Group #31915495 — subgroups

Subgroups — #31915495

Order 18176.

318 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 131 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×#17985825	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985774	C₂	1	1	yes	maximal, characteristic
3	#22589071	C₂	1	2	yes	maximal
4	#22586431	C₂	1	2	yes	maximal
5	C₂×#367×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14556062	C₂²	1	2	yes	—
8	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
9	#17985895	—	2	2	no	—
10	#17985825	C₄	1	4	yes	complemented
11	#17985774	—	2	2	no	—
12	#17985761	—	2	2	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#107×C₇₁	C₂²	1	2	yes	—
15	#367×C₇₁	—	2	2	no	—
16	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇₁	—	2	2	no	—
18	C₂×Dic₇₁⋊C₄	—	4	2	no	—
19	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	—	2	1	no	—
22	C₂×C₇₁⋊C₁₆	Q₈	1	2	yes	—
23	#14556062	—	2	2	no	—
24	#14556062	C₂×C₄	1	4	yes	—
25	C₈×Dic₇₁	—	2	1	no	—
26	C₈×Dic₇₁	—	2	1	no	—
27	C₇₁⋊#95	—	4	2	no	—
28	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
32	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
33	#107×C₇₁	—	2	2	no	—
34	#107×C₇₁	—	2	2	no	—
35	C₂²×Dic₇₁	—	4	2	no	—
36	C₂²×Dic₇₁	D₈	1	2	yes	—
37	C₂×C₇₁⋊C₈	—	2	1	no	—
38	C₂×C₇₁⋊C₈	—	2	1	no	—
39	C₂×C₇₁⋊C₈	Q₁₆	1	2	yes	—
40	C₄⋊Dic₇₁	—	2	4	no	—
41	C₄⋊Dic₇₁	—	4	2	no	—
42	Dic₇₁⋊C₄	—	8	4	no	—
43	C₄×Dic₇₁	—	2	1	no	—
44	C₄×Dic₇₁	—	2	1	no	—
45	C₇₁⋊C₁₆	—	2	2	no	—
46	C₂²×C₄×C₇₁	—	4	2	no	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₇₁	—	4	2	no	—
55	C₄⋊C₄×C₇₁	—	4	2	no	—
56	C₁₆×C₇₁	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	—	8	2	no	—
59	C₂×C₄×C₇₁	—	8	2	no	—
60	C₂×C₄×C₇₁	—	8	2	no	—
61	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
67	C₂×Dic₇₁	—	2	2	no	—
68	C₂×Dic₇₁	—	2	2	no	—
69	C₂×Dic₇₁	—	2	2	no	—
70	C₂×Dic₇₁	—	4	4	no	—
71	C₂×Dic₇₁	—	8	2	no	—
72	C₂×Dic₇₁	—	8	2	no	—
73	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₇₁	#125	1	1	yes	characteristic
77	C₇₁⋊C₈	—	2	2	no	—
78	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₇₁	#466	1	1	yes	characteristic
80	C₂²×C₇₁	#500	1	1	yes	characteristic
81	C₂²×C₇₁	#509	1	2	yes	—
82	C₂²×C₇₁	#510	1	2	yes	—
83	C₄×C₇₁	—	16	2	no	—
84	C₄×C₇₁	#389	1	1	yes	characteristic
85	C₄×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
86	C₄×C₇₁	#467	1	1	yes	characteristic
87	C₄×C₇₁	#499	1	1	yes	characteristic
88	Dic₇₁	—	4	2	no	—
89	Dic₇₁	—	8	4	no	—
90	#13168	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	#1942	1	1	yes	characteristic
92	C₂×C₇₁	#2176	1	1	yes	characteristic
93	C₂×C₇₁	#2177	1	1	yes	characteristic
94	C₂×C₇₁	#2220	1	2	yes	—
95	C₂×C₇₁	#2255	1	2	yes	—
96	C₂×#367	D₇₁	1	1	yes	characteristic
97	C₂×#366	—	71	1	no	—
98	C₂×#346	—	71	1	no	—
99	#1942	—	71	2	no	—
100	#1380	—	71	2	no	—
101	C₇₁	#13168	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	—	71	1	no	—
103	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
104	C₂×#107	—	71	2	no	—
105	C₂×#107	C₂×D₇₁	1	2	yes	—
106	C₂×C₄×C₈	—	71	1	no	—
107	#389	—	142	2	no	—
108	#367	—	2	2	no	—
109	#366	—	71	4	no	—
110	#346	—	142	2	no	—
111	#340	—	142	2	no	—
112	C₂²×C₈	—	71	1	no	—
113	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	284	2	no	—
116	C₂×C₄⋊C₄	—	142	2	no	—
117	C₂×C₄²	—	71	1	no	—
118	C₂×C₁₆	—	71	2	no	—
119	C₂×C₁₆	—	142	1	no	—
120	C₂×C₁₆	—	142	1	no	—
121	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
122	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	71	4	no	—
127	#107	—	142	2	no	—
128	C₄×C₈	—	142	1	no	—
129	C₄×C₈	—	142	1	no	—
130	#95	—	284	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	284	2	no	—
133	C₂²×C₄	—	71	2	no	—
134	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
135	C₂×C₈	—	71	2	no	—
136	C₂×C₈	—	142	1	no	—
137	C₂×C₈	—	142	1	no	—
138	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
142	C₂×C₈	#11765864	1	1	yes	characteristic
143	C₂×C₈	#11765866	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	284	2	no	—
147	C₄⋊C₄	—	568	4	no	—
148	C₄⋊C₄	—	142	4	no	—
149	C₄²	—	142	1	no	—
150	C₄²	—	142	1	no	—
151	C₁₆	—	142	2	no	—
152	C₁₆	Dic₇₁⋊C₄	1	4	yes	complemented
153	C₂³	D₈×D₇₁	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	284	4	no	—
158	C₂×C₄	—	568	2	no	—
159	C₂×C₄	—	568	2	no	—
160	C₂×C₄	—	142	2	no	—
161	C₂×C₄	—	142	2	no	—
162	C₂×C₄	—	142	2	no	—
163	C₂×C₄	#14556090	1	1	yes	characteristic
164	C₂×C₄	#14556099	1	1	yes	characteristic
165	C₂×C₄	#14556127	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
167	C₂×C₄	#14556147	1	1	yes	characteristic
168	C₂×C₄	#14556151	1	1	yes	characteristic
169	C₈	—	142	2	no	—
170	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
171	C₈	#14556093	1	1	yes	characteristic
172	C₈	#14556104	1	1	yes	characteristic
173	C₈	#14556119	1	1	yes	characteristic
174	C₂²	#17986081	1	1	yes	characteristic
175	C₂²	#17986232	1	1	yes	characteristic
176	C₂²	#17986403	1	1	yes	characteristic
177	C₂²	#17986436	1	2	yes	—
178	C₂²	#17986438	1	2	yes	—
179	C₄	—	16	2	no	—
180	C₄	—	284	2	no	—
181	C₄	—	568	4	no	—
182	C₄	#17985895	1	1	yes	characteristic
183	C₄	#17986093	1	1	yes	characteristic
184	C₄	#17986237	1	1	yes	characteristic
185	C₄	#17986400	1	1	yes	characteristic
186	C₂	#22589071	1	1	yes	characteristic
187	C₂	#22590210	1	1	yes	characteristic
188	C₂	#22590215	1	1	yes	characteristic
189	C₂	#22590450	1	2	yes	—
190	C₂	#22590625	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985825

C₂

yes

maximal, characteristic

C₂×#17985774

C₂

yes

maximal, characteristic

#22589071

C₂

yes

maximal

#22586431

C₂

yes

maximal

C₂×#367×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×#14556062

C₂²

yes

—

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

#17985895

—

#17985825

C₄

yes

complemented

#17985774

—

#17985761

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#107×C₇₁

C₂²

yes

—

#367×C₇₁

—

C₂²×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₇₁

—

C₂×Dic₇₁⋊C₄

—

C₂×C₄×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

Q₈

yes

—

#14556062

—

#14556062

C₂×C₄

yes

—

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

Q₈

yes

—

#107×C₇₁

—

#107×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₈

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

Q₁₆

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

C₄⋊C₄

yes

complemented

C₂³×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

characteristic

C₈×C₇₁

C₂²⋊Q₈

yes

characteristic

C₈×C₇₁

#125

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₄×D₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#510

yes

—

C₄×C₇₁

—

C₄×C₇₁

#389

yes

characteristic

C₄×C₇₁

C₄×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2255

yes

—

C₂×#367

D₇₁

yes

—

—

—

100

#1380

—

101

C₇₁

#13168

yes

characteristic, complemented, Sylow

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

104

C₂×#107

—

105

C₂×#107

C₂×D₇₁

yes

—

106

C₂×C₄×C₈

—

107

#389

—

142

—

108

#367

—

109

#366

—

110

#346

—

142

—

111

#340

—

142

—

112

C₂²×C₈

—

113

C₂²×C₈

C₂²×D₇₁

yes

characteristic

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

284

—

116

C₂×C₄⋊C₄

—

142

—

117

C₂×C₄²

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

142

—

120

C₂×C₁₆

—

142

—

121

C₂×C₁₆

Dic₁₄₂

yes

—

122

C₂×C₁₆

C₄×D₇₁

yes

—

123

C₂×C₁₆

C₂²⋊D₇₁

yes

—

124

#107

—

125

#107

—

126

#107

—

127

#107

—

142

—

128

C₄×C₈

—

142

—

129

C₄×C₈

—

142

—

130

#95

—

284

—

131

C₂²×C₄

—

132

C₂²×C₄

—

284

—

133

C₂²×C₄

—

134

C₂²×C₄

D₄×D₇₁

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

142

—

137

C₂×C₈

—

142

—

138

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

139

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

140

C₂×C₈

Q₈×D₇₁

yes

characteristic

141

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

142

C₂×C₈

#11765864

yes

characteristic

143

C₂×C₈

#11765866

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

284

—

147

C₄⋊C₄

—

568

—

148

C₄⋊C₄

—

142

—

149

C₄²

—

142

—

150

C₄²

—

142

—

151

C₁₆

—

142

—

152

C₁₆

Dic₇₁⋊C₄

yes

complemented

153

C₂³

D₈×D₇₁

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

284

—

158

C₂×C₄

—

568

—

159

C₂×C₄

—

568

—

160

C₂×C₄

—

142

—

161

C₂×C₄

—

142

—

162

C₂×C₄

—

142

—

163

C₂×C₄

#14556090

yes

characteristic

164

C₂×C₄

#14556099

yes

characteristic

165

C₂×C₄

#14556127

yes

characteristic

166

C₂×C₄

Q₁₆×D₇₁

yes

characteristic

167

C₂×C₄

#14556147

yes

characteristic

168

C₂×C₄

#14556151

yes

characteristic

169

C₈

—

142

—

170

C₈

C₂×Dic₇₁⋊C₄

yes

characteristic

171

C₈

#14556093

yes

characteristic

172

C₈

#14556104

yes

characteristic

173

C₈

#14556119

yes

characteristic

174

C₂²

#17986081

yes

characteristic

175

C₂²

#17986232

yes

characteristic

176

C₂²

#17986403

yes

characteristic

177

C₂²

#17986436

yes

—

178

C₂²

#17986438

yes

—

179

C₄

—

180

C₄

—

284

—

181

C₄

—

568

—

182

C₄

#17985895

yes

characteristic

183

C₄

#17986093

yes

characteristic

184

C₄

#17986237

yes

characteristic

185

C₄

#17986400

yes

characteristic

186

C₂

#22589071

yes

characteristic

187

C₂

#22590210

yes

characteristic

188

C₂

#22590215

yes

characteristic

189

C₂

#22590450

yes

—

190

C₂

#22590625

yes

—

	subgroup	G/H
center	C₂³	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#367×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#367	D₇₁
hypocenter	C₇₁	#13168
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂³

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#367×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂×D₈

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
10. #17985825	C₄	semidirect
56. C₁₆×C₇₁	C₄⋊C₄	semidirect
101. C₇₁	#13168	semidirect
152. C₁₆	Dic₇₁⋊C₄	semidirect

H ≅ G/N

type

10. #17985825

C₄

semidirect

56. C₁₆×C₇₁

C₄⋊C₄

semidirect

101. C₇₁

#13168

semidirect

152. C₁₆

Dic₇₁⋊C₄

semidirect

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

subgroup	index
C₂×#17985825	2
C₂×#17985774	2
#22589071	2
#22586431	2
C₂×#367×C₇₁	2
#13168	71

subgroup

index

C₂×#367×C₇₁

Subgroups — #31915495

Conjugacy classes of subgroups

Complements

Maximal subgroups