Group #31915937 — subgroups

Subgroups — #31915937

Order 18176.

462 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 143 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	#22590096	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985815	C₂	1	4	yes	maximal
3	C₄×#14556064	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	C₂²	1	2	yes	—
6	C₂×#14556064	C₂²	1	2	yes	—
7	C₄×#11765849	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
9	#17985815	—	2	8	no	—
10	#17985764	—	2	1	no	—
11	C₂²×Q₁₆×C₇₁	—	2	1	no	—
12	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
14	#494×C₇₁	—	2	2	no	—
15	#494×C₇₁	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#11765849	—	2	2	no	—
18	#14556068	—	2	4	no	—
19	C₂×C₇₁⋊C₁₆	—	2	2	no	—
20	#14556064	—	2	2	no	—
21	#14556064	D₄	1	4	yes	—
22	#14556039	—	4	1	no	—
23	C₄×C₇₁⋊C₈	—	4	1	no	—
24	C₂²×Q₈×C₇₁	—	2	2	no	—
25	C₂×Q₁₆×C₇₁	—	2	4	no	—
26	C₂×Q₁₆×C₇₁	—	2	4	no	—
27	C₂×Q₁₆×C₇₁	—	2	2	no	—
28	C₂×Q₁₆×C₇₁	—	2	4	no	—
29	C₂×Q₁₆×C₇₁	—	4	4	no	—
30	C₂×Q₁₆×C₇₁	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₇₁	—	2	4	no	—
34	C₄⋊Q₈×C₇₁	—	4	1	no	—
35	C₄⋊Q₈×C₇₁	—	4	1	no	—
36	C₂×C₄⋊C₄×C₇₁	—	4	2	no	—
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₁	D₄	1	2	yes	—
40	#11765849	—	2	4	no	—
41	#11765849	—	4	2	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₇₁⋊C₁₆	—	4	2	no	—
44	C₂×Q₈×C₇₁	—	2	4	no	—
45	C₂×Q₈×C₇₁	—	4	2	no	—
46	C₂×Q₈×C₇₁	—	4	4	no	—
47	C₂×Q₈×C₇₁	—	4	4	no	—
48	C₂²×C₄×C₇₁	—	4	2	no	—
49	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
50	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	Q₁₆×C₇₁	—	2	8	no	—
52	Q₁₆×C₇₁	—	4	8	no	—
53	Q₁₆×C₇₁	—	4	2	no	—
54	Q₁₆×C₇₁	—	4	2	no	—
55	C₂×C₈×C₇₁	—	2	2	no	—
56	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	#41	1	2	yes	—
61	C₄⋊C₄×C₇₁	—	4	4	no	—
62	C₄⋊C₄×C₇₁	—	4	4	no	—
63	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₇₁	D₈	1	1	yes	characteristic
65	C₄²×C₇₁	D₈	1	1	yes	characteristic
66	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₇₁	—	4	4	no	—
68	Q₈×C₇₁	—	4	4	no	—
69	Q₈×C₇₁	—	8	4	no	—
70	C₂×C₄×C₇₁	—	2	4	no	—
71	C₂×C₄×C₇₁	—	4	4	no	—
72	C₂×C₄×C₇₁	—	8	4	no	—
73	C₂×C₄×C₇₁	#102	1	2	yes	—
74	C₂×C₄×C₇₁	#102	1	2	yes	—
75	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#121	1	2	yes	—
77	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
79	C₈×C₇₁	—	2	2	no	—
80	C₈×C₇₁	C₄×D₄	1	2	yes	—
81	C₈×C₇₁	#124	1	1	yes	characteristic
82	C₈×C₇₁	#127	1	1	yes	characteristic
83	C₇₁⋊C₈	—	4	4	no	—
84	C₂²×C₇₁	—	2	2	no	—
85	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
86	C₂²×C₇₁	#460	1	2	yes	—
87	C₄×C₇₁	—	2	2	no	—
88	C₄×C₇₁	—	8	4	no	—
89	C₄×C₇₁	#362	1	4	yes	complemented
90	C₄×C₇₁	#390	1	1	yes	characteristic
91	C₄×C₇₁	#418	1	1	yes	characteristic
92	C₄×C₇₁	#459	1	2	yes	—
93	#13256	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	—	2	2	no	—
95	C₂×C₇₁	#1982	1	1	yes	characteristic
96	C₂×C₇₁	C₂×#362	1	2	yes	—
97	C₂×#494	D₇₁	1	1	yes	characteristic
98	#2150	—	71	1	no	—
99	C₂×#362	—	71	4	no	—
100	C₄×#108	—	71	1	no	—
101	C₇₁	#13256	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₁₆	—	2	1	no	—
103	C₂²×Q₁₆	C₂×D₇₁	1	2	yes	—
104	C₂×C₄⋊Q₈	—	2	1	no	—
105	C₂×M₃₂	—	71	2	no	—
106	#494	—	2	2	no	—
107	#494	Dic₇₁	1	4	yes	—
108	C₂×#108	—	71	2	no	—
109	C₄×M₁₆	—	142	1	no	—
110	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
111	C₄⋊C₁₆	—	142	2	no	—
112	#362	—	142	8	no	—
113	#341	—	142	1	no	—
114	C₂²×Q₈	—	2	2	no	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	4	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	—	2	4	no	—
119	C₂×Q₁₆	—	4	4	no	—
120	C₂×Q₁₆	C₂×Dic₇₁	1	4	yes	—
121	C₂×M₁₆	—	142	2	no	—
122	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	1	no	—
126	C₄⋊Q₈	—	4	1	no	—
127	C₂×C₄⋊C₄	—	4	2	no	—
128	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
129	M₃₂	—	142	4	no	—
130	C₂×C₁₆	—	142	2	no	—
131	#108	—	71	4	no	—
132	#108	—	142	2	no	—
133	#97	—	284	1	no	—
134	C₄×C₈	—	284	1	no	—
135	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
136	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	#11765853	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	4	8	no	—
146	Q₁₆	—	4	2	no	—
147	Q₁₆	—	4	2	no	—
148	M₁₆	—	284	2	no	—
149	M₁₆	—	142	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	284	2	no	—
152	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
153	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
154	C₂×C₈	D₄×D₇₁	1	2	yes	—
155	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
156	C₂×C₈	#11765864	1	2	yes	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
160	C₄²	#11765850	1	1	yes	characteristic
161	C₄²	#11765850	1	1	yes	characteristic
162	C₁₆	—	284	2	no	—
163	C₂³	C₂×#11765853	1	1	yes	characteristic
164	Q₈	—	4	4	no	—
165	Q₈	—	4	4	no	—
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	#14556049	1	2	yes	—
171	C₂×C₄	#14556049	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
173	C₂×C₄	#14556099	1	2	yes	—
174	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
175	C₂×C₄	#14556154	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	284	4	no	—
178	C₈	D₄×Dic₇₁	1	2	yes	—
179	C₈	#14556115	1	1	yes	characteristic
180	C₈	#14556124	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₂×#14556049	1	1	yes	characteristic
183	C₂²	#17986196	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	#17985815	1	4	yes	complemented
187	C₄	#17985902	1	1	yes	characteristic
188	C₄	#17986004	1	1	yes	characteristic
189	C₄	#17986191	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	#22589268	1	1	yes	characteristic
192	C₂	C₂×#17985815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590096

C₂

yes

maximal, characteristic

C₂×#17985815

C₂

yes

maximal

C₄×#14556064

C₂

yes

maximal, characteristic

C₂×#494×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

—

C₂×#14556064

C₂²

yes

—

C₄×#11765849

—

C₄⋊(C₇₁⋊C₁₆)

—

#17985815

—

#17985764

—

C₂²×Q₁₆×C₇₁

—

C₂²×Q₁₆×C₇₁

C₂²

yes

—

C₂×C₄⋊Q₈×C₇₁

—

#494×C₇₁

—

#494×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

#14556064

—

#14556064

D₄

yes

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

SD₁₆

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₂³×C₇₁

C₂×SD₁₆

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#121

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₄×D₄

yes

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#102

yes

characteristic

C₂²×C₇₁

#460

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#13256

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1982

yes

characteristic

C₂×C₇₁

C₂×#362

yes

—

C₂×#494

D₇₁

yes

characteristic

#2150

—

C₂×#362

—

100

C₄×#108

—

101

C₇₁

#13256

yes

characteristic, complemented, Sylow

102

C₂²×Q₁₆

—

103

C₂²×Q₁₆

C₂×D₇₁

yes

—

104

C₂×C₄⋊Q₈

—

105

C₂×M₃₂

—

106

#494

—

107

#494

Dic₇₁

yes

—

108

C₂×#108

—

109

C₄×M₁₆

—

142

—

110

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

111

C₄⋊C₁₆

—

142

—

112

#362

—

142

—

113

#341

—

142

—

114

C₂²×Q₈

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

C₂×Dic₇₁

yes

—

121

C₂×M₁₆

—

142

—

122

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

123

C₂²×C₈

C₂²×D₇₁

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

129

M₃₂

—

142

—

130

C₂×C₁₆

—

142

—

131

#108

—

132

#108

—

142

—

133

#97

—

284

—

134

C₄×C₈

—

284

—

135

C₄×C₈

C₂×Dic₇₁

yes

—

136

C₄×C₈

C₂²⋊D₇₁

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

#11765853

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

144

Q₁₆

—

145

Q₁₆

—

146

Q₁₆

—

147

Q₁₆

—

148

M₁₆

—

284

—

149

M₁₆

—

142

—

150

C₂×C₈

—

151

C₂×C₈

—

284

—

152

C₂×C₈

C₂²⋊Dic₇₁

yes

—

153

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

154

C₂×C₈

D₄×D₇₁

yes

—

155

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

156

C₂×C₈

#11765864

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

C₂²⋊Dic₇₁

yes

—

160

C₄²

#11765850

yes

characteristic

161

C₄²

#11765850

yes

characteristic

162

C₁₆

—

284

—

163

C₂³

C₂×#11765853

yes

characteristic

164

Q₈

—

165

Q₈

—

166

Q₈

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#14556049

yes

—

171

C₂×C₄

#14556049

yes

—

172

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

173

C₂×C₄

#14556099

yes

—

174

C₂×C₄

C₂×#11765850

yes

characteristic

175

C₂×C₄

#14556154

yes

—

176

C₈

—

177

C₈

—

284

—

178

C₈

D₄×Dic₇₁

yes

—

179

C₈

#14556115

yes

characteristic

180

C₈

#14556124

yes

characteristic

181

C₂²

—

182

C₂²

C₂×#14556049

yes

characteristic

183

C₂²

#17986196

yes

—

184

C₄

—

185

C₄

—

186

C₄

#17985815

yes

complemented

187

C₄

#17985902

yes

characteristic

188

C₄

#17986004

yes

characteristic

189

C₄

#17986191

yes

—

190

C₂

—

191

C₂

#22589268

yes

characteristic

192

C₂

C₂×#17985815

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556049
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#494×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#494	D₇₁
hypocenter	C₇₁	#13256
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂²

C₂×#14556049

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#494×C₇₁

C₂

socle

C₂²×C₇₁

C₂×#102

prime radical

C₂³×C₇₁

C₂×SD₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
89. C₄×C₇₁	#362	semidirect
101. C₇₁	#13256	semidirect
186. C₄	#17985815	semidirect

H ≅ G/N

type

89. C₄×C₇₁

#362

semidirect

101. C₇₁

#13256

semidirect

186. C₄

#17985815

semidirect

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

subgroup	index
#22590096	2
C₂×#17985815	2
C₄×#14556064	2
C₂×#494×C₇₁	2
#13256	71

subgroup

index

C₂×#494×C₇₁

Subgroups — #31915937

Conjugacy classes of subgroups

Complements

Maximal subgroups