Group #30760626 — subgroups

Subgroups — #30760626

Order 17152.

310 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22561029	C₂	1	1	yes	maximal, characteristic
2	#22560384	C₂	1	1	yes	maximal, characteristic
3	#22558880	C₂	1	2	yes	maximal
4	#22558068	C₂	1	2	yes	maximal
5	C₆₇×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	1	yes	characteristic
7	C₂×#14523979	—	2	1	no	—
8	C₂×#14523979	—	2	1	no	—
9	#16863420	C₂²	1	2	yes	—
10	C₂×#14523968	—	2	1	no	—
11	C₂×#14523968	—	2	1	no	—
12	#16863262	C₂²	1	1	yes	characteristic
13	#16863141	—	2	4	no	—
14	#431×C₆₇	C₂²	1	2	yes	—
15	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₃₄	—	2	1	no	—
17	C₂×#11761612	—	2	2	no	—
18	#14524043	—	2	2	no	—
19	#14524043	—	2	1	no	—
20	#14524043	D₄	1	2	yes	—
21	C₂×C₄⋊Dic₆₇	—	2	1	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	#14523979	—	4	2	no	—
24	#14523979	—	4	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	#14523968	—	4	2	no	—
28	#14523968	—	4	2	no	—
29	C₆₇⋊#95	—	2	2	no	—
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	—	2	2	no	—
34	C₂²⋊C₈×C₆₇	—	2	2	no	—
35	#97×C₆₇	—	2	2	no	—
36	C₄×C₈×C₆₇	—	2	2	no	—
37	C₂×Dic₁₃₄	—	2	2	no	—
38	C₂×Dic₁₃₄	—	4	1	no	—
39	C₂×Dic₁₃₄	—	4	1	no	—
40	C₂×Dic₁₃₄	—	4	1	no	—
41	C₂×Dic₁₃₄	—	4	1	no	—
42	C₂×Dic₁₃₄	—	4	1	no	—
43	C₂²×Dic₆₇	—	4	1	no	—
44	C₂²×Dic₆₇	—	4	1	no	—
45	#11761612	—	4	2	no	—
46	#11761612	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₄⋊Dic₆₇	—	2	4	no	—
49	C₄⋊Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	4	1	no	—
51	C₄⋊Dic₆₇	—	4	1	no	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	#41	1	2	yes	—
54	C₂×C₈×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	—	2	1	no	—
56	C₂×C₈×C₆₇	—	2	1	no	—
57	C₂×C₈×C₆₇	—	2	1	no	—
58	C₂×C₈×C₆₇	—	4	2	no	—
59	C₂×C₈×C₆₇	D₈	1	2	yes	—
60	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
61	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₆₇	#41	1	2	yes	—
63	C₄²×C₆₇	#41	1	1	yes	characteristic
64	C₂³×C₆₇	#124	1	1	yes	characteristic
65	C₂×C₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#123	1	2	yes	—
68	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#126	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
74	C₂×Dic₆₇	—	4	2	no	—
75	C₂×Dic₆₇	—	8	1	no	—
76	C₂×Dic₆₇	—	8	1	no	—
77	C₂×Dic₆₇	—	8	2	no	—
78	C₂×Dic₆₇	—	8	1	no	—
79	Dic₁₃₄	—	4	2	no	—
80	Dic₁₃₄	—	4	2	no	—
81	Dic₁₃₄	—	8	1	no	—
82	Dic₁₃₄	—	8	1	no	—
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	—	2	2	no	—
85	C₈×C₆₇	—	8	2	no	—
86	C₆₇⋊C₈	—	8	2	no	—
87	C₂²×C₆₇	—	2	2	no	—
88	C₂²×C₆₇	#397	1	1	yes	characteristic
89	C₂²×C₆₇	#486	1	1	yes	characteristic
90	C₂²×C₆₇	#489	1	1	yes	characteristic
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	#459	1	1	yes	characteristic
94	C₄×C₆₇	#461	1	1	yes	characteristic
95	C₄×C₆₇	#480	1	1	yes	characteristic
96	C₄×C₆₇	#481	1	1	yes	characteristic
97	Dic₆₇	—	8	2	no	—
98	Dic₆₇	—	16	1	no	—
99	#10632	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	—	2	1	no	—
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	#2071	1	1	yes	characteristic
103	C₂×C₆₇	#2082	1	2	yes	—
104	#1960	—	67	1	no	—
105	#1849	—	67	1	no	—
106	#1546	—	67	2	no	—
107	#1286	—	67	2	no	—
108	#1281	D₆₇	1	1	yes	characteristic
109	C₆₇	#10632	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	—	67	1	no	—
111	#431	C₂×D₆₇	1	2	yes	—
112	C₂×#108	—	134	1	no	—
113	C₂×#108	—	134	1	no	—
114	#423	—	67	2	no	—
115	C₂×#103	—	134	1	no	—
116	C₂×#103	—	134	1	no	—
117	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
118	#382	—	67	1	no	—
119	#332	—	134	4	no	—
120	C₂²×Q₈	—	134	1	no	—
121	C₂×M₁₆	—	134	2	no	—
122	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
123	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
124	C₄⋊Q₈	—	67	2	no	—
125	C₄⋊Q₈	—	134	2	no	—
126	C₄⋊Q₈	—	134	1	no	—
127	C₂×C₄⋊C₄	—	134	1	no	—
128	C₂×C₄⋊C₄	—	134	2	no	—
129	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
130	#108	—	268	2	no	—
131	#108	—	268	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	134	2	no	—
134	C₄⋊C₈	—	134	2	no	—
135	#103	—	268	2	no	—
136	#103	—	268	2	no	—
137	C₂²⋊C₈	—	2	2	no	—
138	#97	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	#95	—	134	2	no	—
141	C₂×Q₈	—	268	1	no	—
142	C₂×Q₈	—	268	1	no	—
143	C₂×Q₈	—	268	1	no	—
144	C₂×Q₈	—	268	1	no	—
145	C₂×Q₈	—	268	1	no	—
146	C₂×Q₈	—	134	2	no	—
147	C₂²×C₄	—	268	1	no	—
148	C₂²×C₄	—	268	1	no	—
149	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
150	C₂²×C₄	#11761628	1	2	yes	—
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	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	268	2	no	—
159	C₂×C₈	D₅₃₆	1	2	yes	—
160	C₂×C₈	#11761617	1	2	yes	—
161	C₄⋊C₄	—	268	2	no	—
162	C₄⋊C₄	—	268	1	no	—
163	C₄⋊C₄	—	268	1	no	—
164	C₄⋊C₄	—	134	4	no	—
165	C₄²	D₄×D₆₇	1	1	yes	characteristic
166	C₄²	#11761627	1	2	yes	—
167	C₄²	#11761629	1	1	yes	characteristic
168	C₂³	#14524031	1	1	yes	characteristic
169	Q₈	—	268	2	no	—
170	Q₈	—	268	2	no	—
171	Q₈	—	536	1	no	—
172	Q₈	—	536	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	268	2	no	—
175	C₂×C₄	—	536	1	no	—
176	C₂×C₄	—	536	1	no	—
177	C₂×C₄	—	536	2	no	—
178	C₂×C₄	—	536	1	no	—
179	C₂×C₄	#14524018	1	1	yes	characteristic
180	C₂×C₄	#14524025	1	2	yes	—
181	C₂×C₄	#14524027	1	1	yes	characteristic
182	C₂×C₄	#14524038	1	2	yes	—
183	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
184	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
185	C₂×C₄	#14524071	1	1	yes	characteristic
186	C₂×C₄	#14524076	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	536	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	#16863323	1	1	yes	characteristic
193	C₂²	#16863743	1	1	yes	characteristic
194	C₂²	#16863756	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	536	2	no	—
198	C₄	—	1072	1	no	—
199	C₄	#16863592	1	1	yes	characteristic
200	C₄	#16863602	1	1	yes	characteristic
201	C₄	#16863710	1	1	yes	characteristic
202	C₄	#16863715	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#22561614	1	1	yes	characteristic
206	C₂	#22561675	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561029

C₂

yes

maximal, characteristic

#22560384

C₂

yes

maximal, characteristic

#22558880

C₂

yes

maximal

#22558068

C₂

yes

maximal

C₆₇×#1281

C₂

yes

maximal, characteristic

C₂×#14524043

C₂²

yes

—

—

C₂²

yes

—

C₂×#14523968

—

C₂×#14523968

—

#16863262

C₂²

yes

characteristic

#16863141

—

#431×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂²×Dic₁₃₄

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523979

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523968

—

#14523968

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#123

yes

characteristic

C₂×C₄×C₆₇

#126

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

#10632

—

maximal, Sylow

100

C₂×C₆₇

—

101

C₂×C₆₇

—

102

C₂×C₆₇

#2071

yes

characteristic

103

C₂×C₆₇

#2082

yes

—

104

#1960

—

105

#1849

—

106

#1546

—

107

#1286

—

108

#1281

D₆₇

yes

characteristic

109

C₆₇

#10632

yes

characteristic, complemented, Sylow

110

C₂×C₄⋊Q₈

—

111

#431

C₂×D₆₇

yes

—

112

C₂×#108

—

134

—

113

C₂×#108

—

134

—

114

#423

—

115

C₂×#103

—

134

—

116

C₂×#103

—

134

—

117

C₂×#97

C₂×D₆₇

yes

characteristic

118

#382

—

119

#332

—

134

—

120

C₂²×Q₈

—

134

—

121

C₂×M₁₆

—

134

—

122

C₂²×C₈

D₂₆₈

yes

characteristic

123

C₂²×C₈

D₂₆₈

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

134

—

126

C₄⋊Q₈

—

134

—

127

C₂×C₄⋊C₄

—

134

—

128

C₂×C₄⋊C₄

—

134

—

129

C₂×C₄²

C₂²×D₆₇

yes

characteristic

130

#108

—

268

—

131

#108

—

268

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

—

134

C₄⋊C₈

—

134

—

135

#103

—

268

—

136

#103

—

268

—

137

C₂²⋊C₈

—

138

#97

—

139

C₄×C₈

—

140

#95

—

134

—

141

C₂×Q₈

—

268

—

142

C₂×Q₈

—

268

—

143

C₂×Q₈

—

268

—

144

C₂×Q₈

—

268

—

145

C₂×Q₈

—

268

—

146

C₂×Q₈

—

134

—

147

C₂²×C₄

—

268

—

148

C₂²×C₄

—

268

—

149

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

150

C₂²×C₄

#11761628

yes

—

151

M₁₆

—

268

—

152

M₁₆

—

268

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

268

—

159

C₂×C₈

D₅₃₆

yes

—

160

C₂×C₈

#11761617

yes

—

161

C₄⋊C₄

—

268

—

162

C₄⋊C₄

—

268

—

163

C₄⋊C₄

—

268

—

164

C₄⋊C₄

—

134

—

165

C₄²

D₄×D₆₇

yes

characteristic

166

C₄²

#11761627

yes

—

167

C₄²

#11761629

yes

characteristic

168

C₂³

#14524031

yes

characteristic

169

Q₈

—

268

—

170

Q₈

—

268

—

171

Q₈

—

536

—

172

Q₈

—

536

—

173

C₂×C₄

—

174

C₂×C₄

—

268

—

175

C₂×C₄

—

536

—

176

C₂×C₄

—

536

—

177

C₂×C₄

—

536

—

178

C₂×C₄

—

536

—

179

C₂×C₄

#14524018

yes

characteristic

180

C₂×C₄

#14524025

yes

—

181

C₂×C₄

#14524027

yes

characteristic

182

C₂×C₄

#14524038

yes

—

183

C₂×C₄

C₂×D₅₃₆

yes

characteristic

184

C₂×C₄

C₂×#11761617

yes

characteristic

185

C₂×C₄

#14524071

yes

characteristic

186

C₂×C₄

#14524076

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

536

—

191

C₂²

—

192

C₂²

#16863323

yes

characteristic

193

C₂²

#16863743

yes

characteristic

194

C₂²

#16863756

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

536

—

198

C₄

—

1072

—

199

C₄

#16863592

yes

characteristic

200

C₄

#16863602

yes

characteristic

201

C₄

#16863710

yes

characteristic

202

C₄

#16863715

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

#22561614

yes

characteristic

206

C₂

#22561675

yes

—

	subgroup	G/H
center	C₂²	#16863323
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1281	C₂
socle	C₂²×C₆₇	#397
prime radical	C₂³×C₆₇	#124
hypercenter	#1281	D₆₇
hypocenter	C₆₇	#10632
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#16863323

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1281

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
109. C₆₇	#10632	semidirect

H ≅ G/N

type

109. C₆₇

#10632

semidirect

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

subgroup	index
#22561029	2
#22560384	2
#22558880	2
#22558068	2
C₆₇×#1281	2
#10632	67

subgroup

index

C₆₇×#1281

Subgroups — #30760626

Conjugacy classes of subgroups

Complements

Maximal subgroups