Group #31897412 — subgroups

Subgroups — #31897412

Order 18176.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22588478	C₂	1	1	yes	maximal, characteristic
2	#22586795	C₂	1	2	yes	maximal
3	#22586795	C₂	1	2	yes	maximal
4	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#3160	C₂	1	1	yes	maximal, characteristic
6	#17986023	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#17985764	—	2	2	no	—
10	Q₈⋊(C₇₁⋊C₈)	—	2	4	no	—
11	Q₈⋊(C₇₁⋊C₈)	—	2	4	no	—
12	#17985721	—	2	2	no	—
13	#533×C₇₁	C₂²	1	2	yes	—
14	#500×C₇₁	C₄	1	1	yes	characteristic
15	#500×C₇₁	C₄	1	1	yes	characteristic
16	#499×C₇₁	—	2	2	no	—
17	#498×C₇₁	C₄	1	2	yes	—
18	#427×C₇₁	—	2	2	no	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#11765849	—	2	2	no	—
21	C₂²×C₇₁⋊C₈	—	2	1	no	—
22	C₂²×C₇₁⋊C₈	—	2	1	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	C₄×C₇₁⋊C₈	—	2	1	no	—
26	C₄×C₇₁⋊C₈	—	2	1	no	—
27	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
28	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
29	C₂²×C₈×C₇₁	D₄	1	2	yes	—
30	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
32	#125×C₇₁	—	2	2	no	—
33	C₂²⋊Q₈×C₇₁	—	2	2	no	—
34	C₂²⋊Q₈×C₇₁	—	2	2	no	—
35	C₄×Q₈×C₇₁	—	2	2	no	—
36	C₄×Q₈×C₇₁	—	2	2	no	—
37	#117×C₇₁	—	2	2	no	—
38	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	#107×C₇₁	—	2	2	no	—
40	#107×C₇₁	—	2	2	no	—
41	#106×C₇₁	—	2	4	no	—
42	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	D₄	1	2	yes	—
45	#11765849	—	4	4	no	—
46	C₂×C₇₁⋊C₈	—	2	4	no	—
47	C₂×C₇₁⋊C₈	—	2	2	no	—
48	C₂×C₇₁⋊C₈	—	2	2	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₂×Q₈×C₇₁	—	2	2	no	—
52	C₂×Q₈×C₇₁	—	2	2	no	—
53	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
56	C₂×C₈×C₇₁	—	2	4	no	—
57	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₇₁	—	2	2	no	—
60	C₄⋊C₄×C₇₁	—	2	2	no	—
61	C₄⋊C₄×C₇₁	—	2	2	no	—
62	C₄⋊C₄×C₇₁	—	2	2	no	—
63	C₄⋊C₄×C₇₁	—	4	2	no	—
64	C₄⋊C₄×C₇₁	—	4	2	no	—
65	C₂²⋊C₄×C₇₁	—	4	2	no	—
66	C₄²×C₇₁	—	4	2	no	—
67	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₇₁	#124	1	1	yes	characteristic
72	Q₈×C₇₁	—	4	4	no	—
73	Q₈×C₇₁	—	4	4	no	—
74	C₂×C₄×C₇₁	—	2	2	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	—	4	2	no	—
77	C₂×C₄×C₇₁	—	4	2	no	—
78	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
82	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
83	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#137	1	4	yes	—
85	C₈×C₇₁	—	2	2	no	—
86	C₈×C₇₁	—	2	2	no	—
87	C₇₁⋊C₈	—	2	4	no	—
88	C₇₁⋊C₈	—	4	2	no	—
89	C₇₁⋊C₈	—	8	2	no	—
90	C₂²×C₇₁	—	2	1	no	—
91	C₂²×C₇₁	—	2	1	no	—
92	C₂²×C₇₁	#390	1	1	yes	characteristic
93	C₂²×C₇₁	#491	1	2	yes	—
94	C₄×C₇₁	—	2	1	no	—
95	C₄×C₇₁	—	2	1	no	—
96	C₄×C₇₁	—	8	2	no	—
97	C₄×C₇₁	—	8	2	no	—
98	C₄×C₇₁	#419	1	2	yes	—
99	C₄×C₇₁	#443	1	2	yes	—
100	C₄×C₇₁	#444	1	2	yes	—
101	C₄×C₇₁	#497	1	2	yes	—
102	#9859	—	71	1	no	maximal, Sylow
103	C₂×C₇₁	—	2	2	no	—
104	C₂×C₇₁	#1960	1	1	yes	characteristic
105	C₂×C₇₁	#1974	1	1	yes	characteristic
106	C₂×C₇₁	#1975	1	1	yes	characteristic
107	#3160	D₇₁	1	1	yes	characteristic
108	#1837	—	71	1	no	—
109	#1493	—	71	2	no	—
110	#1493	—	71	2	no	—
111	C₂×#322	—	71	1	no	—
112	C₇₁	#9859	1	1	yes	characteristic, complemented, Sylow
113	#533	C₂×D₇₁	1	2	yes	—
114	#500	Dic₇₁	1	1	yes	characteristic
115	#500	Dic₇₁	1	1	yes	characteristic
116	#499	—	2	2	no	—
117	#498	Dic₇₁	1	2	yes	—
118	#427	—	2	2	no	—
119	#423	—	71	2	no	—
120	C₂×C₄×C₈	—	71	1	no	—
121	C₂×C₄×C₈	—	71	1	no	—
122	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
123	#341	—	142	2	no	—
124	Q₈⋊C₈	—	142	4	no	—
125	Q₈⋊C₈	—	142	4	no	—
126	#322	—	142	2	no	—
127	C₂×M₁₆	—	142	2	no	—
128	C₂²×C₈	—	142	1	no	—
129	C₂²×C₈	—	142	1	no	—
130	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
131	C₄⋊Q₈	C₂×Dic₇₁	1	2	yes	—
132	C₄⋊Q₈	C₂×Dic₇₁	1	2	yes	—
133	#125	—	2	2	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₄×Q₈	—	2	2	no	—
138	#117	—	2	2	no	—
139	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
140	#107	—	2	2	no	—
141	#107	—	2	2	no	—
142	#106	—	2	4	no	—
143	C₄⋊C₈	—	142	2	no	—
144	C₄⋊C₈	—	142	2	no	—
145	C₄×C₈	—	71	2	no	—
146	C₄×C₈	—	71	2	no	—
147	C₄×C₈	—	142	1	no	—
148	C₄×C₈	—	142	1	no	—
149	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
150	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
155	C₂²×C₄	#11765864	1	1	yes	characteristic
156	C₂²×C₄	#11765864	1	1	yes	characteristic
157	M₁₆	—	284	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	284	2	no	—
160	C₂×C₈	—	284	2	no	—
161	C₂×C₈	—	142	2	no	—
162	C₂×C₈	—	142	4	no	—
163	C₂×C₈	—	142	2	no	—
164	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
165	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
175	C₄²	D₄×D₇₁	1	1	yes	characteristic
176	C₄²	D₄×D₇₁	1	1	yes	characteristic
177	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
178	C₂³	#14556115	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	Q₈	—	4	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
186	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
188	C₂×C₄	#14556099	1	1	yes	characteristic
189	C₂×C₄	#14556099	1	1	yes	characteristic
190	C₂×C₄	#14556124	1	1	yes	characteristic
191	C₂×C₄	#14556159	1	4	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	284	2	no	—
195	C₈	—	568	2	no	—
196	C₈	—	142	4	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#17985902	1	1	yes	characteristic
200	C₂²	#17986367	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	8	2	no	—
204	C₄	—	8	2	no	—
205	C₄	#17986008	1	2	yes	—
206	C₄	#17986116	1	2	yes	—
207	C₄	#17986121	1	2	yes	—
208	C₄	#17986391	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	#22589161	1	1	yes	characteristic
211	C₂	#22589229	1	1	yes	characteristic
212	C₂	#22589234	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588478

C₂

yes

maximal, characteristic

#22586795

C₂

yes

maximal

#22586795

C₂

yes

maximal

C₂×#17985721

C₂

yes

maximal, characteristic

C₇₁×#3160

C₂

yes

maximal, characteristic

#17986023

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

#17985764

—

Q₈⋊(C₇₁⋊C₈)

—

Q₈⋊(C₇₁⋊C₈)

—

#17985721

—

#533×C₇₁

C₂²

yes

—

#500×C₇₁

C₄

yes

characteristic

#500×C₇₁

C₄

yes

characteristic

#499×C₇₁

—

#498×C₇₁

C₄

yes

—

#427×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄⋊Q₈×C₇₁

C₂×C₄

yes

—

C₄⋊Q₈×C₇₁

C₂×C₄

yes

—

#125×C₇₁

—

C₂²⋊Q₈×C₇₁

—

C₂²⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

#117×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

#107×C₇₁

—

#106×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

C₂²×C₄×C₇₁

yes

C₂²×C₄×C₇₁

yes

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

#124

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

#491

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#419

yes

—

C₄×C₇₁

#443

yes

—

100

C₄×C₇₁

#444

yes

—

101

C₄×C₇₁

#497

yes

—

102

#9859

—

maximal, Sylow

103

C₂×C₇₁

—

104

C₂×C₇₁

#1960

yes

characteristic

105

C₂×C₇₁

#1974

yes

characteristic

106

C₂×C₇₁

#1975

yes

characteristic

107

#3160

D₇₁

yes

characteristic

108

#1837

—

109

#1493

—

110

#1493

—

111

C₂×#322

—

112

C₇₁

#9859

yes

characteristic, complemented, Sylow

113

#533

C₂×D₇₁

yes

—

114

#500

Dic₇₁

yes

characteristic

115

#500

Dic₇₁

yes

characteristic

116

#499

—

117

#498

Dic₇₁

yes

—

118

#427

—

119

#423

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

—

122

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

123

#341

—

142

—

124

Q₈⋊C₈

—

142

—

125

Q₈⋊C₈

—

142

—

126

#322

—

142

—

127

C₂×M₁₆

—

142

—

128

C₂²×C₈

—

142

—

129

C₂²×C₈

—

142

—

130

C₂²×C₈

C₂²⋊D₇₁

yes

—

131

C₄⋊Q₈

C₂×Dic₇₁

yes

—

132

C₄⋊Q₈

C₂×Dic₇₁

yes

—

133

#125

—

134

C₂²⋊Q₈

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

137

C₄×Q₈

—

138

#117

—

139

C₂×C₄²

C₂²×D₇₁

yes

characteristic

140

#107

—

141

#107

—

142

#106

—

143

C₄⋊C₈

—

142

—

144

C₄⋊C₈

—

142

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

142

—

148

C₄×C₈

—

142

—

149

C₄×C₈

C₂×Dic₇₁

yes

characteristic

150

C₄×C₈

C₂×Dic₇₁

yes

characteristic

151

C₄×C₈

C₂²⋊D₇₁

yes

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

155

C₂²×C₄

#11765864

yes

characteristic

156

C₂²×C₄

#11765864

yes

characteristic

157

M₁₆

—

284

—

158

C₂×C₈

—

159

C₂×C₈

—

284

—

160

C₂×C₈

—

284

—

161

C₂×C₈

—

142

—

162

C₂×C₈

—

142

—

163

C₂×C₈

—

142

—

164

C₂×C₈

C₂²⋊Dic₇₁

yes

—

165

C₂×C₈

C₂²⋊Dic₇₁

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₄²

C₂²×Dic₇₁

yes

characteristic

175

C₄²

D₄×D₇₁

yes

characteristic

176

C₄²

D₄×D₇₁

yes

characteristic

177

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

178

C₂³

#14556115

yes

characteristic

179

Q₈

—

180

Q₈

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

186

C₂×C₄

D₄×Dic₇₁

yes

characteristic

187

C₂×C₄

D₄×Dic₇₁

yes

characteristic

188

C₂×C₄

#14556099

yes

characteristic

189

C₂×C₄

#14556099

yes

characteristic

190

C₂×C₄

#14556124

yes

characteristic

191

C₂×C₄

#14556159

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

284

—

195

C₈

—

568

—

196

C₈

—

142

—

197

C₂²

—

198

C₂²

—

199

C₂²

#17985902

yes

characteristic

200

C₂²

#17986367

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

#17986008

yes

—

206

C₄

#17986116

yes

—

207

C₄

#17986121

yes

—

208

C₄

#17986391

yes

—

209

C₂

—

210

C₂

#22589161

yes

characteristic

211

C₂

#22589229

yes

characteristic

212

C₂

#22589234

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#3160	C₂
socle	C₂²×C₇₁	#390
prime radical	C₂³×C₇₁	#124
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#3160	D₇₁
hypocenter	C₇₁	#9859
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#14556124

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#3160

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
112. C₇₁	#9859	semidirect

H ≅ G/N

type

112. C₇₁

#9859

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22588478	2
#22586795	2
C₂×#17985721	2
C₇₁×#3160	2
#9859	71

subgroup

index

C₇₁×#3160

Subgroups — #31897412

Conjugacy classes of subgroups

Complements

Maximal subgroups