Group #22559974 — subgroups

Subgroups — #22559974

Order 8576.

336 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 121 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×#14524000	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₆₇)⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×(C₂×D₆₇)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523953	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863151	C₂	1	1	yes	maximal, characteristic, complemented
7	#336×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	#14524000	—	2	2	no	—
12	#14524000	—	2	2	no	—
13	C₂×Dic₆₇⋊C₄	C₄	1	2	yes	—
14	C₂×(C₂×D₆₇)⋊C₄	C₄	1	2	yes	—
15	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
16	(C₂×D₆₇)⋊C₈	C₄	1	4	yes	complemented
17	(C₂×D₆₇)⋊C₈	C₄	1	4	yes	complemented
18	#14523953	—	2	2	no	—
19	C₈×Dic₆₇	—	2	2	no	—
20	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂³×D₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₆₇	—	2	1	no	—
27	C₂×C₄×D₆₇	—	2	1	no	—
28	C₂×C₄×D₆₇	—	2	1	no	—
29	C₂×C₄×D₆₇	—	2	1	no	—
30	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₆₇⋊C₈	—	2	1	no	—
33	C₂×C₆₇⋊C₈	—	2	1	no	—
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₂×C₆₇⋊C₈	—	2	1	no	—
36	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
38	Dic₆₇⋊C₄	—	2	4	no	—
39	(C₂×D₆₇)⋊C₄	—	2	4	no	—
40	C₄×Dic₆₇	D₄	1	2	yes	—
41	C₄×Dic₆₇	D₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	—	2	1	no	—
46	C₂×C₈×C₆₇	—	2	1	no	—
47	C₂×C₈×C₆₇	—	2	1	no	—
48	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₄²×C₆₇	—	2	1	no	—
51	C₄²×C₆₇	—	2	1	no	—
52	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×D₆₇	—	2	1	no	—
54	C₂²×D₆₇	—	2	1	no	—
55	C₂²×D₆₇	—	2	2	no	—
56	C₂²×D₆₇	—	2	2	no	—
57	C₂²×D₆₇	C₄²	1	2	yes	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
67	C₂×Dic₆₇	—	2	1	no	—
68	C₂×Dic₆₇	—	2	1	no	—
69	C₂×Dic₆₇	—	2	2	no	—
70	C₂×Dic₆₇	—	2	2	no	—
71	C₂×Dic₆₇	C₄²	1	2	yes	—
72	C₂×Dic₆₇	C₂²⋊C₄	1	4	yes	—
73	C₄×D₆₇	—	2	2	no	—
74	C₄×D₆₇	—	2	2	no	—
75	C₄×D₆₇	—	2	2	no	—
76	C₄×D₆₇	—	2	2	no	—
77	C₈×C₆₇	—	2	2	no	—
78	C₈×C₆₇	—	2	2	no	—
79	C₆₇⋊C₈	—	2	2	no	—
80	C₆₇⋊C₈	—	2	2	no	—
81	C₂²×C₆₇	C₂×C₄²	1	1	yes	characteristic
82	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₆₇	#117	1	1	yes	characteristic
84	C₂²×C₆₇	#131	1	2	yes	—
85	C₂²×C₆₇	#131	1	2	yes	—
86	C₂×D₆₇	—	2	2	no	—
87	C₂×D₆₇	—	2	2	no	—
88	C₂×D₆₇	—	2	2	no	—
89	C₂×D₆₇	—	4	2	no	—
90	C₂×D₆₇	—	4	2	no	—
91	C₄×C₆₇	—	4	2	no	—
92	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
93	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
94	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
95	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
96	Dic₆₇	—	2	2	no	—
97	Dic₆₇	—	2	4	no	—
98	C₂×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₆₇	#405	1	1	yes	characteristic
100	C₂×C₆₇	#405	1	1	yes	characteristic
101	C₂×C₆₇	#408	1	2	yes	—
102	C₂×C₆₇	#433	1	2	yes	—
103	D₆₇	—	4	2	no	—
104	#1750	—	67	1	no	maximal, Sylow
105	C₆₇	#1750	1	1	yes	characteristic, complemented, Sylow
106	C₂×#117	—	67	1	no	—
107	C₂×C₂²⋊C₈	—	67	1	no	—
108	C₂×C₂²⋊C₈	—	67	1	no	—
109	C₂×#97	—	67	1	no	—
110	C₂×C₄×C₈	—	67	1	no	—
111	#336	—	67	1	no	—
112	#336	D₆₇	1	1	yes	characteristic, complemented
113	C₂³×C₄	—	67	1	no	—
114	C₂²×C₈	—	67	1	no	—
115	C₂²×C₈	—	67	1	no	—
116	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
117	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
118	#117	—	134	2	no	—
119	#117	—	134	2	no	—
120	C₂×C₄⋊C₄	—	67	2	no	—
121	C₂×C₂²⋊C₄	—	67	2	no	—
122	C₂×C₄²	—	67	1	no	—
123	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
124	C₂²⋊C₈	—	67	4	no	—
125	C₂²⋊C₈	—	67	4	no	—
126	#97	—	134	2	no	—
127	C₄×C₈	—	134	2	no	—
128	C₂⁴	—	67	1	no	—
129	C₂²×C₄	—	67	2	no	—
130	C₂²×C₄	—	67	1	no	—
131	C₂²×C₄	—	67	2	no	—
132	C₂²×C₄	—	67	2	no	—
133	C₂²×C₄	—	134	1	no	—
134	C₂²×C₄	—	134	1	no	—
135	C₂²×C₄	—	134	1	no	—
136	C₂²×C₄	—	134	1	no	—
137	C₂²×C₄	C₄×D₆₇	1	2	yes	—
138	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	67	2	no	—
144	C₂×C₈	—	67	2	no	—
145	C₂×C₈	—	134	1	no	—
146	C₂×C₈	—	134	1	no	—
147	C₂×C₈	—	134	1	no	—
148	C₂×C₈	—	134	1	no	—
149	C₂×C₈	C₄×D₆₇	1	2	yes	—
150	C₂×C₈	C₄×D₆₇	1	2	yes	—
151	C₄⋊C₄	—	134	4	no	—
152	C₂²⋊C₄	—	134	4	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	67	2	no	—
156	C₄²	—	67	2	no	—
157	C₂³	—	67	2	no	—
158	C₂³	—	134	1	no	—
159	C₂³	—	134	1	no	—
160	C₂³	—	134	2	no	—
161	C₂³	—	134	2	no	—
162	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	67	2	no	—
167	C₂×C₄	—	67	4	no	—
168	C₂×C₄	—	134	1	no	—
169	C₂×C₄	—	134	1	no	—
170	C₂×C₄	—	134	2	no	—
171	C₂×C₄	—	134	2	no	—
172	C₂×C₄	—	134	2	no	—
173	C₂×C₄	—	134	2	no	—
174	C₂×C₄	—	134	2	no	—
175	C₂×C₄	—	134	2	no	—
176	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
178	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
179	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
180	C₂×C₄	#11761627	1	1	yes	characteristic
181	C₂×C₄	#11761629	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	2	2	no	—
184	C₈	—	134	2	no	—
185	C₈	—	134	2	no	—
186	C₂²	—	268	2	no	—
187	C₂²	—	268	2	no	—
188	C₂²	—	134	2	no	—
189	C₂²	—	134	2	no	—
190	C₂²	—	134	2	no	—
191	C₂²	C₄²×D₆₇	1	1	yes	characteristic
192	C₂²	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
193	C₂²	#14524001	1	1	yes	characteristic
194	C₂²	#14524049	1	2	yes	—
195	C₂²	#14524051	1	2	yes	—
196	C₄	—	4	2	no	—
197	C₄	—	134	2	no	—
198	C₄	—	134	4	no	—
199	C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
200	C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
201	C₄	#14524005	1	1	yes	characteristic
202	C₄	#14524005	1	1	yes	characteristic
203	C₂	—	268	2	no	—
204	C₂	#16863243	1	1	yes	characteristic
205	C₂	#16863350	1	1	yes	characteristic
206	C₂	#16863353	1	1	yes	characteristic
207	C₂	#16863361	1	2	yes	—
208	C₂	#16863456	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524000

C₂

yes

maximal, characteristic

C₂×(C₂×D₆₇)⋊C₈

C₂

yes

maximal, characteristic

C₂×(C₂×D₆₇)⋊C₈

C₂

yes

maximal, characteristic

C₂×#14523953

C₂

yes

maximal, characteristic, complemented

C₂×C₈×Dic₆₇

C₂

yes

maximal, characteristic, complemented

#16863151

C₂

yes

maximal, characteristic, complemented

#336×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524000

—

#14524000

—

C₂×Dic₆₇⋊C₄

C₄

yes

—

C₂×(C₂×D₆₇)⋊C₄

C₄

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

(C₂×D₆₇)⋊C₈

C₄

yes

complemented

(C₂×D₆₇)⋊C₈

C₄

yes

complemented

#14523953

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂³×D₆₇

C₂×C₄

yes

characteristic

C₂²×Dic₆₇

C₂×C₄

yes

characteristic

C₂²×Dic₆₇

C₂×C₄

yes

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

C₂×C₄

yes

—

C₂×C₄×D₆₇

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

Dic₆₇⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

C₄²

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₄²

yes

—

C₂×Dic₆₇

C₂²⋊C₄

yes

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₄²

yes

characteristic

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#131

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₆₇

#405

yes

characteristic

100

C₂×C₆₇

#405

yes

characteristic

101

C₂×C₆₇

#408

yes

—

102

C₂×C₆₇

#433

yes

—

103

D₆₇

—

104

#1750

—

maximal, Sylow

105

C₆₇

#1750

yes

characteristic, complemented, Sylow

106

C₂×#117

—

107

C₂×C₂²⋊C₈

—

108

C₂×C₂²⋊C₈

—

109

C₂×#97

—

110

C₂×C₄×C₈

—

111

#336

—

112

#336

D₆₇

yes

characteristic, complemented

113

C₂³×C₄

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

C₂×D₆₇

yes

characteristic

117

C₂²×C₈

C₂×D₆₇

yes

characteristic

118

#117

—

134

—

119

#117

—

134

—

120

C₂×C₄⋊C₄

—

121

C₂×C₂²⋊C₄

—

122

C₂×C₄²

—

123

C₂×C₄²

C₂×D₆₇

yes

characteristic

124

C₂²⋊C₈

—

125

C₂²⋊C₈

—

126

#97

—

134

—

127

C₄×C₈

—

134

—

128

C₂⁴

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

—

134

C₂²×C₄

—

134

—

135

C₂²×C₄

—

134

—

136

C₂²×C₄

—

134

—

137

C₂²×C₄

C₄×D₆₇

yes

—

138

C₂²×C₄

C₂²×D₆₇

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

134

—

146

C₂×C₈

—

134

—

147

C₂×C₈

—

134

—

148

C₂×C₈

—

134

—

149

C₂×C₈

C₄×D₆₇

yes

—

150

C₂×C₈

C₄×D₆₇

yes

—

151

C₄⋊C₄

—

134

—

152

C₂²⋊C₄

—

134

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₂³

—

158

C₂³

—

134

—

159

C₂³

—

134

—

160

C₂³

—

134

—

161

C₂³

—

134

—

162

C₂³

C₂×C₄×D₆₇

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

134

—

169

C₂×C₄

—

134

—

170

C₂×C₄

—

134

—

171

C₂×C₄

—

134

—

172

C₂×C₄

—

134

—

173

C₂×C₄

—

134

—

174

C₂×C₄

—

134

—

175

C₂×C₄

—

134

—

176

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

177

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

178

C₂×C₄

D₄×D₆₇

yes

characteristic

179

C₂×C₄

D₄×D₆₇

yes

characteristic

180

C₂×C₄

#11761627

yes

characteristic

181

C₂×C₄

#11761629

yes

characteristic

182

C₈

—

183

C₈

—

184

C₈

—

134

—

185

C₈

—

134

—

186

C₂²

—

268

—

187

C₂²

—

268

—

188

C₂²

—

134

—

189

C₂²

—

134

—

190

C₂²

—

134

—

191

C₂²

C₄²×D₆₇

yes

characteristic

192

C₂²

C₂²⋊C₄×D₆₇

yes

characteristic

193

C₂²

#14524001

yes

characteristic

194

C₂²

#14524049

yes

—

195

C₂²

#14524051

yes

—

196

C₄

—

197

C₄

—

134

—

198

C₄

—

134

—

199

C₄

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

200

C₄

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

201

C₄

#14524005

yes

characteristic

202

C₄

#14524005

yes

characteristic

203

C₂

—

268

—

204

C₂

#16863243

yes

characteristic

205

C₂

#16863350

yes

characteristic

206

C₂

#16863353

yes

characteristic

207

C₂

#16863361

yes

—

208

C₂

#16863456

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄²
Fitting	#336×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₄
hypercenter	#336	D₆₇
hypocenter	C₆₇	#1750
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₆₇

commutator

C₂²×C₆₇

C₂×C₄²

Fitting

#336×C₆₇

C₂

socle

C₂³×C₆₇

C₂²×C₄

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
4. C₂×#14523953	C₂	semidirect
5. C₂×C₈×Dic₆₇	C₂	semidirect
6. #16863151	C₂	semidirect
7. #336×C₆₇	C₂	semidirect
16. (C₂×D₆₇)⋊C₈	C₄	semidirect
17. (C₂×D₆₇)⋊C₈	C₄	semidirect
105. C₆₇	#1750	semidirect
112. #336	D₆₇	semidirect

H ≅ G/N

type

4. C₂×#14523953

C₂

semidirect

5. C₂×C₈×Dic₆₇

C₂

semidirect

6. #16863151

C₂

semidirect

7. #336×C₆₇

C₂

semidirect

16. (C₂×D₆₇)⋊C₈

C₄

semidirect

17. (C₂×D₆₇)⋊C₈

C₄

semidirect

105. C₆₇

#1750

semidirect

112. #336

D₆₇

semidirect

Subgroups — #22559974

Conjugacy classes of subgroups

Complements