Group #22589062 — subgroups

Subgroups — #22589062

Order 9088.

248 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 105 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⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556038	C₂	1	1	yes	maximal, characteristic
5	C₄×Dic₇₁⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#17985753	C₂	1	1	yes	maximal, characteristic
7	#336×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₇₁⋊C₄	C₄	1	1	yes	characteristic
11	C₂×Dic₇₁⋊C₄	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
15	Dic₇₁⋊C₈	—	2	2	no	—
16	Dic₇₁⋊C₈	—	2	2	no	—
17	#14556038	—	2	2	no	—
18	C₇₁⋊#95	C₄	1	1	yes	characteristic
19	C₇₁⋊#95	C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₇₁⋊C₈	—	2	1	no	—
30	C₂×C₇₁⋊C₈	—	2	1	no	—
31	C₂×C₇₁⋊C₈	—	2	1	no	—
32	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
33	Dic₇₁⋊C₄	—	2	2	no	—
34	Dic₇₁⋊C₄	—	2	2	no	—
35	C₄×Dic₇₁	—	2	1	no	—
36	C₄×Dic₇₁	—	2	1	no	—
37	C₄×Dic₇₁	D₄	1	2	yes	—
38	C₄×Dic₇₁	Q₈	1	2	yes	—
39	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₂×C₈×C₇₁	—	2	1	no	—
46	C₂×C₈×C₇₁	—	2	1	no	—
47	C₂×C₈×C₇₁	Q₈	1	2	yes	—
48	C₄²×C₇₁	—	2	1	no	—
49	C₄²×C₇₁	—	2	1	no	—
50	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	—	2	1	no	—
53	C₂×C₄×C₇₁	—	2	1	no	—
54	C₂×C₄×C₇₁	—	2	1	no	—
55	C₂×C₄×C₇₁	—	2	1	no	—
56	C₂×C₄×C₇₁	M₁₆	1	2	yes	—
57	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
63	C₂×Dic₇₁	—	2	1	no	—
64	C₂×Dic₇₁	—	2	1	no	—
65	C₂×Dic₇₁	—	2	1	no	—
66	C₂×Dic₇₁	—	2	1	no	—
67	C₂×Dic₇₁	—	2	1	no	—
68	C₂×Dic₇₁	—	2	1	no	—
69	C₂×Dic₇₁	—	2	1	no	—
70	C₂×Dic₇₁	—	2	1	no	—
71	C₂×Dic₇₁	—	2	1	no	—
72	C₂×Dic₇₁	C₄⋊C₄	1	2	yes	—
73	C₂×Dic₇₁	C₄⋊C₄	1	2	yes	—
74	C₂×Dic₇₁	M₁₆	1	2	yes	—
75	C₈×C₇₁	—	2	2	no	—
76	C₈×C₇₁	—	4	1	no	—
77	C₇₁⋊C₈	—	2	2	no	—
78	C₇₁⋊C₈	—	4	1	no	—
79	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₇₁	C₄×D₄	1	1	yes	characteristic
81	C₂²×C₇₁	C₄×Q₈	1	1	yes	characteristic
82	C₂²×C₇₁	C₂×M₁₆	1	1	yes	characteristic
83	C₂²×C₇₁	C₂×M₁₆	1	1	yes	characteristic
84	C₂²×C₇₁	#131	1	1	yes	characteristic
85	C₂²×C₇₁	#131	1	1	yes	characteristic
86	C₄×C₇₁	—	2	2	no	—
87	C₄×C₇₁	—	4	1	no	—
88	C₄×C₇₁	#121	1	1	yes	characteristic
89	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
90	C₄×C₇₁	#125	1	1	yes	characteristic
91	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
92	Dic₇₁	—	2	2	no	—
93	Dic₇₁	—	2	2	no	—
94	Dic₇₁	—	2	2	no	—
95	Dic₇₁	—	4	1	no	—
96	C₂×C₇₁	#389	1	1	yes	characteristic
97	C₂×C₇₁	#423	1	1	yes	characteristic
98	C₂×C₇₁	#424	1	1	yes	characteristic
99	C₂×C₇₁	#435	1	1	yes	characteristic
100	C₂×C₇₁	#435	1	1	yes	characteristic
101	C₂×C₇₁	#446	1	1	yes	characteristic
102	C₂×C₇₁	#446	1	1	yes	characteristic
103	#1940	—	71	1	no	maximal, Sylow
104	C₇₁	#1940	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄⋊C₈	—	71	1	no	—
106	C₂×C₄⋊C₈	—	71	1	no	—
107	C₂×C₄⋊C₈	—	71	1	no	—
108	C₂×#97	—	71	1	no	—
109	C₄×C₄⋊C₄	—	71	1	no	—
110	#336	—	71	1	no	—
111	#336	D₇₁	1	1	yes	characteristic
112	C₂²×C₈	—	71	1	no	—
113	C₂²×C₈	—	71	1	no	—
114	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
115	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	71	1	no	—
117	C₂×C₄⋊C₄	—	71	1	no	—
118	C₂×C₄²	—	71	1	no	—
119	C₂×C₄²	—	71	1	no	—
120	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
121	C₄⋊C₈	—	142	2	no	—
122	C₄⋊C₈	—	142	2	no	—
123	C₄⋊C₈	—	142	2	no	—
124	#97	—	142	2	no	—
125	#95	—	71	1	no	—
126	#95	—	71	1	no	—
127	C₂²×C₄	—	71	1	no	—
128	C₂²×C₄	—	71	1	no	—
129	C₂²×C₄	—	71	1	no	—
130	C₂²×C₄	—	71	1	no	—
131	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
132	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	71	2	no	—
140	C₂×C₈	—	142	1	no	—
141	C₂×C₈	—	142	1	no	—
142	C₂×C₈	—	142	1	no	—
143	C₂×C₈	—	142	1	no	—
144	C₂×C₈	—	142	1	no	—
145	C₂×C₈	Dic₁₄₂	1	2	yes	—
146	C₄⋊C₄	—	142	2	no	—
147	C₄⋊C₄	—	142	2	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	71	2	no	—
151	C₄²	—	71	2	no	—
152	C₄²	—	142	1	no	—
153	C₄²	—	142	1	no	—
154	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	71	2	no	—
161	C₂×C₄	—	71	2	no	—
162	C₂×C₄	—	71	2	no	—
163	C₂×C₄	—	142	1	no	—
164	C₂×C₄	—	142	1	no	—
165	C₂×C₄	—	142	1	no	—
166	C₂×C₄	—	142	1	no	—
167	C₂×C₄	—	142	1	no	—
168	C₂×C₄	—	142	1	no	—
169	C₂×C₄	—	142	1	no	—
170	C₂×C₄	—	142	1	no	—
171	C₂×C₄	—	142	1	no	—
172	C₂×C₄	#11765848	1	2	yes	—
173	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
174	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
175	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
176	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
177	C₂×C₄	#11765864	1	1	yes	characteristic
178	C₂×C₄	#11765865	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	4	1	no	—
181	C₈	—	284	1	no	—
182	C₈	—	142	2	no	—
183	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
184	C₂²	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
185	C₂²	C₄×Dic₁₄₂	1	1	yes	characteristic
186	C₂²	M₁₆×D₇₁	1	1	yes	characteristic
187	C₂²	C₂×#11765848	1	1	yes	characteristic
188	C₂²	#14556134	1	1	yes	characteristic
189	C₂²	#14556136	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	4	1	no	—
192	C₄	—	284	1	no	—
193	C₄	—	142	2	no	—
194	C₄	—	142	2	no	—
195	C₄	—	142	2	no	—
196	C₄	#14556101	1	1	yes	characteristic
197	C₄	#14556105	1	1	yes	characteristic
198	C₄	#14556118	1	1	yes	characteristic
199	C₄	#14556127	1	1	yes	characteristic
200	C₂	#17985896	1	1	yes	characteristic
201	C₂	#17986021	1	1	yes	characteristic
202	C₂	#17986024	1	1	yes	characteristic
203	C₂	#17986067	1	1	yes	characteristic
204	C₂	#17986073	1	1	yes	characteristic
205	C₂	#17986127	1	1	yes	characteristic
206	C₂	#17986129	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₇₁⋊C₈

C₂

yes

maximal, characteristic

C₂×Dic₇₁⋊C₈

C₂

yes

maximal, characteristic

C₂×#14556038

C₂

yes

maximal, characteristic

C₄×Dic₇₁⋊C₄

C₂

yes

maximal, characteristic

#17985753

C₂

yes

maximal, characteristic

#336×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×Dic₇₁⋊C₄

C₄

yes

characteristic

C₂×Dic₇₁⋊C₄

C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

Dic₇₁⋊C₈

—

#14556038

—

C₇₁⋊#95

C₄

yes

characteristic

C₇₁⋊#95

C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

C₂×C₄

yes

characteristic

C₂²×Dic₇₁

C₂×C₄

yes

characteristic

C₂²×Dic₇₁

C₂×C₄

yes

characteristic

C₂²×Dic₇₁

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

Q₈

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

M₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₄×D₄

yes

characteristic

C₂²×C₇₁

C₄×Q₈

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#121

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

#125

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

100

C₂×C₇₁

#435

yes

characteristic

101

C₂×C₇₁

#446

yes

characteristic

102

C₂×C₇₁

yes

103

—

104

C₇₁

#1940

yes

characteristic, complemented, Sylow

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×#97

—

109

C₄×C₄⋊C₄

—

110

#336

—

111

#336

D₇₁

yes

characteristic

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂×D₇₁

yes

characteristic

115

C₂²×C₈

C₂×D₇₁

yes

characteristic

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

C₂×D₇₁

yes

characteristic

121

C₄⋊C₈

—

142

—

122

C₄⋊C₈

—

142

—

123

C₄⋊C₈

—

142

—

124

#97

—

142

—

125

#95

—

126

#95

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₄×D₇₁

yes

characteristic

132

C₂²×C₄

C₄×D₇₁

yes

characteristic

133

C₂²×C₄

C₂²×D₇₁

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

142

—

141

C₂×C₈

—

142

—

142

C₂×C₈

—

142

—

143

C₂×C₈

—

142

—

144

C₂×C₈

—

142

—

145

C₂×C₈

Dic₁₄₂

yes

—

146

C₄⋊C₄

—

142

—

147

C₄⋊C₄

—

142

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

142

—

153

C₄²

—

142

—

154

C₂³

C₂×C₄×D₇₁

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

142

—

164

C₂×C₄

—

142

—

165

C₂×C₄

—

142

—

166

C₂×C₄

—

142

—

167

C₂×C₄

—

142

—

168

C₂×C₄

—

142

—

169

C₂×C₄

—

142

—

170

C₂×C₄

—

142

—

171

C₂×C₄

—

142

—

172

C₂×C₄

#11765848

yes

—

173

C₂×C₄

D₄×D₇₁

yes

characteristic

174

C₂×C₄

D₄×D₇₁

yes

characteristic

175

C₂×C₄

Q₈×D₇₁

yes

characteristic

176

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

177

C₂×C₄

#11765864

yes

characteristic

178

C₂×C₄

#11765865

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

—

284

—

182

C₈

—

142

—

183

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

184

C₂²

(C₂²⋊D₇₁)⋊C₄

yes

characteristic

185

C₂²

C₄×Dic₁₄₂

yes

characteristic

186

C₂²

M₁₆×D₇₁

yes

characteristic

187

C₂²

C₂×#11765848

yes

characteristic

188

C₂²

#14556134

yes

characteristic

189

C₂²

#14556136

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

284

—

193

C₄

—

142

—

194

C₄

—

142

—

195

C₄

—

142

—

196

C₄

#14556101

yes

characteristic

197

C₄

#14556105

yes

characteristic

198

C₄

#14556118

yes

characteristic

199

C₄

#14556127

yes

characteristic

200

C₂

#17985896

yes

characteristic

201

C₂

#17986021

yes

characteristic

202

C₂

#17986024

yes

characteristic

203

C₂

#17986067

yes

characteristic

204

C₂

#17986073

yes

characteristic

205

C₂

#17986127

yes

characteristic

206

C₂

#17986129

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator, socle	C₂³×C₇₁	C₂²×C₄
Fitting	#336×C₇₁	C₂
hypercenter	#336	D₇₁
hypocenter	C₇₁	#1940
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator, socle

C₂³×C₇₁

C₂²×C₄

Fitting

#336×C₇₁

C₂

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
104. C₇₁	#1940	semidirect

H ≅ G/N

type

104. C₇₁

#1940

semidirect

Subgroups — #22589062

Conjugacy classes of subgroups

Complements