Group #31915510 — subgroups

Subgroups — #31915510

Order 18176.

274 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 115 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×#17985829	C₂	1	1	yes	maximal, characteristic
2	#22589936	C₂	1	1	yes	maximal, characteristic
3	#22589123	C₂	1	2	yes	maximal
4	#22586448	C₂	1	2	yes	maximal
5	C₂×#366×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	1	yes	characteristic
7	C₂×#14556064	C₂²	1	2	yes	—
8	#17986039	—	2	2	no	—
9	#17985954	C₂²	1	1	yes	characteristic
10	#17985829	C₄	1	4	yes	complemented
11	#17985774	—	2	1	no	—
12	C₁₆×Dic₇₁	—	2	1	no	—
13	#17985768	—	2	2	no	—
14	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₇₁	C₂²	1	2	yes	—
16	#366×C₇₁	—	2	1	no	—
17	#366×C₇₁	—	2	1	no	—
18	C₂×#11765849	—	2	2	no	—
19	C₂×#11765849	D₄	1	1	yes	characteristic
20	#14556087	D₄	1	1	yes	characteristic
21	#14556068	—	2	1	no	—
22	#14556068	Q₈	1	2	yes	—
23	C₂×C₇₁⋊C₁₆	—	2	1	no	—
24	#14556064	—	2	2	no	—
25	#14556064	C₂×C₄	1	4	yes	—
26	#14556061	—	4	2	no	—
27	#14556059	—	4	2	no	—
28	#14556038	—	2	1	no	—
29	C₈×Dic₇₁	—	2	1	no	—
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
36	#107×C₇₁	—	2	2	no	—
37	#107×C₇₁	—	2	2	no	—
38	#11765849	—	2	1	no	—
39	#11765849	—	2	4	no	—
40	#11765849	—	4	2	no	—
41	#11765849	Q₁₆	1	2	yes	—
42	C₂×C₇₁⋊C₈	—	2	1	no	—
43	C₂×C₇₁⋊C₈	—	4	2	no	—
44	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
45	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
46	C₂²⋊Dic₇₁	—	2	1	no	—
47	C₄×Dic₇₁	—	2	1	no	—
48	C₇₁⋊C₁₆	—	2	2	no	—
49	C₂²×C₄×C₇₁	—	4	2	no	—
50	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
56	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₇₁	—	4	2	no	—
58	C₄⋊C₄×C₇₁	—	4	2	no	—
59	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
61	C₂³×C₇₁	#135	1	1	yes	characteristic
62	C₂×C₄×C₇₁	—	8	2	no	—
63	C₂×C₄×C₇₁	—	16	2	no	—
64	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
70	C₂×Dic₇₁	—	2	1	no	—
71	C₂×Dic₇₁	—	2	1	no	—
72	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
74	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₈×C₇₁	#125	1	1	yes	characteristic
76	C₇₁⋊C₈	—	2	2	no	—
77	C₇₁⋊C₈	—	4	4	no	—
78	C₂²×C₇₁	—	2	1	no	—
79	C₂²×C₇₁	—	2	1	no	—
80	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
81	C₂²×C₇₁	#466	1	1	yes	characteristic
82	C₂²×C₇₁	#499	1	1	yes	characteristic
83	C₄×C₇₁	—	16	2	no	—
84	C₄×C₇₁	#389	1	1	yes	characteristic
85	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
86	C₄×C₇₁	#467	1	1	yes	characteristic
87	C₄×C₇₁	#500	1	1	yes	characteristic
88	Dic₇₁	—	4	1	no	—
89	Dic₇₁	—	4	1	no	—
90	#13171	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	—	2	1	no	—
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	#1942	1	1	yes	characteristic
94	C₂×C₇₁	#2221	1	1	yes	characteristic
95	C₂×C₇₁	#2222	1	1	yes	characteristic
96	C₂×#368	—	71	1	no	—
97	C₂×#366	D₇₁	1	1	yes	characteristic
98	#2108	—	71	1	no	—
99	#1952	—	71	2	no	—
100	#1387	—	71	2	no	—
101	C₇₁	#13171	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	—	71	1	no	—
103	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
104	C₂×#108	—	71	2	no	—
105	#428	—	142	2	no	—
106	C₂×#107	C₂×D₇₁	1	2	yes	—
107	#405	—	71	1	no	—
108	#368	—	71	4	no	—
109	#366	—	2	1	no	—
110	#366	—	2	1	no	—
111	#346	—	142	1	no	—
112	C₄×C₁₆	—	142	1	no	—
113	#343	—	142	2	no	—
114	C₂×M₁₆	—	71	1	no	—
115	C₂×M₁₆	—	142	2	no	—
116	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	#117	—	71	1	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	M₃₂	—	71	2	no	—
120	M₃₂	—	142	1	no	—
121	C₂×C₁₆	—	142	1	no	—
122	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
123	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
125	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
126	#108	—	71	4	no	—
127	#108	—	142	2	no	—
128	#107	—	2	2	no	—
129	#107	—	2	2	no	—
130	#107	—	284	2	no	—
131	#106	—	284	2	no	—
132	#97	—	142	1	no	—
133	C₄×C₈	—	142	1	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
136	M₁₆	—	284	2	no	—
137	M₁₆	—	71	2	no	—
138	M₁₆	—	142	4	no	—
139	M₁₆	—	142	1	no	—
140	C₂×C₈	—	284	2	no	—
141	C₂×C₈	—	142	1	no	—
142	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
146	C₂×C₈	#11765864	1	1	yes	characteristic
147	C₂×C₈	#11765866	1	1	yes	characteristic
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	71	1	no	—
151	C₄⋊C₄	—	71	1	no	—
152	C₂²⋊C₄	—	142	1	no	—
153	C₄²	—	142	1	no	—
154	C₁₆	—	142	2	no	—
155	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
156	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
157	C₂³	#14556147	1	1	yes	characteristic
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	16	2	no	—
160	C₂×C₄	—	142	1	no	—
161	C₂×C₄	—	142	1	no	—
162	C₂×C₄	#14556090	1	1	yes	characteristic
163	C₂×C₄	#14556099	1	1	yes	characteristic
164	C₂×C₄	#14556127	1	1	yes	characteristic
165	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
167	C₂×C₄	#14556151	1	1	yes	characteristic
168	C₈	—	284	4	no	—
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	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#17986093	1	1	yes	characteristic
177	C₂²	#17986232	1	1	yes	characteristic
178	C₂²	#17986400	1	1	yes	characteristic
179	C₄	—	16	2	no	—
180	C₄	—	284	1	no	—
181	C₄	—	284	1	no	—
182	C₄	#17985895	1	1	yes	characteristic
183	C₄	#17986081	1	1	yes	characteristic
184	C₄	#17986237	1	1	yes	characteristic
185	C₄	#17986403	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#22589071	1	1	yes	characteristic
189	C₂	#22590457	1	1	yes	characteristic
190	C₂	#22590462	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985829

C₂

yes

maximal, characteristic

#22589936

C₂

yes

maximal, characteristic

#22589123

C₂

yes

maximal

#22586448

C₂

yes

maximal

C₂×#366×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

characteristic

C₂×#14556064

C₂²

yes

—

#17986039

—

#17985954

C₂²

yes

characteristic

#17985829

C₄

yes

complemented

#17985774

—

C₁₆×Dic₇₁

—

#17985768

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#107×C₇₁

C₂²

yes

—

#366×C₇₁

—

#366×C₇₁

—

C₂×#11765849

—

C₂×#11765849

D₄

yes

characteristic

#14556087

D₄

yes

characteristic

#14556068

—

#14556068

Q₈

yes

—

C₂×C₇₁⋊C₁₆

—

#14556064

—

#14556064

C₂×C₄

yes

—

#14556061

—

#14556059

—

#14556038

—

C₈×Dic₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

Q₈

yes

—

#107×C₇₁

—

#107×C₇₁

—

—

—

—

Q₁₆

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₈

yes

characteristic

C₄⋊Dic₇₁

D₈

yes

characteristic

C₂²⋊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

—

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

#135

yes

characteristic

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×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

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

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×Q₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

#389

yes

characteristic

C₄×C₇₁

C₄×D₈

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#13171

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

#2108

—

#1952

—

100

#1387

—

101

C₇₁

#13171

yes

characteristic, complemented, Sylow

102

C₂×M₃₂

—

103

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

104

C₂×#108

—

105

#428

—

142

—

106

C₂×#107

C₂×D₇₁

yes

—

107

#405

—

108

#368

—

109

#366

—

110

#366

—

111

#346

—

142

—

112

C₄×C₁₆

—

142

—

113

#343

—

142

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

142

—

116

C₂²×C₈

C₂²×D₇₁

yes

characteristic

117

#117

—

118

C₂×C₄⋊C₄

—

119

M₃₂

—

120

M₃₂

—

142

—

121

C₂×C₁₆

—

142

—

122

C₂×C₁₆

Dic₁₄₂

yes

—

123

C₂×C₁₆

C₄×D₇₁

yes

—

124

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

125

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

126

#108

—

127

#108

—

142

—

128

#107

—

129

#107

—

130

#107

—

284

—

131

#106

—

284

—

132

#97

—

142

—

133

C₄×C₈

—

142

—

134

C₂²×C₄

—

135

C₂²×C₄

D₄×D₇₁

yes

characteristic

136

M₁₆

—

284

—

137

M₁₆

—

138

M₁₆

—

142

—

139

M₁₆

—

142

—

140

C₂×C₈

—

284

—

141

C₂×C₈

—

142

—

142

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

143

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

144

C₂×C₈

Q₈×D₇₁

yes

characteristic

145

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

146

C₂×C₈

#11765864

yes

characteristic

147

C₂×C₈

#11765866

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₂²⋊C₄

—

142

—

153

C₄²

—

142

—

154

C₁₆

—

142

—

155

C₁₆

Dic₇₁⋊C₄

yes

—

156

C₁₆

Dic₇₁⋊C₄

yes

—

157

C₂³

#14556147

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

142

—

161

C₂×C₄

—

142

—

162

C₂×C₄

#14556090

yes

characteristic

163

C₂×C₄

#14556099

yes

characteristic

164

C₂×C₄

#14556127

yes

characteristic

165

C₂×C₄

D₈×D₇₁

yes

characteristic

166

C₂×C₄

Q₁₆×D₇₁

yes

characteristic

167

C₂×C₄

#14556151

yes

characteristic

168

C₈

—

284

—

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

—

175

C₂²

—

176

C₂²

#17986093

yes

characteristic

177

C₂²

#17986232

yes

characteristic

178

C₂²

#17986400

yes

characteristic

179

C₄

—

180

C₄

—

284

—

181

C₄

—

284

—

182

C₄

#17985895

yes

characteristic

183

C₄

#17986081

yes

characteristic

184

C₄

#17986237

yes

characteristic

185

C₄

#17986403

yes

characteristic

186

C₂

—

187

C₂

—

188

C₂

#22589071

yes

characteristic

189

C₂

#22590457

yes

characteristic

190

C₂

#22590462

yes

characteristic

	subgroup	G/H
center	C₂²	#17986232
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#366×C₇₁	C₂
socle	C₂²×C₇₁	#466
prime radical	C₂³×C₇₁	#135
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#366	D₇₁
hypocenter	C₇₁	#13171
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂²

#17986232

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#366×C₇₁

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
10. #17985829	C₄	semidirect
101. C₇₁	#13171	semidirect

H ≅ G/N

type

10. #17985829

C₄

semidirect

101. C₇₁

#13171

semidirect

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

subgroup	index
C₂×#17985829	2
#22589936	2
#22589123	2
#22586448	2
C₂×#366×C₇₁	2
#13171	71

subgroup

index

C₂×#366×C₇₁

Subgroups — #31915510

Conjugacy classes of subgroups

Complements

Maximal subgroups