Group #22560939 — subgroups

Subgroups — #22560939

Order 8576.

248 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 107 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×#14523976	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523976	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#16863294	C₂	1	1	yes	maximal, characteristic
5	#16863158	C₂	1	1	yes	maximal, characteristic
6	#16863158	C₂	1	1	yes	maximal, characteristic
7	#389×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₆₇⋊C₈	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	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₄	—	2	1	no	—
13	#14523976	—	2	2	no	—
14	#14523976	C₄	1	4	yes	complemented
15	C₄×C₆₇⋊C₈	—	2	2	no	—
16	C₆₇⋊#95	—	2	1	no	—
17	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#95×C₆₇	—	2	1	no	—
22	C₂²×Dic₆₇	—	2	1	no	—
23	C₂²×Dic₆₇	—	2	1	no	—
24	C₂×C₆₇⋊C₈	—	2	1	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
28	C₂×C₆₇⋊C₈	Q₈	1	2	yes	—
29	C₂×C₆₇⋊C₈	Q₈	1	2	yes	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	—	2	1	no	—
33	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
34	Dic₆₇⋊C₄	—	4	2	no	—
35	C₂²×C₄×C₆₇	—	2	1	no	—
36	C₂²×C₄×C₆₇	—	2	1	no	—
37	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	—	2	1	no	—
41	C₄⋊C₄×C₆₇	—	2	1	no	—
42	C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₄⋊C₄×C₆₇	—	4	2	no	—
44	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₄²×C₆₇	—	2	1	no	—
46	C₄²×C₆₇	—	2	1	no	—
47	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	1	no	—
49	C₂×C₄×C₆₇	—	2	1	no	—
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	—	2	1	no	—
52	C₂×C₄×C₆₇	—	2	2	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	—	4	1	no	—
58	C₂×C₄×C₆₇	D₈	1	2	yes	—
59	C₂×C₄×C₆₇	Q₁₆	1	2	yes	—
60	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
66	C₂×Dic₆₇	—	2	2	no	—
67	C₂×Dic₆₇	—	4	1	no	—
68	C₂×Dic₆₇	—	4	1	no	—
69	C₂×Dic₆₇	—	4	1	no	—
70	C₂×Dic₆₇	—	4	1	no	—
71	C₂×Dic₆₇	—	4	1	no	—
72	C₆₇⋊C₈	—	2	2	no	—
73	C₆₇⋊C₈	C₄⋊C₄	1	4	yes	complemented
74	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
75	C₂²×C₆₇	#121	1	1	yes	characteristic
76	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
77	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
78	C₂²×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂²×C₆₇	#135	1	1	yes	characteristic
80	C₂²×C₆₇	#135	1	1	yes	characteristic
81	C₄×C₆₇	—	2	2	no	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	—	4	2	no	—
84	C₄×C₆₇	—	8	1	no	—
85	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
87	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
88	C₄×C₆₇	#125	1	1	yes	characteristic
89	Dic₆₇	—	4	2	no	—
90	Dic₆₇	—	8	1	no	—
91	C₂×C₆₇	#389	1	1	yes	characteristic
92	C₂×C₆₇	C₄×D₈	1	1	yes	characteristic
93	C₂×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
94	C₂×C₆₇	#466	1	1	yes	characteristic
95	C₂×C₆₇	#467	1	1	yes	characteristic
96	C₂×C₆₇	#499	1	1	yes	characteristic
97	C₂×C₆₇	#500	1	1	yes	characteristic
98	#1942	—	67	1	no	maximal, Sylow
99	C₆₇	#1942	1	1	yes	characteristic, complemented, Sylow
100	C₂×#107	—	67	1	no	—
101	C₂×#107	—	67	1	no	—
102	C₂×C₄×C₈	—	67	1	no	—
103	#389	—	67	1	no	—
104	#389	D₆₇	1	1	yes	characteristic
105	#340	—	67	1	no	—
106	#340	—	67	1	no	—
107	C₂²×C₈	—	67	1	no	—
108	C₂²×C₈	—	67	1	no	—
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₄⋊C₄	—	67	1	no	—
111	C₂×C₄⋊C₄	—	67	1	no	—
112	C₂×C₄⋊C₄	—	134	1	no	—
113	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
116	#107	—	67	4	no	—
117	#107	—	134	2	no	—
118	C₄×C₈	—	134	2	no	—
119	#95	—	2	1	no	—
120	#95	—	134	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	134	1	no	—
124	C₂²×C₄	—	134	1	no	—
125	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
128	C₂×C₈	—	67	2	no	—
129	C₂×C₈	—	67	2	no	—
130	C₂×C₈	—	67	2	no	—
131	C₂×C₈	—	67	2	no	—
132	C₂×C₈	—	134	1	no	—
133	C₂×C₈	—	134	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	268	2	no	—
139	C₄⋊C₄	—	67	2	no	—
140	C₄⋊C₄	—	134	1	no	—
141	C₄⋊C₄	—	134	1	no	—
142	C₄⋊C₄	—	134	1	no	—
143	C₄⋊C₄	C₄×D₆₇	1	2	yes	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	1	no	—
146	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	268	1	no	—
158	C₂×C₄	—	268	1	no	—
159	C₂×C₄	—	268	1	no	—
160	C₂×C₄	—	268	1	no	—
161	C₂×C₄	—	268	1	no	—
162	C₂×C₄	—	134	2	no	—
163	C₂×C₄	#11761613	1	2	yes	—
164	C₂×C₄	#11761618	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
166	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
167	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
168	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
169	C₂×C₄	#11761627	1	1	yes	characteristic
170	C₂×C₄	#11761628	1	1	yes	characteristic
171	C₈	—	67	4	no	—
172	C₈	—	134	2	no	—
173	C₂²	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
174	C₂²	#14524015	1	1	yes	characteristic
175	C₂²	#14524041	1	1	yes	characteristic
176	C₂²	C₂×#11761613	1	1	yes	characteristic
177	C₂²	C₂×#11761618	1	1	yes	characteristic
178	C₂²	#14524063	1	1	yes	characteristic
179	C₂²	#14524063	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	268	2	no	—
185	C₄	—	536	1	no	—
186	C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
187	C₄	#14524008	1	1	yes	characteristic
188	C₄	#14524021	1	1	yes	characteristic
189	C₄	#14524033	1	1	yes	characteristic
190	C₂	#16863291	1	1	yes	characteristic
191	C₂	C₄×#11761613	1	1	yes	characteristic
192	C₂	C₄×#11761618	1	1	yes	characteristic
193	C₂	#16863630	1	1	yes	characteristic
194	C₂	#16863635	1	1	yes	characteristic
195	C₂	#16863795	1	1	yes	characteristic
196	C₂	#16863799	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523976

C₂

yes

maximal, characteristic

C₂×#14523976

C₂

yes

maximal, characteristic

C₂×C₄×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#16863294

C₂

yes

maximal, characteristic

#16863158

C₂

yes

maximal, characteristic

#16863158

C₂

yes

maximal, characteristic

#389×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₄

—

#14523976

—

#14523976

C₄

yes

complemented

C₄×C₆₇⋊C₈

—

C₆₇⋊#95

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#95×C₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

Q₈

yes

—

C₂×C₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

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

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

D₈

yes

—

C₂×C₄×C₆₇

Q₁₆

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₂²×C₆₇

C₂×Q₁₆

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₄×Q₈

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

#125

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#389

yes

characteristic

C₂×C₆₇

C₄×D₈

yes

characteristic

C₂×C₆₇

C₄×Q₁₆

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

100

C₂×#107

—

101

C₂×#107

—

102

C₂×C₄×C₈

—

103

#389

—

104

#389

D₆₇

yes

characteristic

105

#340

—

106

#340

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

134

—

113

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

114

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

115

C₂×C₄²

C₂×D₆₇

yes

characteristic

116

#107

—

117

#107

—

134

—

118

C₄×C₈

—

134

—

119

#95

—

120

#95

—

134

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

134

—

124

C₂²×C₄

—

134

—

125

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

126

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

127

C₂²×C₄

C₂²×D₆₇

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

134

—

133

C₂×C₈

—

134

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

268

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

134

—

141

C₄⋊C₄

—

134

—

142

C₄⋊C₄

—

134

—

143

C₄⋊C₄

C₄×D₆₇

yes

—

144

C₄²

—

145

C₄²

—

146

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

268

—

158

C₂×C₄

—

268

—

159

C₂×C₄

—

268

—

160

C₂×C₄

—

268

—

161

C₂×C₄

—

268

—

162

C₂×C₄

—

134

—

163

C₂×C₄

#11761613

yes

—

164

C₂×C₄

#11761618

yes

—

165

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

166

C₂×C₄

D₄×D₆₇

yes

characteristic

167

C₂×C₄

Q₈×D₆₇

yes

characteristic

168

C₂×C₄

Q₈×D₆₇

yes

characteristic

169

C₂×C₄

#11761627

yes

characteristic

170

C₂×C₄

#11761628

yes

characteristic

171

C₈

—

172

C₈

—

134

—

173

C₂²

C₄×C₂²⋊D₆₇

yes

characteristic

174

C₂²

#14524015

yes

characteristic

175

C₂²

#14524041

yes

characteristic

176

C₂²

C₂×#11761613

yes

characteristic

177

C₂²

C₂×#11761618

yes

characteristic

178

C₂²

#14524063

yes

characteristic

179

C₂²

#14524063

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

268

—

185

C₄

—

536

—

186

C₄

C₄⋊C₄×D₆₇

yes

characteristic

187

C₄

#14524008

yes

characteristic

188

C₄

#14524021

yes

characteristic

189

C₄

#14524033

yes

characteristic

190

C₂

#16863291

yes

characteristic

191

C₂

C₄×#11761613

yes

characteristic

192

C₂

C₄×#11761618

yes

characteristic

193

C₂

#16863630

yes

characteristic

194

C₂

#16863635

yes

characteristic

195

C₂

#16863795

yes

characteristic

196

C₂

#16863799

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#389×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#389	D₆₇
hypocenter	C₆₇	#1942
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#389×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
14. #14523976	C₄	semidirect
73. C₆₇⋊C₈	C₄⋊C₄	semidirect
99. C₆₇	#1942	semidirect

H ≅ G/N

type

14. #14523976

C₄

semidirect

73. C₆₇⋊C₈

C₄⋊C₄

semidirect

99. C₆₇

#1942

semidirect

Subgroups — #22560939

Conjugacy classes of subgroups

Complements