Group #22569692 — subgroups

Subgroups — #22569692

Order 8576.

480 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14524074	C₂	1	2	yes	maximal
2	C₄×#11761628	C₂	1	1	yes	maximal, characteristic
3	#16863603	C₂	1	4	yes	maximal, complemented
4	#16863598	C₂	1	4	yes	maximal, complemented
5	#16863420	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863405	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761628	C₂²	1	1	yes	characteristic
9	#14524074	—	2	8	no	—
10	C₂×#11761618	C₂²	1	4	yes	—
11	C₂×#11761615	C₂²	1	4	yes	—
12	C₂×#11761612	C₂²	1	2	yes	—
13	C₄×Dic₁₃₄	C₂²	1	2	yes	—
14	C₄×D₂₆₈	C₂²	1	2	yes	—
15	C₄×C₂²⋊D₆₇	—	2	2	no	—
16	#14524002	C₂²	1	1	yes	characteristic
17	#14524000	—	2	1	no	—
18	C₄²×D₆₇	—	2	1	no	—
19	C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
20	#14523967	C₂²	1	8	yes	—
21	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
22	C₄⋊Q₈×C₆₇	—	2	1	no	—
23	C₄⋊Q₈×C₆₇	—	2	1	no	—
24	C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#11761628	—	2	2	no	—
28	#11761628	D₄	1	4	yes	—
29	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
30	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₆₇	—	2	1	no	—
32	C₂×C₄×D₆₇	—	2	1	no	—
33	C₂×C₄×D₆₇	—	2	2	no	—
34	#11761618	—	2	8	no	—
35	#11761615	—	2	8	no	—
36	#11761612	—	2	4	no	—
37	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
38	C₄⋊Dic₆₇	C₂³	1	2	yes	—
39	Dic₆₇⋊C₄	—	2	2	no	—
40	C₂²⋊Dic₆₇	—	2	1	no	—
41	(C₂×D₆₇)⋊C₄	—	2	2	no	—
42	C₄×Dic₆₇	—	2	1	no	—
43	C₄×Dic₆₇	—	2	2	no	—
44	C₂×Q₈×C₆₇	—	2	2	no	—
45	C₂×Q₈×C₆₇	—	2	4	no	—
46	C₂×Q₈×C₆₇	—	2	4	no	—
47	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₄²×C₆₇	D₄	1	1	yes	characteristic
55	C₄²×C₆₇	C₂³	1	2	yes	—
56	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×D₆₇	—	2	1	no	—
58	Q₈×C₆₇	—	2	4	no	—
59	Q₈×C₆₇	—	2	4	no	—
60	Q₈×C₆₇	—	4	4	no	—
61	C₂×C₄×C₆₇	—	2	4	no	—
62	C₂×C₄×C₆₇	—	2	4	no	—
63	C₂×C₄×C₆₇	—	4	4	no	—
64	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₆₇	#41	1	2	yes	—
69	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₂²⋊D₆₇	—	2	4	no	—
71	C₂×Dic₆₇	—	2	1	no	—
72	C₂×Dic₆₇	—	2	2	no	—
73	D₂₆₈	—	2	1	no	—
74	D₂₆₈	C₂×D₄	1	2	yes	—
75	C₄×D₆₇	—	2	4	no	—
76	C₄×D₆₇	—	4	4	no	—
77	Dic₁₃₄	—	2	1	no	—
78	Dic₁₃₄	C₂×D₄	1	2	yes	—
79	C₆₇⋊C₈	—	2	4	no	—
80	C₂²×C₆₇	—	2	2	no	—
81	C₂²×C₆₇	#121	1	2	yes	—
82	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₆₇	—	2	2	no	—
84	C₂×D₆₇	—	4	2	no	—
85	C₄×C₆₇	—	2	2	no	—
86	C₄×C₆₇	—	4	4	no	—
87	C₄×C₆₇	#121	1	2	yes	—
88	C₄×C₆₇	#137	1	4	yes	complemented
89	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
91	Dic₆₇	—	2	2	no	—
92	Dic₆₇	—	4	2	no	—
93	C₂×C₆₇	—	2	2	no	—
94	C₂×C₆₇	C₂×#121	1	1	yes	characteristic
95	C₂×C₆₇	C₂×#137	1	2	yes	—
96	D₆₇	—	4	2	no	—
97	#3039	—	67	1	no	maximal, Sylow
98	C₆₇	#3039	1	1	yes	characteristic, complemented, Sylow
99	C₂×#137	—	67	2	no	—
100	C₂×C₄⋊Q₈	D₆₇	1	1	yes	characteristic, complemented
101	C₄×#41	—	67	1	no	—
102	#462	—	67	4	no	—
103	#461	—	67	4	no	—
104	#423	—	67	1	no	—
105	#419	—	67	2	no	—
106	C₂×#41	—	67	1	no	—
107	C₂²×Q₈	C₂×D₆₇	1	2	yes	—
108	#137	—	134	8	no	—
109	C₂×Q₁₆	—	67	4	no	—
110	C₂×SD₁₆	—	67	4	no	—
111	C₂×M₁₆	—	67	2	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	C₂×D₆₇	1	4	yes	—
115	C₄×Q₈	—	67	2	no	—
116	C₄×D₄	—	67	2	no	—
117	C₄×D₄	—	134	2	no	—
118	#117	—	67	1	no	—
119	#117	—	134	1	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	—	134	1	no	—
122	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
123	C₄⋊C₈	—	67	4	no	—
124	#103	—	67	8	no	—
125	#41	—	67	4	no	—
126	#41	—	134	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	—	67	1	no	—
131	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
132	C₂×D₄	—	67	1	no	—
133	C₂×D₄	—	134	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	134	2	no	—
136	C₂²×C₄	—	134	1	no	—
137	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
138	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
139	Q₁₆	—	134	8	no	—
140	SD₁₆	—	134	8	no	—
141	M₁₆	—	134	4	no	—
142	C₂×C₈	—	67	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	67	2	no	—
146	C₄⋊C₄	—	134	2	no	—
147	C₂²⋊C₄	—	134	1	no	—
148	C₂²⋊C₄	—	134	2	no	—
149	C₄²	—	134	1	no	—
150	C₄²	—	134	2	no	—
151	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
152	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
153	C₄²	C₂²×D₆₇	1	2	yes	—
154	C₂³	—	134	1	no	—
155	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
156	Q₈	—	2	4	no	—
157	Q₈	—	2	4	no	—
158	Q₈	—	4	4	no	—
159	Q₈	—	67	2	no	—
160	Q₈	—	134	1	no	—
161	D₄	—	67	2	no	—
162	D₄	—	134	4	no	—
163	D₄	—	134	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	268	4	no	—
168	C₂×C₄	—	134	2	no	—
169	C₂×C₄	—	134	1	no	—
170	C₂×C₄	—	134	4	no	—
171	C₂×C₄	D₄×D₆₇	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
175	C₂×C₄	#11761627	1	2	yes	—
176	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
177	C₈	—	134	4	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	268	2	no	—
180	C₂²	—	134	2	no	—
181	C₂²	#14524014	1	2	yes	—
182	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	268	2	no	—
186	C₄	—	134	2	no	—
187	C₄	#14524014	1	2	yes	—
188	C₄	#14524074	1	4	yes	complemented
189	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
190	C₄	C₂×#11761627	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	268	2	no	—
193	C₂	C₂×#14524014	1	1	yes	characteristic
194	C₂	C₂×#14524074	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524074

C₂

yes

maximal

C₄×#11761628

C₂

yes

maximal, characteristic

#16863603

C₂

yes

maximal, complemented

#16863598

C₂

yes

maximal, complemented

#16863420

C₂

yes

maximal, characteristic, complemented

#16863405

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

characteristic

#14524074

—

C₂×#11761618

C₂²

yes

—

C₂×#11761615

C₂²

yes

—

C₂×#11761612

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×D₂₆₈

C₂²

yes

—

C₄×C₂²⋊D₆₇

—

#14524002

C₂²

yes

characteristic

#14524000

—

C₄²×D₆₇

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523967

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#11761628

—

#11761628

D₄

yes

—

C₂×Dic₁₃₄

C₂³

yes

characteristic

C₂×D₂₆₈

C₂³

yes

characteristic

C₂×C₂²⋊D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

#11761618

—

#11761615

—

#11761612

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

C₂×D₄

yes

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

C₂×D₄

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

—

C₄×C₆₇

#137

yes

complemented

C₄×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#121

yes

characteristic

C₂×C₆₇

C₂×#137

yes

—

D₆₇

—

#3039

—

maximal, Sylow

C₆₇

#3039

yes

characteristic, complemented, Sylow

C₂×#137

—

100

C₂×C₄⋊Q₈

D₆₇

yes

characteristic, complemented

101

C₄×#41

—

102

#462

—

103

#461

—

104

#423

—

105

#419

—

106

C₂×#41

—

107

C₂²×Q₈

C₂×D₆₇

yes

—

108

#137

—

134

—

109

C₂×Q₁₆

—

110

C₂×SD₁₆

—

111

C₂×M₁₆

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×D₆₇

yes

—

115

C₄×Q₈

—

116

C₄×D₄

—

117

C₄×D₄

—

134

—

118

#117

—

119

#117

—

134

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

—

134

—

122

C₂×C₄²

C₂×D₆₇

yes

characteristic

123

C₄⋊C₈

—

124

#103

—

125

#41

—

126

#41

—

134

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

C₂²×D₆₇

yes

—

132

C₂×D₄

—

133

C₂×D₄

—

134

—

134

C₂²×C₄

—

135

C₂²×C₄

—

134

—

136

C₂²×C₄

—

134

—

137

C₂²×C₄

C₂²⋊D₆₇

yes

—

138

C₂²×C₄

C₂²×D₆₇

yes

characteristic

139

Q₁₆

—

134

—

140

SD₁₆

—

134

—

141

M₁₆

—

134

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

134

—

147

C₂²⋊C₄

—

134

—

148

C₂²⋊C₄

—

134

—

149

C₄²

—

134

—

150

C₄²

—

134

—

151

C₄²

C₂²⋊D₆₇

yes

characteristic

152

C₄²

C₂²⋊D₆₇

yes

characteristic

153

C₄²

C₂²×D₆₇

yes

—

154

C₂³

—

134

—

155

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

156

Q₈

—

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

134

—

161

D₄

—

162

D₄

—

134

—

163

D₄

—

134

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

268

—

168

C₂×C₄

—

134

—

169

C₂×C₄

—

134

—

170

C₂×C₄

—

134

—

171

C₂×C₄

D₄×D₆₇

yes

—

172

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

173

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

175

C₂×C₄

#11761627

yes

—

176

C₂×C₄

C₂³×D₆₇

yes

characteristic

177

C₈

—

134

—

178

C₂²

—

179

C₂²

—

268

—

180

C₂²

—

134

—

181

C₂²

#14524014

yes

—

182

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

268

—

186

C₄

—

134

—

187

C₄

#14524014

yes

—

188

C₄

#14524074

yes

complemented

189

C₄

C₂×D₄×D₆₇

yes

characteristic

190

C₄

C₂×#11761627

yes

characteristic

191

C₂

—

192

C₂

—

268

—

193

C₂

C₂×#14524014

yes

characteristic

194

C₂

C₂×#14524074

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄⋊Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#3039
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄⋊Q₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₇

hypocenter

C₆₇

#3039

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #16863603	C₂	semidirect
4. #16863598	C₂	semidirect
5. #16863420	C₂	semidirect
6. #16863405	C₂	semidirect
7. C₂×C₄⋊Q₈×C₆₇	C₂	semidirect
88. C₄×C₆₇	#137	semidirect
98. C₆₇	#3039	semidirect
100. C₂×C₄⋊Q₈	D₆₇	semidirect
188. C₄	#14524074	semidirect

H ≅ G/N

type

3. #16863603

C₂

semidirect

4. #16863598

C₂

semidirect

5. #16863420

C₂

semidirect

6. #16863405

C₂

semidirect

7. C₂×C₄⋊Q₈×C₆₇

C₂

semidirect

88. C₄×C₆₇

#137

semidirect

98. C₆₇

#3039

semidirect

100. C₂×C₄⋊Q₈

D₆₇

semidirect

188. C₄

#14524074

semidirect

Subgroups — #22569692

Conjugacy classes of subgroups

Complements