Group #22560849 — subgroups

Subgroups — #22560849

Order 8576.

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

normal subgroups only

This group has 254 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×#14523968	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523966	C₂	1	1	yes	maximal, characteristic
3	#16863301	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863158	C₂	1	1	yes	maximal, characteristic
6	#16863150	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₆₇	—	2	1	no	—
12	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
13	#14523968	C₄	1	4	yes	complemented
14	#14523966	—	2	2	no	—
15	C₆₇⋊#95	—	2	1	no	—
16	C₆₇⋊#95	—	2	1	no	—
17	C₆₇⋊#95	—	2	1	no	—
18	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	#107×C₆₇	—	2	2	no	—
22	C₂×Dic₁₃₄	—	2	1	no	—
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₁₃₄	—	2	1	no	—
25	C₂×Dic₁₃₄	—	2	1	no	—
26	C₂×Dic₁₃₄	—	2	1	no	—
27	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
28	C₂²×Dic₆₇	—	2	1	no	—
29	C₂²×Dic₆₇	—	2	1	no	—
30	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
31	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
32	C₂×C₆₇⋊C₈	—	2	1	no	—
33	C₂×C₆₇⋊C₈	—	2	1	no	—
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₄⋊Dic₆₇	—	2	1	no	—
36	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
37	C₄×Dic₆₇	—	2	1	no	—
38	C₄×Dic₆₇	—	2	1	no	—
39	C₄×Dic₆₇	—	4	2	no	—
40	C₂²×C₄×C₆₇	—	2	1	no	—
41	C₂²×C₄×C₆₇	—	2	1	no	—
42	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₆₇	—	2	1	no	—
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄⋊C₄×C₆₇	—	2	1	no	—
47	C₄⋊C₄×C₆₇	—	2	1	no	—
48	C₄⋊C₄×C₆₇	—	2	1	no	—
49	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
50	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	4	1	no	—
53	C₂×C₄×C₆₇	—	4	1	no	—
54	C₂×C₄×C₆₇	—	4	1	no	—
55	C₂×C₄×C₆₇	—	4	1	no	—
56	C₂×C₄×C₆₇	—	4	1	no	—
57	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
63	C₂×Dic₆₇	—	2	1	no	—
64	C₂×Dic₆₇	—	2	1	no	—
65	C₂×Dic₆₇	—	2	2	no	—
66	C₂×Dic₆₇	—	2	2	no	—
67	C₂×Dic₆₇	—	2	1	no	—
68	C₂×Dic₆₇	—	2	1	no	—
69	C₂×Dic₆₇	—	2	1	no	—
70	C₂×Dic₆₇	—	4	1	no	—
71	C₂×Dic₆₇	—	4	1	no	—
72	C₂×Dic₆₇	—	4	1	no	—
73	C₂×Dic₆₇	—	4	1	no	—
74	C₂×Dic₆₇	Q₁₆	1	2	yes	—
75	Dic₁₃₄	—	2	2	no	—
76	Dic₁₃₄	—	2	2	no	—
77	Dic₁₃₄	—	4	1	no	—
78	Dic₁₃₄	—	4	1	no	—
79	C₈×C₆₇	—	2	2	no	—
80	C₆₇⋊C₈	—	4	1	no	—
81	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
82	C₂²×C₆₇	#121	1	1	yes	characteristic
83	C₂²×C₆₇	#123	1	1	yes	characteristic
84	C₂²×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
85	C₂²×C₆₇	#135	1	1	yes	characteristic
86	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂²×C₆₇	#137	1	1	yes	characteristic
88	C₄×C₆₇	—	4	2	no	—
89	C₄×C₆₇	—	8	1	no	—
90	C₄×C₆₇	#117	1	1	yes	characteristic
91	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₄×C₆₇	#124	1	1	yes	characteristic
93	C₄×C₆₇	#126	1	1	yes	characteristic
94	Dic₆₇	—	2	2	no	—
95	Dic₆₇	—	4	2	no	—
96	Dic₆₇	—	4	2	no	—
97	Dic₆₇	—	4	1	no	—
98	C₂×C₆₇	#388	1	1	yes	characteristic
99	C₂×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
100	C₂×C₆₇	#440	1	1	yes	characteristic
101	C₂×C₆₇	#462	1	1	yes	characteristic
102	C₂×C₆₇	#463	1	1	yes	characteristic
103	C₂×C₆₇	#484	1	1	yes	characteristic
104	C₂×C₆₇	#485	1	1	yes	characteristic
105	#1928	—	67	1	no	maximal, Sylow
106	C₆₇	#1928	1	1	yes	characteristic, complemented, Sylow
107	C₂×#107	D₆₇	1	1	yes	characteristic
108	C₂×#103	—	67	1	no	—
109	C₂×#103	—	67	1	no	—
110	#391	—	67	1	no	—
111	C₄×C₄⋊C₄	—	67	1	no	—
112	#340	—	67	1	no	—
113	#336	—	67	1	no	—
114	C₂²×Q₈	—	67	1	no	—
115	C₂²×C₈	—	67	1	no	—
116	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	67	1	no	—
118	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
120	C₂×C₄²	—	67	1	no	—
121	C₂×C₄²	—	134	1	no	—
122	#107	—	2	2	no	—
123	#103	—	67	4	no	—
124	#103	—	134	2	no	—
125	#95	—	134	1	no	—
126	#95	—	134	1	no	—
127	#95	—	134	1	no	—
128	C₂×Q₈	—	67	2	no	—
129	C₂×Q₈	—	134	1	no	—
130	C₂×Q₈	—	134	1	no	—
131	C₂×Q₈	—	134	1	no	—
132	C₂×Q₈	—	134	1	no	—
133	C₂×Q₈	—	134	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	67	1	no	—
137	C₂²×C₄	—	67	1	no	—
138	C₂²×C₄	—	134	1	no	—
139	C₂²×C₄	—	134	1	no	—
140	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	134	1	no	—
144	C₂×C₈	—	134	1	no	—
145	C₂×C₈	—	134	1	no	—
146	C₂×C₈	C₄×D₆₇	1	2	yes	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	67	2	no	—
151	C₄⋊C₄	—	134	1	no	—
152	C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
153	C₄²	—	268	2	no	—
154	C₄²	—	134	1	no	—
155	C₄²	—	134	1	no	—
156	C₂³	D₄×D₆₇	1	1	yes	characteristic
157	Q₈	—	268	1	no	—
158	Q₈	—	268	1	no	—
159	Q₈	—	134	2	no	—
160	Q₈	—	134	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	268	1	no	—
168	C₂×C₄	—	268	1	no	—
169	C₂×C₄	—	268	1	no	—
170	C₂×C₄	—	268	1	no	—
171	C₂×C₄	—	67	2	no	—
172	C₂×C₄	—	134	2	no	—
173	C₂×C₄	—	134	2	no	—
174	C₂×C₄	—	134	1	no	—
175	C₂×C₄	—	134	1	no	—
176	C₂×C₄	—	134	1	no	—
177	C₂×C₄	—	134	1	no	—
178	C₂×C₄	—	134	1	no	—
179	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂×C₄	#11761627	1	1	yes	characteristic
182	C₂×C₄	#11761628	1	1	yes	characteristic
183	C₂×C₄	#11761629	1	1	yes	characteristic
184	C₂×C₄	#11761629	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	268	1	no	—
187	C₂²	#14524005	1	1	yes	characteristic
188	C₂²	#14524015	1	1	yes	characteristic
189	C₂²	#14524028	1	1	yes	characteristic
190	C₂²	Q₁₆×D₆₇	1	1	yes	characteristic
191	C₂²	#14524062	1	1	yes	characteristic
192	C₂²	#14524067	1	1	yes	characteristic
193	C₂²	#14524075	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	8	1	no	—
196	C₄	—	268	2	no	—
197	C₄	—	268	2	no	—
198	C₄	—	268	1	no	—
199	C₄	—	134	2	no	—
200	C₄	#14524001	1	1	yes	characteristic
201	C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
202	C₄	#14524032	1	1	yes	characteristic
203	C₄	#14524038	1	1	yes	characteristic
204	C₂	#16863286	1	1	yes	characteristic
205	C₂	#16863490	1	1	yes	characteristic
206	C₂	#16863495	1	1	yes	characteristic
207	C₂	#16863604	1	1	yes	characteristic
208	C₂	#16863609	1	1	yes	characteristic
209	C₂	#16863733	1	1	yes	characteristic
210	C₂	#16863739	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523968

C₂

yes

maximal, characteristic

C₂×#14523966

C₂

yes

maximal, characteristic

#16863301

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₆₇

C₂

yes

maximal, characteristic

#16863158

C₂

yes

maximal, characteristic

#16863150

C₂

yes

maximal, characteristic

C₂×#107×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523968

C₄

yes

complemented

#14523966

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

#107×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

characteristic

C₂²×Dic₆₇

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

Q₁₆

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×D₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

C₂×Q₁₆

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#388

yes

characteristic

C₂×C₆₇

C₄×Q₁₆

yes

characteristic

100

C₂×C₆₇

#440

yes

characteristic

101

C₂×C₆₇

#462

yes

characteristic

102

C₂×C₆₇

#463

yes

characteristic

103

C₂×C₆₇

#484

yes

characteristic

104

C₂×C₆₇

yes

105

—

106

C₆₇

#1928

yes

characteristic, complemented, Sylow

107

C₂×#107

D₆₇

yes

characteristic

108

C₂×#103

—

109

C₂×#103

—

110

#391

—

111

C₄×C₄⋊C₄

—

112

#340

—

113

#336

—

114

C₂²×Q₈

—

115

C₂²×C₈

—

116

C₂²×C₈

C₂×D₆₇

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

120

C₂×C₄²

—

121

C₂×C₄²

—

134

—

122

#107

—

123

#103

—

124

#103

—

134

—

125

#95

—

134

—

126

#95

—

134

—

127

#95

—

134

—

128

C₂×Q₈

—

129

C₂×Q₈

—

134

—

130

C₂×Q₈

—

134

—

131

C₂×Q₈

—

134

—

132

C₂×Q₈

—

134

—

133

C₂×Q₈

—

134

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

134

—

139

C₂²×C₄

—

134

—

140

C₂²×C₄

C₂²×D₆₇

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

134

—

144

C₂×C₈

—

134

—

145

C₂×C₈

—

134

—

146

C₂×C₈

C₄×D₆₇

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

134

—

152

C₄⋊C₄

C₂²⋊D₆₇

yes

—

153

C₄²

—

268

—

154

C₄²

—

134

—

155

C₄²

—

134

—

156

C₂³

D₄×D₆₇

yes

characteristic

157

Q₈

—

268

—

158

Q₈

—

268

—

159

Q₈

—

134

—

160

Q₈

—

134

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

268

—

168

C₂×C₄

—

268

—

169

C₂×C₄

—

268

—

170

C₂×C₄

—

268

—

171

C₂×C₄

—

172

C₂×C₄

—

134

—

173

C₂×C₄

—

134

—

174

C₂×C₄

—

134

—

175

C₂×C₄

—

134

—

176

C₂×C₄

—

134

—

177

C₂×C₄

—

134

—

178

C₂×C₄

—

134

—

179

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

180

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

181

C₂×C₄

#11761627

yes

characteristic

182

C₂×C₄

#11761628

yes

characteristic

183

C₂×C₄

#11761629

yes

characteristic

184

C₂×C₄

#11761629

yes

characteristic

185

C₈

—

186

C₈

—

268

—

187

C₂²

#14524005

yes

characteristic

188

C₂²

#14524015

yes

characteristic

189

C₂²

#14524028

yes

characteristic

190

C₂²

Q₁₆×D₆₇

yes

characteristic

191

C₂²

#14524062

yes

characteristic

192

C₂²

#14524067

yes

characteristic

193

C₂²

#14524075

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

268

—

197

C₄

—

268

—

198

C₄

—

268

—

199

C₄

—

134

—

200

C₄

#14524001

yes

characteristic

201

C₄

C₄×C₂²⋊D₆₇

yes

characteristic

202

C₄

#14524032

yes

characteristic

203

C₄

#14524038

yes

characteristic

204

C₂

#16863286

yes

characteristic

205

C₂

#16863490

yes

characteristic

206

C₂

#16863495

yes

characteristic

207

C₂

#16863604

yes

characteristic

208

C₂

#16863609

yes

characteristic

209

C₂

#16863733

yes

characteristic

210

C₂

#16863739

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#107×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#107	D₆₇
hypocenter	C₆₇	#1928
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#107×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
13. #14523968	C₄	semidirect
106. C₆₇	#1928	semidirect

H ≅ G/N

type

13. #14523968

C₄

semidirect

106. C₆₇

#1928

semidirect

Subgroups — #22560849

Conjugacy classes of subgroups

Complements