Group #22560220 — subgroups

Subgroups — #22560220

Order 8576.

356 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₄×D₂₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523965	C₂	1	2	yes	maximal
3	#16863262	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863159	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄×D₆₇	—	2	1	no	—
8	C₄×D₂₆₈	—	2	2	no	—
9	C₄×D₂₆₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
13	#14523965	—	2	4	no	—
14	C₆₇⋊#95	—	2	1	no	—
15	C₆₇⋊#95	—	2	1	no	—
16	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₆₇	—	2	2	no	—
19	C₂³×D₆₇	—	2	1	no	—
20	C₂×D₂₆₈	—	2	1	no	—
21	C₂×D₂₆₈	—	2	1	no	—
22	C₂×D₂₆₈	—	2	1	no	—
23	C₂×D₂₆₈	C₂×C₄	1	2	yes	—
24	C₂×D₂₆₈	D₄	1	2	yes	—
25	C₂×D₂₆₈	Q₈	1	2	yes	—
26	C₂²×Dic₆₇	—	2	1	no	—
27	C₂²×Dic₆₇	—	2	2	no	—
28	C₂×C₄×D₆₇	—	2	4	no	—
29	C₂×C₄×D₆₇	—	4	4	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	2	no	—
32	C₄⋊Dic₆₇	—	2	2	no	—
33	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
34	(C₂×D₆₇)⋊C₄	—	2	4	no	—
35	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₆₇	—	2	2	no	—
39	C₂×C₈×C₆₇	—	2	2	no	—
40	C₂×C₈×C₆₇	—	2	2	no	—
41	C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₄²×C₆₇	D₄	1	2	yes	—
43	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×D₆₇	—	2	2	no	—
45	C₂²×D₆₇	—	2	2	no	—
46	C₂²×D₆₇	—	2	2	no	—
47	C₂²×D₆₇	—	4	2	no	—
48	C₂²×D₆₇	—	4	2	no	—
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
59	C₂×Dic₆₇	—	2	2	no	—
60	C₂×Dic₆₇	—	4	2	no	—
61	C₂×Dic₆₇	—	4	2	no	—
62	C₂×Dic₆₇	—	4	2	no	—
63	C₂×Dic₆₇	—	4	2	no	—
64	D₂₆₈	—	2	2	no	—
65	D₂₆₈	—	2	2	no	—
66	D₂₆₈	—	2	2	no	—
67	D₂₆₈	C₄⋊C₄	1	4	yes	—
68	C₄×D₆₇	—	4	8	no	—
69	C₈×C₆₇	—	4	2	no	—
70	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₆₇	#121	1	1	yes	characteristic
72	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂²×C₆₇	#135	1	2	yes	—
74	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×D₆₇	—	2	4	no	—
76	C₂×D₆₇	—	4	2	no	—
77	C₂×D₆₇	—	4	2	no	—
78	C₂×D₆₇	—	4	4	no	—
79	C₂×D₆₇	—	4	4	no	—
80	C₄×C₆₇	—	2	2	no	—
81	C₄×C₆₇	—	2	2	no	—
82	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₆₇	#117	1	1	yes	characteristic
84	C₄×C₆₇	#121	1	1	yes	characteristic
85	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
86	Dic₆₇	—	4	2	no	—
87	Dic₆₇	—	8	2	no	—
88	C₂×C₆₇	#380	1	1	yes	characteristic
89	C₂×C₆₇	#416	1	1	yes	characteristic
90	C₂×C₆₇	#418	1	1	yes	characteristic
91	C₂×C₆₇	#463	1	2	yes	—
92	C₂×C₆₇	#478	1	2	yes	—
93	D₆₇	—	4	4	no	—
94	#1804	—	67	1	no	maximal, Sylow
95	C₆₇	#1804	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×D₄	—	67	1	no	—
97	C₂×C₄⋊C₈	D₆₇	1	1	yes	characteristic, complemented
98	C₂×#102	—	67	2	no	—
99	#382	—	67	1	no	—
100	#340	—	67	2	no	—
101	C₂²×D₄	—	67	1	no	—
102	C₂³×C₄	—	134	1	no	—
103	C₂²×C₈	C₂×D₆₇	1	2	yes	—
104	C₄×D₄	—	67	4	no	—
105	C₄×D₄	—	134	2	no	—
106	C₂×C₄⋊C₄	—	67	1	no	—
107	C₂×C₄⋊C₄	—	67	2	no	—
108	C₂×C₂²⋊C₄	—	134	1	no	—
109	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
110	C₄⋊C₈	—	2	2	no	—
111	#102	—	134	4	no	—
112	#95	—	134	1	no	—
113	#95	—	134	1	no	—
114	C₂⁴	—	134	1	no	—
115	C₂×D₄	—	67	2	no	—
116	C₂×D₄	—	67	2	no	—
117	C₂×D₄	—	67	2	no	—
118	C₂×D₄	—	134	1	no	—
119	C₂×D₄	—	134	1	no	—
120	C₂×D₄	—	134	1	no	—
121	C₂²×C₄	—	268	4	no	—
122	C₂²×C₄	—	134	2	no	—
123	C₂²×C₄	—	134	1	no	—
124	C₂²×C₄	—	134	4	no	—
125	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
126	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₄⋊C₄	—	67	2	no	—
132	C₄⋊C₄	—	134	2	no	—
133	C₄⋊C₄	—	134	2	no	—
134	C₄⋊C₄	—	134	1	no	—
135	C₂²⋊C₄	—	134	4	no	—
136	C₄²	C₄×D₆₇	1	2	yes	—
137	C₄²	C₂²⋊D₆₇	1	2	yes	—
138	C₂³	—	268	2	no	—
139	C₂³	—	268	2	no	—
140	C₂³	—	134	2	no	—
141	C₂³	—	134	2	no	—
142	C₂³	—	134	2	no	—
143	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
144	D₄	—	67	4	no	—
145	D₄	—	134	2	no	—
146	D₄	—	134	2	no	—
147	D₄	—	134	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	268	8	no	—
151	C₂×C₄	—	268	2	no	—
152	C₂×C₄	—	268	2	no	—
153	C₂×C₄	—	268	2	no	—
154	C₂×C₄	—	268	2	no	—
155	C₂×C₄	—	134	2	no	—
156	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
157	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
159	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
160	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
161	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
162	C₂×C₄	#11761627	1	1	yes	characteristic
163	C₂×C₄	#11761629	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₂²	—	268	4	no	—
166	C₂²	—	268	2	no	—
167	C₂²	—	268	2	no	—
168	C₂²	—	268	4	no	—
169	C₂²	—	134	4	no	—
170	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
171	C₂²	#14524018	1	1	yes	characteristic
172	C₂²	#14524023	1	1	yes	characteristic
173	C₂²	#14524065	1	2	yes	—
174	C₂²	#14524071	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	268	2	no	—
178	C₄	—	536	2	no	—
179	C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
180	C₄	#14524001	1	1	yes	characteristic
181	C₄	#14524014	1	1	yes	characteristic
182	C₄	#14524019	1	1	yes	characteristic
183	C₂	—	268	4	no	—
184	C₂	#16863253	1	1	yes	characteristic
185	C₂	#16863393	1	1	yes	characteristic
186	C₂	#16863402	1	1	yes	characteristic
187	C₂	#16863612	1	2	yes	—
188	C₂	#16863697	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₂₆₈

C₂

yes

maximal, characteristic

C₂×#14523965

C₂

yes

maximal

#16863262

C₂

yes

maximal, characteristic, complemented

#16863159

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

characteristic

C₂²×C₄×D₆₇

—

C₄×D₂₆₈

—

C₄×D₂₆₈

C₄

yes

complemented

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×(C₂×D₆₇)⋊C₄

—

#14523965

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

C₂×C₄

yes

—

C₂×D₂₆₈

D₄

yes

—

C₂×D₂₆₈

Q₈

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

(C₂×D₆₇)⋊C₄

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

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

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

C₄⋊C₄

yes

—

C₄×D₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

#135

yes

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#478

yes

—

D₆₇

—

#1804

—

maximal, Sylow

C₆₇

#1804

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

—

C₂×C₄⋊C₈

D₆₇

yes

characteristic, complemented

C₂×#102

—

#382

—

100

#340

—

101

C₂²×D₄

—

102

C₂³×C₄

—

134

—

103

C₂²×C₈

C₂×D₆₇

yes

—

104

C₄×D₄

—

105

C₄×D₄

—

134

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₂²⋊C₄

—

134

—

109

C₂×C₄²

C₂×D₆₇

yes

characteristic

110

C₄⋊C₈

—

111

#102

—

134

—

112

#95

—

134

—

113

#95

—

134

—

114

C₂⁴

—

134

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

134

—

119

C₂×D₄

—

134

—

120

C₂×D₄

—

134

—

121

C₂²×C₄

—

268

—

122

C₂²×C₄

—

134

—

123

C₂²×C₄

—

134

—

124

C₂²×C₄

—

134

—

125

C₂²×C₄

D₂₆₈

yes

characteristic

126

C₂²×C₄

D₂₆₈

yes

characteristic

127

C₂²×C₄

C₂²×D₆₇

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

134

—

133

C₄⋊C₄

—

134

—

134

C₄⋊C₄

—

134

—

135

C₂²⋊C₄

—

134

—

136

C₄²

C₄×D₆₇

yes

—

137

C₄²

C₂²⋊D₆₇

yes

—

138

C₂³

—

268

—

139

C₂³

—

268

—

140

C₂³

—

134

—

141

C₂³

—

134

—

142

C₂³

—

134

—

143

C₂³

C₂×D₂₆₈

yes

characteristic

144

D₄

—

145

D₄

—

134

—

146

D₄

—

134

—

147

D₄

—

134

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

268

—

151

C₂×C₄

—

268

—

152

C₂×C₄

—

268

—

153

C₂×C₄

—

268

—

154

C₂×C₄

—

268

—

155

C₂×C₄

—

134

—

156

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

157

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

158

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

159

C₂×C₄

D₄×D₆₇

yes

characteristic

160

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

161

C₂×C₄

Q₈×D₆₇

yes

characteristic

162

C₂×C₄

#11761627

yes

characteristic

163

C₂×C₄

#11761629

yes

characteristic

164

C₈

—

165

C₂²

—

268

—

166

C₂²

—

268

—

167

C₂²

—

268

—

168

C₂²

—

268

—

169

C₂²

—

134

—

170

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

171

C₂²

#14524018

yes

characteristic

172

C₂²

#14524023

yes

characteristic

173

C₂²

#14524065

yes

—

174

C₂²

#14524071

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

268

—

178

C₄

—

536

—

179

C₄

C₄⋊C₄×D₆₇

yes

characteristic

180

C₄

#14524001

yes

characteristic

181

C₄

#14524014

yes

characteristic

182

C₄

#14524019

yes

characteristic

183

C₂

—

268

—

184

C₂

#16863253

yes

characteristic

185

C₂

#16863393

yes

characteristic

186

C₂

#16863402

yes

characteristic

187

C₂

#16863612

yes

—

188

C₂

#16863697

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×C₄⋊C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₆₇
hypocenter	C₆₇	#1804
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×C₄⋊C₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₆₇

hypocenter

C₆₇

#1804

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #16863262	C₂	semidirect
4. #16863159	C₂	semidirect
5. C₂×C₄⋊C₈×C₆₇	C₂	semidirect
9. C₄×D₂₆₈	C₄	semidirect
95. C₆₇	#1804	semidirect
97. C₂×C₄⋊C₈	D₆₇	semidirect

H ≅ G/N

type

3. #16863262

C₂

semidirect

4. #16863159

C₂

semidirect

5. C₂×C₄⋊C₈×C₆₇

C₂

semidirect

9. C₄×D₂₆₈

C₄

semidirect

95. C₆₇

#1804

semidirect

97. C₂×C₄⋊C₈

D₆₇

semidirect

Subgroups — #22560220

Conjugacy classes of subgroups

Complements