Group #30774393 — subgroups

Subgroups — #30774393

Order 17152.

334 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16863798	C₂	1	1	yes	maximal, characteristic
2	#22560088	C₂	1	1	yes	maximal, characteristic
3	#22558315	C₂	1	2	yes	maximal
4	#22558308	C₂	1	2	yes	maximal
5	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524035	—	2	1	no	—
7	#16863798	—	2	2	no	—
8	#16863798	C₄	1	4	yes	complemented
9	C₂×#14523979	C₂²	1	1	yes	characteristic
10	#16863434	C₂²	1	1	yes	characteristic
11	C₂×#14523977	—	2	1	no	—
12	C₂×#14523975	C₂²	1	2	yes	—
13	#16863351	—	2	1	no	—
14	#16863156	—	2	1	no	—
15	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#364×C₆₇	—	2	2	no	—
18	C₂×#11761612	—	2	1	no	—
19	#14524035	—	2	4	no	—
20	#14524035	—	4	2	no	—
21	#14524002	—	2	1	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	C₂×Dic₆₇⋊C₄	—	4	2	no	—
24	#14523979	—	2	1	no	—
25	#14523979	Q₈	1	2	yes	—
26	#14523977	—	2	4	no	—
27	#14523977	Q₈	1	2	yes	—
28	#14523975	—	2	2	no	—
29	#14523975	—	2	1	no	—
30	#14523975	C₂×C₄	1	4	yes	—
31	#14523953	—	4	1	no	—
32	C₈×Dic₆₇	—	4	1	no	—
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	M₃₂×C₆₇	—	2	2	no	—
37	M₃₂×C₆₇	—	2	2	no	—
38	C₂×C₁₆×C₆₇	—	2	2	no	—
39	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₇	D₄	1	2	yes	—
41	C₂²×Dic₆₇	—	4	2	no	—
42	#11761612	—	2	2	no	—
43	#11761612	—	4	1	no	—
44	C₂×C₆₇⋊C₈	—	4	1	no	—
45	C₄⋊Dic₆₇	—	2	4	no	—
46	C₄⋊Dic₆₇	—	2	2	no	—
47	C₄⋊Dic₆₇	—	4	2	no	—
48	Dic₆₇⋊C₄	—	4	8	no	—
49	C₂²⋊Dic₆₇	—	4	1	no	—
50	C₄×Dic₆₇	—	4	1	no	—
51	C₂²×C₄×C₆₇	—	2	1	no	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
63	C₄²×C₆₇	—	2	1	no	—
64	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₁₆×C₆₇	—	4	2	no	—
66	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	4	2	no	—
69	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×Dic₆₇	—	4	4	no	—
76	C₂×Dic₆₇	—	4	2	no	—
77	C₂×Dic₆₇	—	8	4	no	—
78	C₈×C₆₇	#103	1	2	yes	—
79	C₈×C₆₇	#103	1	2	yes	—
80	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₆₇	#124	1	1	yes	characteristic
83	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
84	C₆₇⋊C₈	—	4	2	no	—
85	C₂²×C₆₇	—	2	2	no	—
86	C₂²×C₆₇	#418	1	1	yes	characteristic
87	C₂²×C₆₇	#476	1	1	yes	characteristic
88	C₂²×C₆₇	#477	1	1	yes	characteristic
89	C₄×C₆₇	—	4	2	no	—
90	C₄×C₆₇	#391	1	1	yes	characteristic
91	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
92	C₄×C₆₇	#476	1	1	yes	characteristic
93	C₄×C₆₇	#477	1	1	yes	characteristic
94	Dic₆₇	—	8	4	no	—
95	Dic₆₇	—	8	2	no	—
96	#13273	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	—	2	1	no	—
98	C₂×C₆₇	—	2	1	no	—
99	C₂×C₆₇	#1987	1	1	yes	characteristic
100	C₂×C₆₇	#2231	1	2	yes	—
101	C₂×#499	—	67	1	no	—
102	C₂×#364	D₆₇	1	1	yes	characteristic
103	#1772	—	67	1	no	—
104	#1386	—	67	2	no	—
105	#1383	—	67	2	no	—
106	C₆₇	#13273	1	1	yes	characteristic, complemented, Sylow
107	C₂×#125	—	134	1	no	—
108	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
109	#499	—	67	4	no	—
110	#499	—	134	2	no	—
111	C₂×#108	—	67	1	no	—
112	#427	—	67	1	no	—
113	C₂×#107	—	134	1	no	—
114	C₂×#106	—	67	2	no	—
115	#405	—	134	1	no	—
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	#364	—	2	2	no	—
118	#339	—	134	1	no	—
119	C₂×M₁₆	—	134	1	no	—
120	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
122	#125	—	268	2	no	—
123	#125	—	134	4	no	—
124	#117	—	134	1	no	—
125	C₂×C₄⋊C₄	—	268	2	no	—
126	C₂×C₄⋊C₄	—	134	2	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	—	67	2	no	—
132	#108	—	134	1	no	—
133	#107	—	67	2	no	—
134	#107	—	134	4	no	—
135	#106	—	67	4	no	—
136	#106	—	134	2	no	—
137	#106	—	134	1	no	—
138	#97	—	268	1	no	—
139	C₄×C₈	—	268	1	no	—
140	C₄×C₈	C₄×D₆₇	1	2	yes	—
141	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	268	2	no	—
144	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
145	M₁₆	—	268	1	no	—
146	M₁₆	—	134	2	no	—
147	C₂×C₈	—	268	1	no	—
148	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
149	C₂×C₈	#11761617	1	1	yes	characteristic
150	C₂×C₈	#11761617	1	1	yes	characteristic
151	C₂×C₈	Dic₂₆₈	1	2	yes	—
152	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
154	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
156	C₂×C₈	#11761627	1	1	yes	characteristic
157	C₂×C₈	#11761627	1	1	yes	characteristic
158	C₄⋊C₄	—	268	8	no	—
159	C₄⋊C₄	—	268	2	no	—
160	C₄⋊C₄	—	134	4	no	—
161	C₄⋊C₄	—	134	2	no	—
162	C₂²⋊C₄	—	268	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	268	1	no	—
165	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
166	C₁₆	—	4	2	no	—
167	C₂³	#14524071	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	268	4	no	—
171	C₂×C₄	—	268	2	no	—
172	C₂×C₄	—	536	4	no	—
173	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	#14524023	1	1	yes	characteristic
175	C₂×C₄	#14524023	1	1	yes	characteristic
176	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
177	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
178	C₂×C₄	#14524071	1	1	yes	characteristic
179	C₈	—	268	2	no	—
180	C₈	#14523968	1	2	yes	—
181	C₈	#14523968	1	2	yes	—
182	C₈	#14524008	1	1	yes	characteristic
183	C₈	#14524008	1	1	yes	characteristic
184	C₈	#14524030	1	1	yes	characteristic
185	C₈	#14524041	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#16863402	1	1	yes	characteristic
188	C₂²	#16863687	1	1	yes	characteristic
189	C₂²	#16863692	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	536	4	no	—
192	C₄	—	536	2	no	—
193	C₄	#16863301	1	1	yes	characteristic
194	C₄	C₂×#14523968	1	1	yes	characteristic
195	C₄	#16863687	1	1	yes	characteristic
196	C₄	#16863692	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#22561156	1	1	yes	characteristic
200	C₂	#22562386	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863798

C₂

yes

maximal, characteristic

#22560088

C₂

yes

maximal, characteristic

#22558315

C₂

yes

maximal

#22558308

C₂

yes

maximal

C₂×#364×C₆₇

C₂

yes

maximal, characteristic

C₂×#14524035

—

#16863798

—

#16863798

C₄

yes

complemented

C₂×#14523979

C₂²

yes

characteristic

#16863434

C₂²

yes

characteristic

C₂×#14523977

—

C₂×#14523975

C₂²

yes

—

#16863351

—

#16863156

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#364×C₆₇

—

—

—

—

—

C₂×C₄⋊Dic₆₇

—

C₂×Dic₆₇⋊C₄

—

#14523979

—

#14523979

Q₈

yes

—

#14523977

—

#14523977

Q₈

yes

—

#14523975

—

#14523975

—

#14523975

C₂×C₄

yes

—

#14523953

—

C₈×Dic₆₇

—

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

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

characteristic

C₂×C₈×C₆₇

SD₁₆

yes

characteristic

C₂×C₈×C₆₇

Q₁₆

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

C₂×C₈×C₆₇

yes

C₂×C₈×C₆₇

yes

C₄²×C₆₇

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

#103

yes

—

C₈×C₆₇

#103

yes

—

C₈×C₆₇

C₄×Q₈

yes

characteristic

C₈×C₆₇

C₄×Q₈

yes

characteristic

C₈×C₆₇

#124

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

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

—

#13273

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1987

yes

characteristic

100

C₂×C₆₇

#2231

yes

—

101

C₂×#499

—

102

C₂×#364

D₆₇

yes

characteristic

103

#1772

—

104

#1386

—

105

#1383

—

106

C₆₇

#13273

yes

characteristic, complemented, Sylow

107

C₂×#125

—

134

—

108

C₂×M₃₂

C₂×D₆₇

yes

—

109

#499

—

110

#499

—

134

—

111

C₂×#108

—

112

#427

—

113

C₂×#107

—

134

—

114

C₂×#106

—

115

#405

—

134

—

116

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

117

#364

—

118

#339

—

134

—

119

C₂×M₁₆

—

134

—

120

C₂²×C₈

D₂₆₈

yes

characteristic

121

C₂²×C₈

C₂²×D₆₇

yes

characteristic

122

#125

—

268

—

123

#125

—

134

—

124

#117

—

134

—

125

C₂×C₄⋊C₄

—

268

—

126

C₂×C₄⋊C₄

—

134

—

127

C₂×C₄²

D₂₆₈

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

134

—

133

#107

—

134

#107

—

134

—

135

#106

—

136

#106

—

134

—

137

#106

—

134

—

138

#97

—

268

—

139

C₄×C₈

—

268

—

140

C₄×C₈

C₄×D₆₇

yes

—

141

C₄×C₈

C₂²⋊D₆₇

yes

—

142

C₂²×C₄

—

143

C₂²×C₄

—

268

—

144

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

145

M₁₆

—

268

—

146

M₁₆

—

134

—

147

C₂×C₈

—

268

—

148

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

149

C₂×C₈

#11761617

yes

characteristic

150

C₂×C₈

#11761617

yes

characteristic

151

C₂×C₈

Dic₂₆₈

yes

—

152

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

153

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

154

C₂×C₈

Q₈×D₆₇

yes

characteristic

155

C₂×C₈

Q₈×D₆₇

yes

characteristic

156

C₂×C₈

#11761627

yes

characteristic

157

C₂×C₈

#11761627

yes

characteristic

158

C₄⋊C₄

—

268

—

159

C₄⋊C₄

—

268

—

160

C₄⋊C₄

—

134

—

161

C₄⋊C₄

—

134

—

162

C₂²⋊C₄

—

268

—

163

C₄²

—

164

C₄²

—

268

—

165

C₄²

(C₂×D₆₇)⋊C₄

yes

—

166

C₁₆

—

167

C₂³

#14524071

yes

characteristic

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

268

—

171

C₂×C₄

—

268

—

172

C₂×C₄

—

536

—

173

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

174

C₂×C₄

#14524023

yes

characteristic

175

C₂×C₄

#14524023

yes

characteristic

176

C₂×C₄

C₂×#11761617

yes

characteristic

177

C₂×C₄

C₂×Dic₂₆₈

yes

characteristic

178

C₂×C₄

#14524071

yes

characteristic

179

C₈

—

268

—

180

C₈

#14523968

yes

—

181

C₈

#14523968

yes

—

182

C₈

#14524008

yes

characteristic

183

C₈

#14524008

yes

characteristic

184

C₈

#14524030

yes

characteristic

185

C₈

#14524041

yes

characteristic

186

C₂²

—

187

C₂²

#16863402

yes

characteristic

188

C₂²

#16863687

yes

characteristic

189

C₂²

#16863692

yes

characteristic

190

C₄

—

191

C₄

—

536

—

192

C₄

—

536

—

193

C₄

#16863301

yes

characteristic

194

C₄

C₂×#14523968

yes

characteristic

195

C₄

#16863687

yes

characteristic

196

C₄

#16863692

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#22561156

yes

characteristic

200

C₂

#22562386

yes

—

	subgroup	G/H
center	C₂²	#16863687
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	#13273
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

subgroup

G/H

center

C₂²

#16863687

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#364×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

N	H ≅ G/N	type
8. #16863798	C₄	semidirect
106. C₆₇	#13273	semidirect

H ≅ G/N

type

8. #16863798

C₄

semidirect

106. C₆₇

#13273

semidirect

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

subgroup	index
C₂×#16863798	2
#22560088	2
#22558315	2
#22558308	2
C₂×#364×C₆₇	2
#13273	67

subgroup

index

C₂×#364×C₆₇

Subgroups — #30774393

Conjugacy classes of subgroups

Complements

Maximal subgroups