Group #22560802 — subgroups

Subgroups — #22560802

Order 8576.

230 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 95 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×#14523977	C₂	1	1	yes	maximal, characteristic
2	#16863294	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863159	C₂	1	1	yes	maximal, characteristic
5	#16863159	C₂	1	1	yes	maximal, characteristic
6	#16863159	C₂	1	1	yes	maximal, characteristic
7	#336×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₆₇	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	C₂×C₄×Dic₆₇	—	2	1	no	—
14	#14523977	C₄	1	4	yes	complemented
15	C₆₇⋊#95	—	2	1	no	—
16	C₆₇⋊#95	—	2	1	no	—
17	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₆₇	—	2	1	no	—
21	C₂²×Dic₆₇	—	2	1	no	—
22	C₂²×Dic₆₇	—	2	1	no	—
23	C₂²×Dic₆₇	—	2	1	no	—
24	C₄⋊Dic₆₇	—	2	1	no	—
25	C₄⋊Dic₆₇	—	2	1	no	—
26	C₄⋊Dic₆₇	—	2	1	no	—
27	C₄⋊Dic₆₇	—	2	1	no	—
28	C₄⋊Dic₆₇	—	2	1	no	—
29	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
30	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₆₇	Q₈	1	2	yes	—
32	Dic₆₇⋊C₄	—	4	2	no	—
33	C₄×Dic₆₇	—	4	2	no	—
34	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₂×C₈×C₆₇	—	2	1	no	—
39	C₂×C₈×C₆₇	—	2	1	no	—
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₄²×C₆₇	—	2	1	no	—
44	C₄²×C₆₇	—	2	1	no	—
45	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	—	2	1	no	—
47	C₂×C₄×C₆₇	—	2	1	no	—
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	Q₁₆	1	2	yes	—
52	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
58	C₂×Dic₆₇	—	2	2	no	—
59	C₂×Dic₆₇	—	2	2	no	—
60	C₂×Dic₆₇	—	2	2	no	—
61	C₂×Dic₆₇	—	4	1	no	—
62	C₂×Dic₆₇	—	4	1	no	—
63	C₂×Dic₆₇	—	4	1	no	—
64	C₂×Dic₆₇	—	4	1	no	—
65	C₂×Dic₆₇	—	4	1	no	—
66	C₂×Dic₆₇	—	4	1	no	—
67	C₂×Dic₆₇	—	4	1	no	—
68	C₂×Dic₆₇	—	4	1	no	—
69	C₂×Dic₆₇	—	4	1	no	—
70	C₈×C₆₇	—	2	2	no	—
71	C₈×C₆₇	—	4	1	no	—
72	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₆₇	#123	1	1	yes	characteristic
75	C₂²×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₆₇	#135	1	1	yes	characteristic
77	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₄×C₆₇	—	2	2	no	—
80	C₄×C₆₇	—	4	1	no	—
81	C₄×C₆₇	#117	1	1	yes	characteristic
82	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
83	C₄×C₆₇	#125	1	1	yes	characteristic
84	C₄×C₆₇	#126	1	1	yes	characteristic
85	Dic₆₇	—	4	2	no	—
86	Dic₆₇	—	4	2	no	—
87	Dic₆₇	—	4	2	no	—
88	Dic₆₇	—	8	1	no	—
89	C₂×C₆₇	#387	1	1	yes	characteristic
90	C₂×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
91	C₂×C₆₇	#442	1	1	yes	characteristic
92	C₂×C₆₇	#477	1	1	yes	characteristic
93	C₂×C₆₇	#478	1	1	yes	characteristic
94	C₂×C₆₇	#482	1	1	yes	characteristic
95	C₂×C₆₇	#484	1	1	yes	characteristic
96	#1921	—	67	1	no	maximal, Sylow
97	C₆₇	#1921	1	1	yes	characteristic, complemented, Sylow
98	C₂×#107	—	67	1	no	—
99	#389	—	67	1	no	—
100	C₄×C₄⋊C₄	—	67	1	no	—
101	#340	—	67	1	no	—
102	#340	—	67	1	no	—
103	#340	—	67	1	no	—
104	#336	D₆₇	1	1	yes	characteristic
105	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
106	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	67	1	no	—
108	C₂×C₄⋊C₄	—	67	1	no	—
109	C₂×C₄⋊C₄	—	67	1	no	—
110	C₂×C₄⋊C₄	—	67	1	no	—
111	C₂×C₄⋊C₄	—	134	1	no	—
112	C₂×C₄²	—	134	1	no	—
113	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
114	#107	—	67	4	no	—
115	#95	—	134	1	no	—
116	#95	—	134	1	no	—
117	C₂²×C₄	—	134	1	no	—
118	C₂²×C₄	—	134	1	no	—
119	C₂²×C₄	—	134	1	no	—
120	C₂²×C₄	—	134	1	no	—
121	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
122	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	C₄×D₆₇	1	2	yes	—
130	C₄⋊C₄	—	268	2	no	—
131	C₄⋊C₄	—	67	2	no	—
132	C₄⋊C₄	—	67	2	no	—
133	C₄⋊C₄	—	67	2	no	—
134	C₄⋊C₄	—	134	1	no	—
135	C₄⋊C₄	—	134	1	no	—
136	C₄⋊C₄	—	134	1	no	—
137	C₄⋊C₄	—	134	1	no	—
138	C₄⋊C₄	—	134	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	268	2	no	—
142	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	268	1	no	—
149	C₂×C₄	—	268	1	no	—
150	C₂×C₄	—	268	1	no	—
151	C₂×C₄	—	268	1	no	—
152	C₂×C₄	—	268	1	no	—
153	C₂×C₄	—	268	1	no	—
154	C₂×C₄	—	268	1	no	—
155	C₂×C₄	—	268	1	no	—
156	C₂×C₄	—	268	1	no	—
157	C₂×C₄	—	134	2	no	—
158	C₂×C₄	—	134	2	no	—
159	C₂×C₄	—	134	2	no	—
160	C₂×C₄	Dic₂₆₈	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
162	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
163	C₂×C₄	#11761627	1	1	yes	characteristic
164	C₂×C₄	#11761627	1	1	yes	characteristic
165	C₂×C₄	#11761627	1	1	yes	characteristic
166	C₂×C₄	#11761628	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	4	1	no	—
169	C₂²	C₄×D₂₆₈	1	1	yes	characteristic
170	C₂²	#14524023	1	1	yes	characteristic
171	C₂²	#14524027	1	1	yes	characteristic
172	C₂²	C₂×Dic₂₆₈	1	1	yes	characteristic
173	C₂²	#14524065	1	1	yes	characteristic
174	C₂²	#14524071	1	1	yes	characteristic
175	C₂²	#14524071	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	4	1	no	—
178	C₄	—	268	2	no	—
179	C₄	—	268	2	no	—
180	C₄	—	268	2	no	—
181	C₄	—	536	1	no	—
182	C₄	#14523999	1	1	yes	characteristic
183	C₄	#14524008	1	1	yes	characteristic
184	C₄	#14524033	1	1	yes	characteristic
185	C₄	#14524036	1	1	yes	characteristic
186	C₂	#16863281	1	1	yes	characteristic
187	C₂	C₄×Dic₂₆₈	1	1	yes	characteristic
188	C₂	#16863509	1	1	yes	characteristic
189	C₂	#16863692	1	1	yes	characteristic
190	C₂	#16863697	1	1	yes	characteristic
191	C₂	#16863721	1	1	yes	characteristic
192	C₂	#16863732	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523977

C₂

yes

maximal, characteristic

#16863294

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₆₇

C₂

yes

maximal, characteristic

#16863159

C₂

yes

maximal, characteristic

#16863159

C₂

yes

maximal, characteristic

#16863159

C₂

yes

maximal, characteristic

#336×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

—

#14523977

C₄

yes

complemented

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊Dic₆₇

Q₈

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

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

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×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

Q₁₆

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

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

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

#123

yes

characteristic

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×Q₈

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#387

yes

characteristic

C₂×C₆₇

C₄×Q₁₆

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#107

—

#389

—

100

C₄×C₄⋊C₄

—

101

#340

—

102

#340

—

103

#340

—

104

#336

D₆₇

yes

characteristic

105

C₂²×C₈

C₂×D₆₇

yes

characteristic

106

C₂²×C₈

C₂×D₆₇

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

134

—

112

C₂×C₄²

—

134

—

113

C₂×C₄²

C₂×D₆₇

yes

characteristic

114

#107

—

115

#95

—

134

—

116

#95

—

134

—

117

C₂²×C₄

—

134

—

118

C₂²×C₄

—

134

—

119

C₂²×C₄

—

134

—

120

C₂²×C₄

—

134

—

121

C₂²×C₄

D₂₆₈

yes

characteristic

122

C₂²×C₄

D₂₆₈

yes

characteristic

123

C₂²×C₄

C₂²×D₆₇

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₄×D₆₇

yes

—

130

C₄⋊C₄

—

268

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

134

—

135

C₄⋊C₄

—

134

—

136

C₄⋊C₄

—

134

—

137

C₄⋊C₄

—

134

—

138

C₄⋊C₄

—

134

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

268

—

142

C₂³

C₂×D₂₆₈

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

268

—

149

C₂×C₄

—

268

—

150

C₂×C₄

—

268

—

151

C₂×C₄

—

268

—

152

C₂×C₄

—

268

—

153

C₂×C₄

—

268

—

154

C₂×C₄

—

268

—

155

C₂×C₄

—

268

—

156

C₂×C₄

—

268

—

157

C₂×C₄

—

134

—

158

C₂×C₄

—

134

—

159

C₂×C₄

—

134

—

160

C₂×C₄

Dic₂₆₈

yes

—

161

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

162

C₂×C₄

Q₈×D₆₇

yes

characteristic

163

C₂×C₄

#11761627

yes

characteristic

164

C₂×C₄

#11761627

yes

characteristic

165

C₂×C₄

#11761627

yes

characteristic

166

C₂×C₄

#11761628

yes

characteristic

167

C₈

—

168

C₈

—

169

C₂²

C₄×D₂₆₈

yes

characteristic

170

C₂²

#14524023

yes

characteristic

171

C₂²

#14524027

yes

characteristic

172

C₂²

C₂×Dic₂₆₈

yes

characteristic

173

C₂²

#14524065

yes

characteristic

174

C₂²

#14524071

yes

characteristic

175

C₂²

#14524071

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

268

—

179

C₄

—

268

—

180

C₄

—

268

—

181

C₄

—

536

—

182

C₄

#14523999

yes

characteristic

183

C₄

#14524008

yes

characteristic

184

C₄

#14524033

yes

characteristic

185

C₄

#14524036

yes

characteristic

186

C₂

#16863281

yes

characteristic

187

C₂

C₄×Dic₂₆₈

yes

characteristic

188

C₂

#16863509

yes

characteristic

189

C₂

#16863692

yes

characteristic

190

C₂

#16863697

yes

characteristic

191

C₂

#16863721

yes

characteristic

192

C₂

#16863732

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#336×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#336	D₆₇
hypocenter	C₆₇	#1921
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#336×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
14. #14523977	C₄	semidirect
97. C₆₇	#1921	semidirect

H ≅ G/N

type

14. #14523977

C₄

semidirect

97. C₆₇

#1921

semidirect

Subgroups — #22560802

Conjugacy classes of subgroups

Complements