Group #22589718 — subgroups

Subgroups — #22589718

Order 9088.

248 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14556059	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556051	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556051	C₂	1	1	yes	maximal, characteristic
5	#17985905	C₂	1	1	yes	maximal, characteristic
6	#17985761	C₂	1	1	yes	maximal, characteristic
7	#389×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₂	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
12	#14556059	—	2	2	no	—
13	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
14	#14556051	—	2	2	no	—
15	#14556051	—	2	2	no	—
16	C₇₁⋊#95	—	2	1	no	—
17	C₇₁⋊#95	—	2	1	no	—
18	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
19	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
22	#95×C₇₁	—	2	1	no	—
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×Dic₁₄₂	—	2	1	no	—
25	C₂×Dic₁₄₂	—	2	1	no	—
26	C₂×Dic₁₄₂	—	2	1	no	—
27	C₂×Dic₁₄₂	—	2	1	no	—
28	C₂×Dic₁₄₂	Q₈	1	2	yes	—
29	C₂²×Dic₇₁	—	2	1	no	—
30	C₂²×Dic₇₁	—	2	1	no	—
31	C₂×C₇₁⋊C₈	—	2	1	no	—
32	C₂×C₇₁⋊C₈	—	2	1	no	—
33	C₂×C₇₁⋊C₈	—	2	1	no	—
34	C₂×C₇₁⋊C₈	—	2	1	no	—
35	C₂×C₇₁⋊C₈	—	2	1	no	—
36	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
37	C₄⋊Dic₇₁	—	2	1	no	—
38	C₄⋊Dic₇₁	—	2	1	no	—
39	C₂²×C₄×C₇₁	—	2	1	no	—
40	C₂²×C₄×C₇₁	—	2	1	no	—
41	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₇₁	—	2	1	no	—
45	C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₄⋊C₄×C₇₁	—	2	1	no	—
47	C₄⋊C₄×C₇₁	—	4	2	no	—
48	C₄⋊C₄×C₇₁	D₄	1	2	yes	—
49	C₄²×C₇₁	—	2	1	no	—
50	C₄²×C₇₁	—	2	1	no	—
51	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
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	2	no	—
56	C₂×C₄×C₇₁	—	2	1	no	—
57	C₂×C₄×C₇₁	—	2	1	no	—
58	C₂×C₄×C₇₁	—	4	1	no	—
59	C₂×C₄×C₇₁	—	4	1	no	—
60	C₂×C₄×C₇₁	—	4	1	no	—
61	C₂×C₄×C₇₁	—	4	1	no	—
62	C₂×C₄×C₇₁	—	4	1	no	—
63	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
64	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
70	C₂×Dic₇₁	—	2	2	no	—
71	C₂×Dic₇₁	—	4	1	no	—
72	C₂×Dic₇₁	—	4	1	no	—
73	C₂×Dic₇₁	—	4	1	no	—
74	C₂×Dic₇₁	—	4	1	no	—
75	C₂×Dic₇₁	—	4	1	no	—
76	Dic₁₄₂	—	2	2	no	—
77	Dic₁₄₂	—	2	2	no	—
78	Dic₁₄₂	—	4	1	no	—
79	Dic₁₄₂	—	4	1	no	—
80	C₇₁⋊C₈	—	2	2	no	—
81	C₇₁⋊C₈	—	4	1	no	—
82	C₂²×C₇₁	#121	1	1	yes	characteristic
83	C₂²×C₇₁	#121	1	1	yes	characteristic
84	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
86	C₂²×C₇₁	#135	1	1	yes	characteristic
87	C₂²×C₇₁	#137	1	1	yes	characteristic
88	C₂²×C₇₁	#137	1	1	yes	characteristic
89	C₄×C₇₁	—	2	2	no	—
90	C₄×C₇₁	—	4	2	no	—
91	C₄×C₇₁	—	4	1	no	—
92	C₄×C₇₁	—	8	1	no	—
93	C₄×C₇₁	#121	1	1	yes	characteristic
94	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
95	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
96	C₄×C₇₁	#126	1	1	yes	characteristic
97	Dic₇₁	—	4	2	no	—
98	Dic₇₁	—	8	1	no	—
99	C₂×C₇₁	#396	1	1	yes	characteristic
100	C₂×C₇₁	#461	1	1	yes	characteristic
101	C₂×C₇₁	#464	1	1	yes	characteristic
102	C₂×C₇₁	#465	1	1	yes	characteristic
103	C₂×C₇₁	#470	1	1	yes	characteristic
104	C₂×C₇₁	#475	1	1	yes	characteristic
105	C₂×C₇₁	#479	1	1	yes	characteristic
106	#2066	—	71	1	no	maximal, Sylow
107	C₇₁	#2066	1	1	yes	characteristic, complemented, Sylow
108	C₂×#106	—	71	1	no	—
109	C₂×C₄⋊C₈	—	71	1	no	—
110	C₂×#103	—	71	1	no	—
111	C₂×#103	—	71	1	no	—
112	#391	—	71	1	no	—
113	#389	D₇₁	1	1	yes	characteristic
114	#340	—	71	1	no	—
115	C₂²×Q₈	—	71	1	no	—
116	C₂²×C₈	—	71	1	no	—
117	C₂²×C₈	—	71	1	no	—
118	C₂×C₄⋊C₄	—	2	1	no	—
119	C₂×C₄⋊C₄	—	71	1	no	—
120	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
122	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
123	#106	—	142	2	no	—
124	C₄⋊C₈	—	142	2	no	—
125	#103	—	142	2	no	—
126	#103	—	142	2	no	—
127	#95	—	2	1	no	—
128	#95	—	142	1	no	—
129	#95	—	142	1	no	—
130	C₂×Q₈	—	71	2	no	—
131	C₂×Q₈	—	142	1	no	—
132	C₂×Q₈	—	142	1	no	—
133	C₂×Q₈	—	142	1	no	—
134	C₂×Q₈	—	142	1	no	—
135	C₂×Q₈	—	142	1	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	142	1	no	—
139	C₂²×C₄	—	142	1	no	—
140	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
141	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
143	C₂×C₈	—	71	2	no	—
144	C₂×C₈	—	142	1	no	—
145	C₂×C₈	—	142	1	no	—
146	C₂×C₈	—	142	1	no	—
147	C₂×C₈	—	142	1	no	—
148	C₂×C₈	—	142	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	142	1	no	—
154	C₄⋊C₄	—	142	1	no	—
155	C₄⋊C₄	D₂₈₄	1	2	yes	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
159	Q₈	—	284	1	no	—
160	Q₈	—	284	1	no	—
161	Q₈	—	142	2	no	—
162	Q₈	—	142	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	284	1	no	—
175	C₂×C₄	—	284	1	no	—
176	C₂×C₄	—	284	1	no	—
177	C₂×C₄	—	284	1	no	—
178	C₂×C₄	—	284	1	no	—
179	C₂×C₄	—	142	2	no	—
180	C₂×C₄	#11765853	1	2	yes	—
181	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
182	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
183	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
184	C₂×C₄	#11765864	1	1	yes	characteristic
185	C₂×C₄	#11765865	1	1	yes	characteristic
186	C₂×C₄	#11765866	1	1	yes	characteristic
187	C₈	—	284	1	no	—
188	C₈	—	142	2	no	—
189	C₂²	#14556100	1	1	yes	characteristic
190	C₂²	#14556102	1	1	yes	characteristic
191	C₂²	#14556104	1	1	yes	characteristic
192	C₂²	C₂×#11765853	1	1	yes	characteristic
193	C₂²	#14556148	1	1	yes	characteristic
194	C₂²	#14556158	1	1	yes	characteristic
195	C₂²	#14556159	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	1	no	—
199	C₄	—	8	1	no	—
200	C₄	—	284	2	no	—
201	C₄	—	568	1	no	—
202	C₄	#14556103	1	1	yes	characteristic
203	C₄	#14556106	1	1	yes	characteristic
204	C₄	#14556108	1	1	yes	characteristic
205	C₄	#14556123	1	1	yes	characteristic
206	C₂	#17985921	1	1	yes	characteristic
207	C₂	#17986204	1	1	yes	characteristic
208	C₂	#17986221	1	1	yes	characteristic
209	C₂	#17986226	1	1	yes	characteristic
210	C₂	#17986255	1	1	yes	characteristic
211	C₂	#17986281	1	1	yes	characteristic
212	C₂	#17986303	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556059

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14556051

C₂

yes

maximal, characteristic

C₂×#14556051

C₂

yes

maximal, characteristic

#17985905

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal, characteristic

#389×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

#14556059

—

C₄⋊(C₇₁⋊C₈)

—

#14556051

—

#14556051

—

C₇₁⋊#95

—

C₇₁⋊#95

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#95×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

Q₈

yes

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₁

C₂×SD₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#121

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

#126

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#396

yes

characteristic

100

C₂×C₇₁

#461

yes

characteristic

101

C₂×C₇₁

#464

yes

characteristic

102

C₂×C₇₁

#465

yes

characteristic

103

C₂×C₇₁

#470

yes

characteristic

104

C₂×C₇₁

#475

yes

characteristic

105

C₂×C₇₁

yes

106

—

107

C₇₁

#2066

yes

characteristic, complemented, Sylow

108

C₂×#106

—

109

C₂×C₄⋊C₈

—

110

C₂×#103

—

111

C₂×#103

—

112

#391

—

113

#389

D₇₁

yes

characteristic

114

#340

—

115

C₂²×Q₈

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

121

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

122

C₂×C₄²

C₂×D₇₁

yes

characteristic

123

#106

—

142

—

124

C₄⋊C₈

—

142

—

125

#103

—

142

—

126

#103

—

142

—

127

#95

—

128

#95

—

142

—

129

#95

—

142

—

130

C₂×Q₈

—

131

C₂×Q₈

—

142

—

132

C₂×Q₈

—

142

—

133

C₂×Q₈

—

142

—

134

C₂×Q₈

—

142

—

135

C₂×Q₈

—

142

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

142

—

139

C₂²×C₄

—

142

—

140

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

141

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

142

C₂²×C₄

C₂²×D₇₁

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

142

—

145

C₂×C₈

—

142

—

146

C₂×C₈

—

142

—

147

C₂×C₈

—

142

—

148

C₂×C₈

—

142

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

142

—

154

C₄⋊C₄

—

142

—

155

C₄⋊C₄

D₂₈₄

yes

—

156

C₄²

—

157

C₄²

—

158

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

159

Q₈

—

284

—

160

Q₈

—

284

—

161

Q₈

—

142

—

162

Q₈

—

142

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

284

—

175

C₂×C₄

—

284

—

176

C₂×C₄

—

284

—

177

C₂×C₄

—

284

—

178

C₂×C₄

—

284

—

179

C₂×C₄

—

142

—

180

C₂×C₄

#11765853

yes

—

181

C₂×C₄

D₄×D₇₁

yes

characteristic

182

C₂×C₄

C₂×D₂₈₄

yes

characteristic

183

C₂×C₄

Q₈×D₇₁

yes

characteristic

184

C₂×C₄

#11765864

yes

characteristic

185

C₂×C₄

#11765865

yes

characteristic

186

C₂×C₄

#11765866

yes

characteristic

187

C₈

—

284

—

188

C₈

—

142

—

189

C₂²

#14556100

yes

characteristic

190

C₂²

#14556102

yes

characteristic

191

C₂²

#14556104

yes

characteristic

192

C₂²

C₂×#11765853

yes

characteristic

193

C₂²

#14556148

yes

characteristic

194

C₂²

#14556158

yes

characteristic

195

C₂²

#14556159

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

284

—

201

C₄

—

568

—

202

C₄

#14556103

yes

characteristic

203

C₄

#14556106

yes

characteristic

204

C₄

#14556108

yes

characteristic

205

C₄

#14556123

yes

characteristic

206

C₂

#17985921

yes

characteristic

207

C₂

#17986204

yes

characteristic

208

C₂

#17986221

yes

characteristic

209

C₂

#17986226

yes

characteristic

210

C₂

#17986255

yes

characteristic

211

C₂

#17986281

yes

characteristic

212

C₂

#17986303

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#389×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	#389	D₇₁
hypocenter	C₇₁	#2066
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#389×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
107. C₇₁	#2066	semidirect

H ≅ G/N

type

107. C₇₁

#2066

semidirect

Subgroups — #22589718

Conjugacy classes of subgroups

Complements