Group #22570078 — subgroups

Subgroups — C₂×#16863707 #22570078

Order 8576.

512 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂²×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863707	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14523976	C₂	1	2	yes	maximal, complemented
4	C₂×#14523964	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₆₇	—	2	1	no	—
8	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₆₇	—	2	1	no	—
10	C₂×C₄⋊Dic₆₇	—	2	2	no	—
11	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	complemented
13	#14523976	C₂²	1	8	yes	complemented
14	#14523964	C₂²	1	8	yes	complemented
15	C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	complemented
16	C₂²×D₄×C₆₇	—	2	1	no	—
17	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
19	#121×C₆₇	—	2	2	no	—
20	#121×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
23	C₂²×Dic₆₇	—	2	2	no	—
24	C₂²×Dic₆₇	—	2	4	no	—
25	C₂²×Dic₆₇	—	4	4	no	—
26	C₂×C₆₇⋊C₈	—	2	4	no	—
27	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
28	C₄⋊Dic₆₇	—	2	2	no	—
29	C₄⋊Dic₆₇	—	2	4	no	—
30	C₄⋊Dic₆₇	C₂³	1	4	yes	—
31	C₂⁴×C₆₇	—	2	1	no	—
32	C₂⁴×C₆₇	D₄	1	1	yes	characteristic
33	C₂×D₄×C₆₇	—	2	4	no	—
34	C₂×D₄×C₆₇	—	2	1	no	—
35	C₂×D₄×C₆₇	—	2	4	no	—
36	C₂×D₄×C₆₇	—	4	4	no	—
37	C₂×D₄×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₆₇	—	2	4	no	—
39	C₂²×C₄×C₆₇	—	2	1	no	—
40	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	D₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₆₇	—	2	4	no	—
47	C₂³×C₆₇	—	2	2	no	—
48	C₂³×C₆₇	—	2	2	no	—
49	C₂³×C₆₇	—	2	2	no	—
50	C₂³×C₆₇	—	4	2	no	—
51	C₂³×C₆₇	—	4	2	no	—
52	C₂³×C₆₇	—	4	2	no	—
53	C₂³×C₆₇	D₈	1	4	yes	—
54	C₂³×C₆₇	C₂×D₄	1	2	yes	—
55	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
56	D₄×C₆₇	—	2	4	no	—
57	D₄×C₆₇	—	4	8	no	—
58	D₄×C₆₇	—	4	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	4	no	—
61	C₂×C₄×C₆₇	—	2	4	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	4	2	no	—
64	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#41	1	4	yes	—
67	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₆₇	—	2	4	no	—
69	C₂×Dic₆₇	—	4	8	no	—
70	C₂×Dic₆₇	—	4	4	no	—
71	C₆₇⋊C₈	—	2	4	no	—
72	C₂²×C₆₇	—	2	2	no	—
73	C₂²×C₆₇	—	2	2	no	—
74	C₂²×C₆₇	—	2	8	no	—
75	C₂²×C₆₇	—	4	2	no	—
76	C₂²×C₆₇	—	4	2	no	—
77	C₂²×C₆₇	—	4	2	no	—
78	C₂²×C₆₇	—	8	4	no	—
79	C₂²×C₆₇	#123	1	2	yes	—
80	C₂²×C₆₇	C₂×D₈	1	4	yes	—
81	C₂²×C₆₇	C₂×D₈	1	2	yes	—
82	C₂²×C₆₇	#137	1	2	yes	—
83	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	4	2	no	—
86	C₄×C₆₇	#123	1	2	yes	—
87	C₄×C₆₇	C₂×#41	1	2	yes	—
88	Dic₆₇	—	4	4	no	—
89	C₂×C₆₇	—	2	4	no	—
90	C₂×C₆₇	—	8	2	no	—
91	C₂×C₆₇	#480	1	4	yes	complemented
92	C₂×C₆₇	C₂×#123	1	1	yes	characteristic
93	C₂×C₆₇	C₂²×D₈	1	1	yes	characteristic
94	C₂×C₆₇	C₂×#137	1	1	yes	characteristic
95	C₂×#480	—	67	1	no	maximal, Sylow
96	C₆₇	C₂×#480	1	1	yes	characteristic, complemented, Sylow
97	C₂×#121	D₆₇	1	1	yes	characteristic
98	C₂²×C₄⋊C₄	—	67	1	no	—
99	#480	—	67	8	no	—
100	C₂×#107	—	67	2	no	—
101	C₂×#102	—	67	2	no	—
102	C₂×C₂²⋊C₈	—	67	1	no	—
103	C₂²×D₄	—	2	1	no	—
104	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
105	C₂³×C₄	—	134	1	no	—
106	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
107	C₂²×C₈	—	67	2	no	—
108	#121	—	2	2	no	—
109	#121	C₂×D₆₇	1	4	yes	—
110	C₂×C₄⋊C₄	—	67	2	no	—
111	C₂×C₄⋊C₄	—	67	4	no	—
112	C₂×C₄⋊C₄	—	134	2	no	—
113	C₂×C₄⋊C₄	—	134	1	no	—
114	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
115	C₂×C₂²⋊C₄	—	2	1	no	—
116	#107	—	67	8	no	—
117	#102	—	67	8	no	—
118	C₂²⋊C₈	—	67	4	no	—
119	C₂⁴	—	2	1	no	—
120	C₂⁴	C₂²⋊D₆₇	1	1	yes	characteristic
121	C₂×D₄	—	2	4	no	—
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	—	2	4	no	—
124	C₂×D₄	—	4	4	no	—
125	C₂×D₄	C₂²×D₆₇	1	2	yes	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	268	4	no	—
129	C₂²×C₄	—	134	4	no	—
130	C₂²×C₄	—	134	2	no	—
131	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
132	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
134	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
135	C₂×C₈	—	67	4	no	—
136	C₂×C₈	—	134	4	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	67	4	no	—
139	C₄⋊C₄	—	134	4	no	—
140	C₄⋊C₄	—	134	2	no	—
141	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
142	C₂²⋊C₄	—	2	4	no	—
143	C₂³	—	2	2	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	4	2	no	—
147	C₂³	—	4	2	no	—
148	C₂³	—	4	2	no	—
149	C₂³	#11761613	1	4	yes	—
150	C₂³	C₂×C₂²⋊D₆₇	1	2	yes	—
151	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
152	D₄	—	2	4	no	—
153	D₄	—	4	8	no	—
154	D₄	—	4	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	268	8	no	—
161	C₂×C₄	—	268	4	no	—
162	C₂×C₄	—	134	4	no	—
163	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
164	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂×C₄	#11761627	1	4	yes	—
166	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
167	C₈	—	134	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	8	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	8	4	no	—
175	C₂²	#14524024	1	2	yes	—
176	C₂²	C₂×#11761613	1	4	yes	—
177	C₂²	C₂×#11761613	1	2	yes	—
178	C₂²	#14524073	1	2	yes	—
179	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	268	4	no	—
183	C₄	#14524024	1	2	yes	—
184	C₄	C₂×#11761627	1	2	yes	—
185	C₂	—	2	4	no	—
186	C₂	—	8	2	no	—
187	C₂	#16863707	1	4	yes	direct product factor
188	C₂	C₂×#14524024	1	1	yes	characteristic
189	C₂	C₂²×#11761613	1	1	yes	characteristic
190	C₂	C₂×#14524073	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₆₇

C₂

yes

maximal, characteristic, complemented

#16863707

C₂

yes

maximal, direct product factor

C₂×#14523976

C₂

yes

maximal, complemented

C₂×#14523964

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×#121×C₆₇

C₂

yes

maximal, characteristic

C₂³×Dic₆₇

—

C₂²×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

complemented

#14523976

C₂²

yes

complemented

#14523964

C₂²

yes

complemented

C₂²⋊(C₆₇⋊C₈)

C₂²

yes

complemented

C₂²×D₄×C₆₇

—

C₂²×D₄×C₆₇

C₂²

yes

characteristic

C₂³×C₄×C₆₇

C₂²

yes

characteristic

#121×C₆₇

—

#121×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

C₂⁴×C₆₇

—

C₂⁴×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

D₈

yes

—

C₂³×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#123

yes

—

C₂²×C₆₇

C₂×D₈

yes

—

C₂²×C₆₇

C₂×D₈

yes

—

C₂²×C₆₇

#137

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#123

yes

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

C₂²×D₈

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#121

D₆₇

yes

characteristic

C₂²×C₄⋊C₄

—

#480

—

100

C₂×#107

—

101

C₂×#102

—

102

C₂×C₂²⋊C₈

—

103

C₂²×D₄

—

104

C₂²×D₄

C₂×D₆₇

yes

characteristic

105

C₂³×C₄

—

134

—

106

C₂³×C₄

C₂×D₆₇

yes

characteristic

107

C₂²×C₈

—

108

#121

—

109

#121

C₂×D₆₇

yes

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

134

—

113

C₂×C₄⋊C₄

—

134

—

114

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

115

C₂×C₂²⋊C₄

—

116

#107

—

117

#102

—

118

C₂²⋊C₈

—

119

C₂⁴

—

120

C₂⁴

C₂²⋊D₆₇

yes

characteristic

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

C₂²×D₆₇

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

268

—

129

C₂²×C₄

—

134

—

130

C₂²×C₄

—

134

—

131

C₂²×C₄

C₂²⋊D₆₇

yes

—

132

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

133

C₂²×C₄

C₂²×D₆₇

yes

—

134

C₂²×C₄

C₂²×D₆₇

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

134

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

134

—

140

C₄⋊C₄

—

134

—

141

C₄⋊C₄

C₂²×D₆₇

yes

—

142

C₂²⋊C₄

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

#11761613

yes

—

150

C₂³

C₂×C₂²⋊D₆₇

yes

—

151

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

152

D₄

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

268

—

161

C₂×C₄

—

268

—

162

C₂×C₄

—

134

—

163

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

164

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

165

C₂×C₄

#11761627

yes

—

166

C₂×C₄

C₂³×D₆₇

yes

characteristic

167

C₈

—

134

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#14524024

yes

—

176

C₂²

C₂×#11761613

yes

—

177

C₂²

C₂×#11761613

yes

—

178

C₂²

#14524073

yes

—

179

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

268

—

183

C₄

#14524024

yes

—

184

C₄

C₂×#11761627

yes

—

185

C₂

—

186

C₂

—

187

C₂

#16863707

yes

direct product factor

188

C₂

C₂×#14524024

yes

characteristic

189

C₂

C₂²×#11761613

yes

characteristic

190

C₂

C₂×#14524073

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×#121×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#121	D₆₇
hypocenter	C₆₇	C₂×#480
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#121

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×#121×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#121

N	H ≅ G/N	type
1. C₂²×C₄⋊Dic₆₇	C₂	semidirect
2. #16863707	C₂	direct (N × H)
3. C₂×#14523976	C₂	semidirect
4. C₂×#14523964	C₂	semidirect
5. C₂×C₂²⋊(C₆₇⋊C₈)	C₂	semidirect
12. C₂×C₄⋊Dic₆₇	C₂²	semidirect
13. #14523976	C₂²	semidirect
14. #14523964	C₂²	semidirect
15. C₂²⋊(C₆₇⋊C₈)	C₂²	semidirect
91. C₂×C₆₇	#480	semidirect
96. C₆₇	C₂×#480	semidirect
187. C₂	#16863707	direct (N × H)

H ≅ G/N

type

1. C₂²×C₄⋊Dic₆₇

C₂

semidirect

2. #16863707

C₂

direct (N × H)

3. C₂×#14523976

C₂

semidirect

4. C₂×#14523964

C₂

semidirect

5. C₂×C₂²⋊(C₆₇⋊C₈)

C₂

semidirect

12. C₂×C₄⋊Dic₆₇

C₂²

semidirect

13. #14523976

C₂²

semidirect

14. #14523964

C₂²

semidirect

15. C₂²⋊(C₆₇⋊C₈)

C₂²

semidirect

91. C₂×C₆₇

#480

semidirect

96. C₆₇

C₂×#480

semidirect

187. C₂

#16863707

direct (N × H)

Subgroups — C₂×#16863707 #22570078

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863707 #22570078

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863707 #22570078