Group #22560638 — subgroups

Subgroups — #22560638

Order 8576.

432 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂³×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523964	C₂	1	2	yes	maximal, complemented
3	#16863256	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863158	C₂	1	2	yes	maximal, complemented
5	C₂×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	—	2	4	no	—
7	C₂²×C₆₇⋊C₈	—	2	2	no	—
8	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
11	#14523964	—	2	4	no	—
12	C₆₇⋊#95	—	2	1	no	—
13	C₂²×D₄×C₆₇	—	2	1	no	—
14	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	#121×C₆₇	—	2	2	no	—
17	#121×C₆₇	C₄	1	4	yes	complemented
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
20	C₂²×Dic₆₇	—	2	2	no	—
21	C₂×C₆₇⋊C₈	—	2	8	no	—
22	C₂×C₆₇⋊C₈	—	2	2	no	—
23	C₂×C₆₇⋊C₈	—	2	2	no	—
24	C₂×C₆₇⋊C₈	D₄	1	4	yes	complemented
25	C₄⋊Dic₆₇	—	2	2	no	—
26	C₄⋊Dic₆₇	—	2	2	no	—
27	C₂²⋊Dic₆₇	—	4	2	no	—
28	C₂⁴×C₆₇	—	2	1	no	—
29	C₂⁴×C₆₇	D₄	1	1	yes	characteristic
30	C₂×D₄×C₆₇	—	2	4	no	—
31	C₂×D₄×C₆₇	—	2	2	no	—
32	C₂×D₄×C₆₇	—	2	1	no	—
33	C₂×D₄×C₆₇	—	2	2	no	—
34	C₂×D₄×C₆₇	—	4	2	no	—
35	C₂×D₄×C₆₇	—	4	2	no	—
36	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₆₇	—	2	2	no	—
38	C₂²×C₄×C₆₇	—	2	2	no	—
39	C₂²×C₄×C₆₇	—	2	1	no	—
40	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	D₄	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₄⋊C₄×C₆₇	C₂×C₄	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	1	no	—
52	C₂³×C₆₇	—	4	1	no	—
53	C₂³×C₆₇	—	4	2	no	—
54	C₂³×C₆₇	C₂²⋊C₄	1	2	yes	—
55	C₂³×C₆₇	D₈	1	2	yes	—
56	C₂³×C₆₇	SD₁₆	1	2	yes	—
57	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
58	D₄×C₆₇	—	2	4	no	—
59	D₄×C₆₇	—	4	8	no	—
60	D₄×C₆₇	—	4	1	no	—
61	D₄×C₆₇	—	4	1	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	2	4	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	4	1	no	—
68	C₂×C₄×C₆₇	—	4	1	no	—
69	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#41	1	2	yes	—
74	C₂×Dic₆₇	—	4	2	no	—
75	C₂×Dic₆₇	—	4	2	no	—
76	C₂×Dic₆₇	—	4	2	no	—
77	C₆₇⋊C₈	—	2	4	no	—
78	C₂²×C₆₇	—	2	2	no	—
79	C₂²×C₆₇	—	2	2	no	—
80	C₂²×C₆₇	—	2	4	no	—
81	C₂²×C₆₇	—	2	4	no	—
82	C₂²×C₆₇	—	4	2	no	—
83	C₂²×C₆₇	—	4	2	no	—
84	C₂²×C₆₇	—	4	2	no	—
85	C₂²×C₆₇	—	8	2	no	—
86	C₂²×C₆₇	—	8	2	no	—
87	C₂²×C₆₇	#102	1	4	yes	complemented
88	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₆₇	#121	1	2	yes	—
90	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
91	C₂²×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
92	C₂²×C₆₇	#135	1	2	yes	—
93	C₄×C₆₇	—	2	2	no	—
94	C₄×C₆₇	—	4	2	no	—
95	C₄×C₆₇	C₄×D₄	1	2	yes	—
96	C₄×C₆₇	#120	1	1	yes	characteristic
97	C₄×C₆₇	#123	1	1	yes	characteristic
98	Dic₆₇	—	8	2	no	—
99	C₂×C₆₇	—	2	4	no	—
100	C₂×C₆₇	—	8	2	no	—
101	C₂×C₆₇	#386	1	1	yes	characteristic
102	C₂×C₆₇	C₂×#102	1	1	yes	characteristic
103	C₂×C₆₇	#416	1	1	yes	characteristic
104	C₂×C₆₇	#465	1	2	yes	—
105	C₂×C₆₇	#466	1	2	yes	—
106	#1893	—	67	1	no	maximal, Sylow
107	C₆₇	#1893	1	1	yes	characteristic, complemented, Sylow
108	C₂³×C₈	—	67	1	no	—
109	C₂×#121	D₆₇	1	1	yes	characteristic
110	C₂×#102	—	67	2	no	—
111	#380	—	67	1	no	—
112	#340	—	67	2	no	—
113	C₂²×D₄	—	2	1	no	—
114	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
115	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
116	C₂²×C₈	—	67	2	no	—
117	C₂²×C₈	—	134	4	no	—
118	C₂²×C₈	—	134	2	no	—
119	#121	—	2	2	no	—
120	#121	Dic₆₇	1	4	yes	complemented
121	C₂×C₄⋊C₄	—	67	2	no	—
122	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
123	C₂×C₂²⋊C₄	—	2	1	no	—
124	C₂×C₂²⋊C₄	—	134	1	no	—
125	#102	—	134	4	no	—
126	#95	—	134	1	no	—
127	C₂⁴	—	2	1	no	—
128	C₂⁴	C₂²⋊D₆₇	1	1	yes	characteristic
129	C₂×D₄	—	2	4	no	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	—	4	2	no	—
134	C₂×D₄	—	4	2	no	—
135	C₂×D₄	C₂×Dic₆₇	1	2	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	134	2	no	—
140	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
141	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
143	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
144	C₂×C₈	—	67	4	no	—
145	C₂×C₈	—	134	8	no	—
146	C₂×C₈	—	134	2	no	—
147	C₂×C₈	—	134	2	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	134	2	no	—
150	C₄⋊C₄	—	134	2	no	—
151	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₂²⋊C₄	—	268	2	no	—
154	C₂³	—	2	2	no	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	4	2	no	—
158	C₂³	—	4	1	no	—
159	C₂³	—	4	1	no	—
160	C₂³	—	4	2	no	—
161	C₂³	C₂²⋊Dic₆₇	1	2	yes	—
162	C₂³	#11761613	1	2	yes	—
163	C₂³	#11761616	1	2	yes	—
164	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
165	D₄	—	2	4	no	—
166	D₄	—	4	8	no	—
167	D₄	—	4	1	no	—
168	D₄	—	4	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	268	2	no	—
177	C₂×C₄	—	268	2	no	—
178	C₂×C₄	—	268	2	no	—
179	C₂×C₄	C₂²⋊Dic₆₇	1	2	yes	—
180	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
181	C₂×C₄	D₄×D₆₇	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
183	C₂×C₄	#11761627	1	2	yes	—
184	C₈	—	134	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	4	no	—
187	C₂²	—	2	4	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	8	2	no	—
193	C₂²	—	8	2	no	—
194	C₂²	#14523964	1	4	yes	complemented
195	C₂²	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
196	C₂²	#14524015	1	2	yes	—
197	C₂²	C₂×#11761613	1	1	yes	characteristic
198	C₂²	C₂×#11761616	1	1	yes	characteristic
199	C₂²	#14524063	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	536	2	no	—
203	C₄	D₄×Dic₆₇	1	2	yes	—
204	C₄	#14524011	1	1	yes	characteristic
205	C₄	#14524024	1	1	yes	characteristic
206	C₂	—	2	4	no	—
207	C₂	—	8	2	no	—
208	C₂	#16863276	1	1	yes	characteristic
209	C₂	C₂×#14523964	1	1	yes	characteristic
210	C₂	#16863392	1	1	yes	characteristic
211	C₂	#16863623	1	2	yes	—
212	C₂	#16863630	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#14523964

C₂

yes

maximal, complemented

#16863256

C₂

yes

maximal, characteristic, complemented

#16863158

C₂

yes

maximal, complemented

C₂×#121×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₂²⋊Dic₆₇

—

#14523964

—

C₆₇⋊#95

—

C₂²×D₄×C₆₇

—

C₂²×D₄×C₆₇

C₂²

yes

characteristic

C₂³×C₄×C₆₇

C₂²

yes

characteristic

#121×C₆₇

—

#121×C₆₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

complemented

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂⁴×C₆₇

—

C₂⁴×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂²⋊C₄

yes

—

C₂³×C₆₇

D₈

yes

—

C₂³×C₆₇

SD₁₆

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

D₄×C₆₇

—

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×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#102

yes

complemented

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

#121

yes

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₂²×C₆₇

C₂×SD₁₆

yes

characteristic

C₂²×C₆₇

#135

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×D₄

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

—

100

C₂×C₆₇

—

101

C₂×C₆₇

#386

yes

characteristic

102

C₂×C₆₇

C₂×#102

yes

characteristic

103

C₂×C₆₇

#416

yes

characteristic

104

C₂×C₆₇

#465

yes

—

105

C₂×C₆₇

#466

yes

—

106

#1893

—

maximal, Sylow

107

C₆₇

#1893

yes

characteristic, complemented, Sylow

108

C₂³×C₈

—

109

C₂×#121

D₆₇

yes

characteristic

110

C₂×#102

—

111

#380

—

112

#340

—

113

C₂²×D₄

—

114

C₂²×D₄

C₂×D₆₇

yes

characteristic

115

C₂³×C₄

C₂×D₆₇

yes

characteristic

116

C₂²×C₈

—

117

C₂²×C₈

—

134

—

118

C₂²×C₈

—

134

—

119

#121

—

120

#121

Dic₆₇

yes

complemented

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

123

C₂×C₂²⋊C₄

—

124

C₂×C₂²⋊C₄

—

134

—

125

#102

—

134

—

126

#95

—

134

—

127

C₂⁴

—

128

C₂⁴

C₂²⋊D₆₇

yes

characteristic

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

C₂×Dic₆₇

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

134

—

140

C₂²×C₄

C₂×Dic₆₇

yes

—

141

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

142

C₂²×C₄

C₂²⋊D₆₇

yes

—

143

C₂²×C₄

C₂²×D₆₇

yes

characteristic

144

C₂×C₈

—

145

C₂×C₈

—

134

—

146

C₂×C₈

—

134

—

147

C₂×C₈

—

134

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

134

—

150

C₄⋊C₄

—

134

—

151

C₄⋊C₄

C₂×Dic₆₇

yes

—

152

C₂²⋊C₄

—

153

C₂²⋊C₄

—

268

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

—

161

C₂³

C₂²⋊Dic₆₇

yes

—

162

C₂³

#11761613

yes

—

163

C₂³

#11761616

yes

—

164

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

165

D₄

—

166

D₄

—

167

D₄

—

168

D₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

268

—

177

C₂×C₄

—

268

—

178

C₂×C₄

—

268

—

179

C₂×C₄

C₂²⋊Dic₆₇

yes

—

180

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

181

C₂×C₄

D₄×D₆₇

yes

—

182

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

183

C₂×C₄

#11761627

yes

—

184

C₈

—

134

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

#14523964

yes

complemented

195

C₂²

C₂×C₂²⋊Dic₆₇

yes

characteristic

196

C₂²

#14524015

yes

—

197

C₂²

C₂×#11761613

yes

characteristic

198

C₂²

C₂×#11761616

yes

characteristic

199

C₂²

#14524063

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

536

—

203

C₄

D₄×Dic₆₇

yes

—

204

C₄

#14524011

yes

characteristic

205

C₄

#14524024

yes

characteristic

206

C₂

—

207

C₂

—

208

C₂

#16863276

yes

characteristic

209

C₂

C₂×#14523964

yes

characteristic

210

C₂

#16863392

yes

characteristic

211

C₂

#16863623

yes

—

212

C₂

#16863630

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#121×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#121	D₆₇
hypocenter	C₆₇	#1893
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

C₂²×C₄

Frattini

C₂²×C₄

C₂²×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. C₂³×C₆₇⋊C₈	C₂	semidirect
2. C₂×#14523964	C₂	semidirect
3. #16863256	C₂	semidirect
4. #16863158	C₂	semidirect
17. #121×C₆₇	C₄	semidirect
24. C₂×C₆₇⋊C₈	D₄	semidirect
87. C₂²×C₆₇	#102	semidirect
107. C₆₇	#1893	semidirect
120. #121	Dic₆₇	semidirect
194. C₂²	#14523964	semidirect

H ≅ G/N

type

1. C₂³×C₆₇⋊C₈

C₂

semidirect

2. C₂×#14523964

C₂

semidirect

3. #16863256

C₂

semidirect

4. #16863158

C₂

semidirect

17. #121×C₆₇

C₄

semidirect

24. C₂×C₆₇⋊C₈

D₄

semidirect

87. C₂²×C₆₇

#102

semidirect

107. C₆₇

#1893

semidirect

120. #121

Dic₆₇

semidirect

194. C₂²

#14523964

semidirect

Subgroups — #22560638

Conjugacy classes of subgroups

Complements