#13488

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

C₂×D₆₇

yes

characteristic

109

C₂×M₃₂

—

110

C₂×#108

—

111

C₂×#108

—

134

—

112

C₂×#106

C₂×D₆₇

yes

characteristic

113

#423

—

134

—

114

C₂×#103

—

115

C₂×#97

C₂×D₆₇

yes

characteristic

116

#391

—

117

#362

—

134

—

118

C₂²×Q₈

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

D₂₆₈

yes

—

123

C₂×M₁₆

—

134

—

124

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

125

C₂²×C₈

C₂²×D₆₇

yes

characteristic

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

128

M₃₂

—

134

—

129

M₃₂

—

134

—

130

C₂×C₁₆

—

134

—

131

#108

—

268

—

132

#108

—

134

—

133

#108

—

134

—

134

#106

—

135

C₄⋊C₈

—

268

—

136

#103

—

137

#97

—

138

#95

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

#11761616

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

149

Q₁₆

—

150

Q₁₆

—

151

M₁₆

—

268

—

152

M₁₆

—

268

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

268

—

156

C₂×C₈

#11761613

yes

—

157

C₂×C₈

C₂×D₂₆₈

yes

characteristic

158

C₂×C₈

#11761627

yes

—

159

C₂×C₈

#11761627

yes

—

160

C₂×C₈

#11761628

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

268

—

166

C₂³

C₂×#11761616

yes

characteristic

167

Q₈

—

168

Q₈

—

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

—

178

C₂×C₄

#14524017

yes

characteristic

179

C₂×C₄

#14524024

yes

characteristic

180

C₂×C₄

#14524030

yes

characteristic

181

C₂×C₄

C₂×#11761613

yes

characteristic

182

C₂×C₄

#14524068

yes

characteristic

183

C₂×C₄

#14524073

yes

characteristic

184

C₈

—

185

C₈

—

536

—

186

C₈

#14524027

yes

—

187

C₈

#14524036

yes

—

188

C₂²

—

189

C₂²

#16863601

yes

characteristic

190

C₂²

#16863712

yes

characteristic

191

C₂²

#16863742

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

#16863321

yes

characteristic

196

C₄

#16863591

yes

characteristic

197

C₄

#16863707

yes

characteristic

198

C₄

#16863755

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#22561612

yes

characteristic

202

C₂

#22562426

yes

—

	subgroup	G/H
center	C₂²	#16863712
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1971	C₂
socle	C₂²×C₆₇	#481
prime radical	C₂³×C₆₇	C₂×SD₁₆
hypercenter	#1971	D₆₇
hypocenter	C₆₇	#13488
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

subgroup

G/H

center

C₂²

#16863712

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1971

C₂

socle

C₂²×C₆₇

#481

prime radical

C₂³×C₆₇

C₂×SD₁₆

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

N	H ≅ G/N	type
107. C₆₇	#13488	semidirect

H ≅ G/N

type

107. C₆₇

#13488

semidirect

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

subgroup	index
C₂×#16863212	2
#22560384	2
#22558307	2
#22558293	2
C₆₇×#1971	2
#13488	67

subgroup

index

C₆₇×#1971

Subgroups — #30775661

Conjugacy classes of subgroups

Complements

Maximal subgroups