Group #30775663 — subgroups

Subgroups — #30775663

Order 17152.

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

normal subgroups only

This group has 302 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	C₂×#16863213	C₂	1	2	yes	maximal
2	#22561080	C₂	1	1	yes	maximal, characteristic
3	#22560384	C₂	1	1	yes	maximal, characteristic
4	#22558308	C₂	1	2	yes	maximal
5	C₆₇×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
7	C₂×#14523979	—	2	1	no	—
8	C₂×#14523979	C₂²	1	2	yes	—
9	C₂×#14523975	C₂²	1	1	yes	characteristic
10	#16863420	—	2	1	no	—
11	C₂×#14523968	—	2	1	no	—
12	#16863302	—	2	1	no	—
13	#16863213	—	2	4	no	—
14	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
15	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₃₄	—	2	1	no	—
17	C₂×Dic₂₆₈	—	2	2	no	—
18	C₂×Dic₂₆₈	—	2	2	no	—
19	C₂×Dic₂₆₈	—	2	1	no	—
20	C₂×Dic₂₆₈	D₄	1	2	yes	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×C₄⋊Dic₆₇	—	2	1	no	—
23	#14523979	—	2	2	no	—
24	#14523979	—	2	2	no	—
25	#14523979	—	4	2	no	—
26	#14523975	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
28	#14523968	—	4	2	no	—
29	C₆₇⋊#95	—	4	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	M₃₂×C₆₇	—	2	2	no	—
34	M₃₂×C₆₇	—	2	2	no	—
35	C₂×C₁₆×C₆₇	—	2	2	no	—
36	#97×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	Dic₂₆₈	—	2	4	no	—
46	Dic₂₆₈	—	4	2	no	—
47	#11761612	—	4	2	no	—
48	#11761612	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₄⋊Dic₆₇	—	4	1	no	—
51	C₄⋊Dic₆₇	—	4	1	no	—
52	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	—	2	1	no	—
56	C₂×C₈×C₆₇	D₈	1	2	yes	—
57	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	#41	1	2	yes	—
59	C₂×C₈×C₆₇	#41	1	2	yes	—
60	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
61	C₄²×C₆₇	—	2	1	no	—
62	C₄²×C₆₇	—	2	1	no	—
63	C₁₆×C₆₇	—	4	2	no	—
64	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
73	C₂×Dic₆₇	—	4	2	no	—
74	C₂×Dic₆₇	—	8	1	no	—
75	C₂×Dic₆₇	—	8	1	no	—
76	C₂×Dic₆₇	—	8	2	no	—
77	C₂×Dic₆₇	—	8	1	no	—
78	Dic₁₃₄	—	4	2	no	—
79	Dic₁₃₄	—	4	2	no	—
80	Dic₁₃₄	—	8	1	no	—
81	Dic₁₃₄	—	8	1	no	—
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	#397	1	1	yes	characteristic
92	C₄×C₆₇	#459	1	1	yes	characteristic
93	C₄×C₆₇	#480	1	1	yes	characteristic
94	C₄×C₆₇	#489	1	1	yes	characteristic
95	Dic₆₇	—	8	2	no	—
96	Dic₆₇	—	16	1	no	—
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	—	67	1	no	—
104	#1849	—	67	1	no	—
105	#1383	—	67	2	no	—
106	#1377	D₆₇	1	1	yes	characteristic
107	C₆₇	#13488	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	—	67	1	no	—
109	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
110	C₂×#108	—	67	2	no	—
111	C₂×#108	—	134	1	no	—
112	C₂×#106	—	67	1	no	—
113	#423	—	134	1	no	—
114	C₂×#103	—	134	1	no	—
115	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
116	#391	—	134	1	no	—
117	#362	—	134	4	no	—
118	C₂²×Q₈	—	134	1	no	—
119	C₂×Q₁₆	—	67	2	no	—
120	C₂×Q₁₆	—	134	2	no	—
121	C₂×Q₁₆	—	134	2	no	—
122	C₂×Q₁₆	—	134	1	no	—
123	C₂×M₁₆	—	134	2	no	—
124	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	134	1	no	—
127	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	268	2	no	—
132	#108	—	134	2	no	—
133	#108	—	134	2	no	—
134	#106	—	134	2	no	—
135	C₄⋊C₈	—	268	2	no	—
136	#103	—	268	2	no	—
137	#97	—	2	2	no	—
138	#95	—	268	2	no	—
139	C₂×Q₈	—	268	1	no	—
140	C₂×Q₈	—	268	1	no	—
141	C₂×Q₈	—	268	1	no	—
142	C₂×Q₈	—	268	1	no	—
143	C₂×Q₈	—	268	1	no	—
144	C₂×Q₈	—	134	2	no	—
145	C₂²×C₄	—	268	1	no	—
146	C₂²×C₄	—	268	1	no	—
147	C₂²×C₄	#11761617	1	2	yes	—
148	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
149	Q₁₆	—	268	2	no	—
150	Q₁₆	—	134	4	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	D₅₃₆	1	2	yes	—
157	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
158	C₂×C₈	#11761627	1	2	yes	—
159	C₂×C₈	#11761628	1	2	yes	—
160	C₂×C₈	#11761629	1	1	yes	characteristic
161	C₄⋊C₄	—	268	1	no	—
162	C₄⋊C₄	—	268	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	4	2	no	—
166	C₂³	C₂×#11761617	1	1	yes	characteristic
167	Q₈	—	268	2	no	—
168	Q₈	—	268	2	no	—
169	Q₈	—	536	1	no	—
170	Q₈	—	536	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	268	2	no	—
174	C₂×C₄	—	536	1	no	—
175	C₂×C₄	—	536	1	no	—
176	C₂×C₄	—	536	2	no	—
177	C₂×C₄	—	536	1	no	—
178	C₂×C₄	#14524018	1	1	yes	characteristic
179	C₂×C₄	#14524027	1	1	yes	characteristic
180	C₂×C₄	#14524031	1	1	yes	characteristic
181	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
182	C₂×C₄	#14524071	1	1	yes	characteristic
183	C₂×C₄	#14524076	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	536	2	no	—
186	C₈	#14524025	1	2	yes	—
187	C₈	#14524038	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#16863602	1	1	yes	characteristic
190	C₂²	#16863715	1	1	yes	characteristic
191	C₂²	#16863743	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	536	2	no	—
194	C₄	—	1072	1	no	—
195	C₄	#16863323	1	1	yes	characteristic
196	C₄	#16863592	1	1	yes	characteristic
197	C₄	#16863710	1	1	yes	characteristic
198	C₄	#16863756	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#22561614	1	1	yes	characteristic
202	C₂	#22562429	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863213

C₂

yes

maximal

#22561080

C₂

yes

maximal, characteristic

#22560384

C₂

yes

maximal, characteristic

#22558308

C₂

yes

maximal

C₆₇×#1377

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

C₂²

yes

characteristic

C₂×#14523979

—

C₂×#14523979

C₂²

yes

—

C₂×#14523975

C₂²

yes

—

—

—

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

D₄

yes

—

C₂×#11761612

—

C₂×C₄⋊Dic₆₇

—

—

—

—

—

C₄⋊(C₆₇⋊C₈)

—

#14523968

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

#97×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

SD₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

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

C₄²×C₆₇

—

C₄²×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

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

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

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

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

#13488

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

100

C₂×C₆₇

#2071

yes

characteristic

101

C₂×C₆₇

#2239

yes

—

102

C₂×#362

—

103

#1971

—

104

#1849

—

105

#1383

—

106

#1377

D₆₇

yes

characteristic

107

C₆₇

#13488

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

—

109

C₂×M₃₂

C₂×D₆₇

yes

—

110

C₂×#108

—

111

C₂×#108

—

134

—

112

C₂×#106

—

113

#423

—

134

—

114

C₂×#103

—

134

—

115

C₂×#97

C₂×D₆₇

yes

characteristic

116

#391

—

134

—

117

#362

—

134

—

118

C₂²×Q₈

—

134

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

134

—

121

C₂×Q₁₆

—

134

—

122

C₂×Q₁₆

—

134

—

123

C₂×M₁₆

—

134

—

124

C₂²×C₈

D₂₆₈

yes

characteristic

125

C₂²×C₈

C₂²×D₆₇

yes

characteristic

126

C₂×C₄⋊C₄

—

134

—

127

C₂×C₄²

D₂₆₈

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

268

—

132

#108

—

134

—

133

#108

—

134

—

134

#106

—

134

—

135

C₄⋊C₈

—

268

—

136

#103

—

268

—

137

#97

—

138

#95

—

268

—

139

C₂×Q₈

—

268

—

140

C₂×Q₈

—

268

—

141

C₂×Q₈

—

268

—

142

C₂×Q₈

—

268

—

143

C₂×Q₈

—

268

—

144

C₂×Q₈

—

134

—

145

C₂²×C₄

—

268

—

146

C₂²×C₄

—

268

—

147

C₂²×C₄

#11761617

yes

—

148

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

149

Q₁₆

—

268

—

150

Q₁₆

—

134

—

151

M₁₆

—

268

—

152

M₁₆

—

268

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

268

—

156

C₂×C₈

D₅₃₆

yes

—

157

C₂×C₈

D₄×D₆₇

yes

characteristic

158

C₂×C₈

#11761627

yes

—

159

C₂×C₈

#11761628

yes

—

160

C₂×C₈

#11761629

yes

characteristic

161

C₄⋊C₄

—

268

—

162

C₄⋊C₄

—

268

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

166

C₂³

C₂×#11761617

yes

characteristic

167

Q₈

—

268

—

168

Q₈

—

268

—

169

Q₈

—

536

—

170

Q₈

—

536

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

268

—

174

C₂×C₄

—

536

—

175

C₂×C₄

—

536

—

176

C₂×C₄

—

536

—

177

C₂×C₄

—

536

—

178

C₂×C₄

#14524018

yes

characteristic

179

C₂×C₄

#14524027

yes

characteristic

180

C₂×C₄

#14524031

yes

characteristic

181

C₂×C₄

C₂×D₅₃₆

yes

characteristic

182

C₂×C₄

#14524071

yes

characteristic

183

C₂×C₄

#14524076

yes

characteristic

184

C₈

—

185

C₈

—

536

—

186

C₈

#14524025

yes

—

187

C₈

#14524038

yes

—

188

C₂²

—

189

C₂²

#16863602

yes

characteristic

190

C₂²

#16863715

yes

characteristic

191

C₂²

#16863743

yes

characteristic

192

C₄

—

193

C₄

—

536

—

194

C₄

—

1072

—

195

C₄

#16863323

yes

characteristic

196

C₄

#16863592

yes

characteristic

197

C₄

#16863710

yes

characteristic

198

C₄

#16863756

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#22561614

yes

characteristic

202

C₂

#22562429

yes

—

	subgroup	G/H
center	C₂²	#16863715
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1377	C₂
socle	C₂²×C₆₇	#481
prime radical	C₂³×C₆₇	C₂×SD₁₆
hypercenter	#1377	D₆₇
hypocenter	C₆₇	#13488
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

subgroup

G/H

center

C₂²

#16863715

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1377

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 ◁ #1377

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×#16863213	2
#22561080	2
#22560384	2
#22558308	2
C₆₇×#1377	2
#13488	67

subgroup

index

C₆₇×#1377

Subgroups — #30775663

Conjugacy classes of subgroups

Complements

Maximal subgroups