Group #30775947 — subgroups

Subgroups — #30775947

Order 17152.

318 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16863220	C₂	1	2	yes	maximal
2	C₂×#16863211	C₂	1	2	yes	maximal
3	#22561080	C₂	1	1	yes	maximal, characteristic
4	#22558293	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
7	C₂×#14523983	C₂²	1	2	yes	—
8	C₂×#14523975	C₂²	1	1	yes	characteristic
9	C₂×#14523968	—	2	1	no	—
10	#16863302	—	2	1	no	—
11	#16863220	—	2	4	no	—
12	#16863211	—	2	4	no	—
13	C₂×#108×C₆₇	—	2	1	no	—
14	C₂×#108×C₆₇	C₂²	1	2	yes	—
15	#423×C₆₇	—	2	1	no	—
16	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₃₄	—	2	1	no	—
18	C₂×Dic₂₆₈	—	2	2	no	—
19	C₂×Dic₂₆₈	—	2	2	no	—
20	C₂×Dic₂₆₈	—	2	1	no	—
21	C₂×Dic₂₆₈	Q₈	1	2	yes	—
22	C₂×C₄⋊Dic₆₇	—	2	1	no	—
23	#14523983	—	2	2	no	—
24	#14523983	D₄	1	4	yes	—
25	C₂×C₆₇⋊C₁₆	—	2	2	no	—
26	#14523975	—	2	2	no	—
27	#14523968	—	4	2	no	—
28	C₆₇⋊#95	—	4	2	no	—
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	#108×C₆₇	—	2	2	no	—
34	#108×C₆₇	—	2	2	no	—
35	#108×C₆₇	—	4	2	no	—
36	C₄⋊C₈×C₆₇	—	4	2	no	—
37	#97×C₆₇	—	2	2	no	—
38	C₂×Dic₁₃₄	—	2	2	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	C₂×Dic₁₃₄	—	4	1	no	—
44	C₂²×Dic₆₇	—	4	1	no	—
45	C₂²×Dic₆₇	—	4	1	no	—
46	Dic₂₆₈	—	2	4	no	—
47	Dic₂₆₈	—	4	2	no	—
48	C₄⋊Dic₆₇	—	4	1	no	—
49	C₄⋊Dic₆₇	—	4	1	no	—
50	C₆₇⋊C₁₆	—	2	4	no	—
51	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	M₁₆×C₆₇	—	4	2	no	—
54	M₁₆×C₆₇	—	4	2	no	—
55	C₂×C₈×C₆₇	—	2	1	no	—
56	C₂×C₈×C₆₇	—	2	1	no	—
57	C₂×C₈×C₆₇	—	4	2	no	—
58	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
59	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₆₇	#41	1	2	yes	—
62	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
63	C₄²×C₆₇	—	2	1	no	—
64	C₄²×C₆₇	—	2	1	no	—
65	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	4	2	no	—
68	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#127	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	AGL(1,ℤ/8)	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	—	8	2	no	—
85	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
86	C₈×C₆₇	#123	1	2	yes	—
87	C₂²×C₆₇	—	2	2	no	—
88	C₂²×C₆₇	#476	1	1	yes	characteristic
89	C₂²×C₆₇	#481	1	1	yes	characteristic
90	C₂²×C₆₇	#492	1	1	yes	characteristic
91	C₄×C₆₇	—	4	2	no	—
92	C₄×C₆₇	#399	1	1	yes	characteristic
93	C₄×C₆₇	#476	1	1	yes	characteristic
94	C₄×C₆₇	#481	1	1	yes	characteristic
95	C₄×C₆₇	#496	1	1	yes	characteristic
96	Dic₆₇	—	8	2	no	—
97	Dic₆₇	—	16	1	no	—
98	#13536	—	67	1	no	maximal, Sylow
99	C₂×C₆₇	—	2	1	no	—
100	C₂×C₆₇	—	2	1	no	—
101	C₂×C₆₇	#2094	1	1	yes	characteristic
102	C₂×C₆₇	#2239	1	2	yes	—
103	C₂×#365	—	67	2	no	—
104	C₂×#362	—	67	2	no	—
105	#1971	—	67	1	no	—
106	#1849	D₆₇	1	1	yes	characteristic
107	#1377	—	67	1	no	—
108	C₆₇	#13536	1	1	yes	characteristic, complemented, Sylow
109	C₂²×Q₁₆	—	67	1	no	—
110	C₂×M₃₂	—	67	2	no	—
111	C₂×#108	—	2	1	no	—
112	C₂×#108	C₂×D₆₇	1	2	yes	—
113	C₂×#106	—	67	1	no	—
114	#423	—	2	1	no	—
115	C₂×#103	—	134	1	no	—
116	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
117	#391	—	134	1	no	—
118	#365	—	134	4	no	—
119	#362	—	134	4	no	—
120	C₂²×Q₈	—	134	1	no	—
121	C₂×Q₁₆	—	67	2	no	—
122	C₂×Q₁₆	—	134	2	no	—
123	C₂×Q₁₆	—	134	2	no	—
124	C₂×Q₁₆	—	134	1	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	134	1	no	—
129	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
130	M₃₂	—	67	4	no	—
131	M₃₂	—	134	2	no	—
132	C₂×C₁₆	—	134	2	no	—
133	#108	—	2	2	no	—
134	#108	—	2	2	no	—
135	#108	—	4	2	no	—
136	#106	—	134	2	no	—
137	C₄⋊C₈	—	4	2	no	—
138	#103	—	268	2	no	—
139	#97	—	2	2	no	—
140	#95	—	268	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	#11761615	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
151	Q₁₆	—	268	2	no	—
152	Q₁₆	—	134	4	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	#11761615	1	2	yes	—
159	C₂×C₈	D₄×D₆₇	1	2	yes	—
160	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
161	C₂×C₈	#11761628	1	2	yes	—
162	C₂×C₈	#11761629	1	1	yes	characteristic
163	C₄⋊C₄	—	268	1	no	—
164	C₄⋊C₄	—	268	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₁₆	—	134	4	no	—
168	C₂³	C₂×#11761615	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	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	268	2	no	—
176	C₂×C₄	—	536	1	no	—
177	C₂×C₄	—	536	1	no	—
178	C₂×C₄	—	536	2	no	—
179	C₂×C₄	—	536	1	no	—
180	C₂×C₄	#14524019	1	1	yes	characteristic
181	C₂×C₄	#14524026	1	1	yes	characteristic
182	C₂×C₄	#14524039	1	1	yes	characteristic
183	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
184	C₂×C₄	#14524069	1	1	yes	characteristic
185	C₂×C₄	#14524069	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	8	2	no	—
188	C₈	#14524021	1	2	yes	—
189	C₈	#14524028	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	#16863683	1	1	yes	characteristic
192	C₂²	#16863714	1	1	yes	characteristic
193	C₂²	#16863770	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	536	2	no	—
196	C₄	—	1072	1	no	—
197	C₄	#16863331	1	1	yes	characteristic
198	C₄	#16863683	1	1	yes	characteristic
199	C₄	#16863714	1	1	yes	characteristic
200	C₄	#16863783	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#22561743	1	1	yes	characteristic
204	C₂	#22562428	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863220

C₂

yes

maximal

C₂×#16863211

C₂

yes

maximal

#22561080

C₂

yes

maximal, characteristic

#22558293

C₂

yes

maximal, characteristic

C₆₇×#1849

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

C₂²

yes

characteristic

C₂×#14523983

C₂²

yes

—

C₂×#14523975

C₂²

yes

—

—

—

—

C₂×#108×C₆₇

—

C₂×#108×C₆₇

C₂²

yes

—

#423×C₆₇

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

Q₈

yes

—

C₂×C₄⋊Dic₆₇

—

#14523983

—

#14523983

D₄

yes

—

C₂×C₆₇⋊C₁₆

—

#14523975

—

#14523968

—

C₆₇⋊#95

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

#108×C₆₇

—

#108×C₆₇

—

#108×C₆₇

—

C₄⋊C₈×C₆₇

—

#97×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

SD₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

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

C₂²⋊Q₈

yes

—

C₈×C₆₇

#123

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

#13536

—

maximal, Sylow

C₂×C₆₇

—

100

C₂×C₆₇

—

101

C₂×C₆₇

#2094

yes

characteristic

102

C₂×C₆₇

#2239

yes

—

103

C₂×#365

—

104

C₂×#362

—

105

#1971

—

106

#1849

D₆₇

yes

characteristic

107

#1377

—

108

C₆₇

#13536

yes

characteristic, complemented, Sylow

109

C₂²×Q₁₆

—

110

C₂×M₃₂

—

111

C₂×#108

—

112

C₂×#108

C₂×D₆₇

yes

—

113

C₂×#106

—

114

#423

—

115

C₂×#103

—

134

—

116

C₂×#97

C₂×D₆₇

yes

characteristic

117

#391

—

134

—

118

#365

—

134

—

119

#362

—

134

—

120

C₂²×Q₈

—

134

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

134

—

123

C₂×Q₁₆

—

134

—

124

C₂×Q₁₆

—

134

—

125

C₂×M₁₆

—

126

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

127

C₂²×C₈

C₂²×D₆₇

yes

characteristic

128

C₂×C₄⋊C₄

—

134

—

129

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

130

M₃₂

—

131

M₃₂

—

134

—

132

C₂×C₁₆

—

134

—

133

#108

—

134

#108

—

135

#108

—

136

#106

—

134

—

137

C₄⋊C₈

—

138

#103

—

268

—

139

#97

—

140

#95

—

268

—

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

#11761615

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

151

Q₁₆

—

268

—

152

Q₁₆

—

134

—

153

M₁₆

—

154

M₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

#11761615

yes

—

159

C₂×C₈

D₄×D₆₇

yes

—

160

C₂×C₈

Q₈×D₆₇

yes

characteristic

161

C₂×C₈

#11761628

yes

—

162

C₂×C₈

#11761629

yes

characteristic

163

C₄⋊C₄

—

268

—

164

C₄⋊C₄

—

268

—

165

C₄²

—

166

C₄²

—

167

C₁₆

—

134

—

168

C₂³

C₂×#11761615

yes

characteristic

169

Q₈

—

268

—

170

Q₈

—

268

—

171

Q₈

—

536

—

172

Q₈

—

536

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

268

—

176

C₂×C₄

—

536

—

177

C₂×C₄

—

536

—

178

C₂×C₄

—

536

—

179

C₂×C₄

—

536

—

180

C₂×C₄

#14524019

yes

characteristic

181

C₂×C₄

#14524026

yes

characteristic

182

C₂×C₄

#14524039

yes

characteristic

183

C₂×C₄

C₂×#11761615

yes

characteristic

184

C₂×C₄

#14524069

yes

characteristic

185

C₂×C₄

#14524069

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

#14524021

yes

—

189

C₈

#14524028

yes

—

190

C₂²

—

191

C₂²

#16863683

yes

characteristic

192

C₂²

#16863714

yes

characteristic

193

C₂²

#16863770

yes

characteristic

194

C₄

—

195

C₄

—

536

—

196

C₄

—

1072

—

197

C₄

#16863331

yes

characteristic

198

C₄

#16863683

yes

characteristic

199

C₄

#16863714

yes

characteristic

200

C₄

#16863783

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

#22561743

yes

characteristic

204

C₂

#22562428

yes

—

	subgroup	G/H
center	C₂²	#16863714
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1849	C₂
socle	C₂²×C₆₇	#481
prime radical	C₂³×C₆₇	C₂×SD₁₆
hypercenter	#1849	D₆₇
hypocenter	C₆₇	#13536
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#16863714

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1849

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 ◁ #1849

N	H ≅ G/N	type
108. C₆₇	#13536	semidirect

H ≅ G/N

type

108. C₆₇

#13536

semidirect

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

subgroup	index
C₂×#16863220	2
C₂×#16863211	2
#22561080	2
#22558293	2
C₆₇×#1849	2
#13536	67

subgroup

index

C₆₇×#1849

Subgroups — #30775947

Conjugacy classes of subgroups

Complements

Maximal subgroups