Group #31915596 — subgroups

Subgroups — #31915596

Order 18176.

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

normal subgroups only

This group has 294 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	C₂×#17985821	C₂	1	2	yes	maximal
2	C₂×#17985820	C₂	1	1	yes	maximal, characteristic
3	#22588219	C₂	1	1	yes	maximal, characteristic
4	#22586447	C₂	1	2	yes	maximal
5	C₇₁×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₂×#14556064	C₂²	1	1	yes	characteristic
8	#17986037	C₂²	1	1	yes	characteristic
9	#17985954	—	2	1	no	—
10	#17985821	—	2	4	no	—
11	#17985820	C₄	1	4	yes	complemented
12	#17985759	—	2	1	no	—
13	C₂×#107×C₇₁	—	2	1	no	—
14	C₂×#106×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#382×C₇₁	—	2	1	no	—
17	C₂×#11765849	—	2	1	no	—
18	#14556087	—	2	1	no	—
19	#14556068	—	2	2	no	—
20	#14556068	C₂×C₄	1	4	yes	—
21	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	#14556064	—	2	1	no	—
23	#14556064	D₄	1	2	yes	—
24	#14556062	Q₈	1	2	yes	—
25	#14556060	—	2	1	no	—
26	#14556038	—	4	1	no	—
27	C₈×Dic₇₁	—	4	1	no	—
28	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
32	#107×C₇₁	—	4	2	no	—
33	#106×C₇₁	—	2	2	no	—
34	#106×C₇₁	—	2	2	no	—
35	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₁	Q₈	1	2	yes	—
37	#95×C₇₁	—	4	2	no	—
38	#11765849	—	2	2	no	—
39	#11765849	—	4	1	no	—
40	C₂×C₇₁⋊C₈	—	4	1	no	—
41	C₄⋊Dic₇₁	—	2	2	no	—
42	C₂²⋊Dic₇₁	—	4	1	no	—
43	C₄×Dic₇₁	—	4	1	no	—
44	C₇₁⋊C₁₆	—	2	4	no	—
45	C₂²×C₄×C₇₁	—	2	1	no	—
46	C₂²×C₄×C₇₁	—	4	2	no	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
52	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
57	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
58	C₄⋊C₄×C₇₁	—	4	2	no	—
59	C₄⋊C₄×C₇₁	—	4	2	no	—
60	C₄²×C₇₁	—	2	1	no	—
61	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
62	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₇₁	—	2	2	no	—
64	C₂×C₄×C₇₁	—	4	2	no	—
65	C₂×C₄×C₇₁	—	8	4	no	—
66	C₂×C₄×C₇₁	—	8	2	no	—
67	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
73	C₂×Dic₇₁	—	4	2	no	—
74	C₈×C₇₁	#106	1	2	yes	—
75	C₈×C₇₁	#106	1	2	yes	—
76	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
77	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
78	C₈×C₇₁	#125	1	1	yes	characteristic
79	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₇₁⋊C₈	—	4	2	no	—
81	C₂²×C₇₁	—	2	2	no	—
82	C₂²×C₇₁	#428	1	1	yes	characteristic
83	C₂²×C₇₁	#460	1	1	yes	characteristic
84	C₂²×C₇₁	#476	1	1	yes	characteristic
85	C₄×C₇₁	—	4	2	no	—
86	C₄×C₇₁	—	16	2	no	—
87	C₄×C₇₁	#389	1	1	yes	characteristic
88	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
89	C₄×C₇₁	#461	1	1	yes	characteristic
90	C₄×C₇₁	#475	1	1	yes	characteristic
91	Dic₇₁	—	8	2	no	—
92	#13187	—	71	1	no	maximal, Sylow
93	C₂×C₇₁	—	2	1	no	—
94	C₂×C₇₁	—	2	1	no	—
95	C₂×C₇₁	#1948	1	1	yes	characteristic
96	C₂×C₇₁	#2231	1	2	yes	—
97	C₂×#365	—	71	2	no	—
98	C₂×#364	—	71	1	no	—
99	#1846	D₇₁	1	1	yes	characteristic
100	#1772	—	71	1	no	—
101	#1386	—	71	2	no	—
102	C₇₁	#13187	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₃₂	—	71	2	no	—
104	C₂×#108	—	71	1	no	—
105	#427	—	71	1	no	—
106	C₂×#107	—	2	1	no	—
107	C₂×#106	C₂×D₇₁	1	2	yes	—
108	#405	—	142	1	no	—
109	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	#382	—	2	1	no	—
111	#365	—	142	4	no	—
112	#364	—	71	4	no	—
113	#339	—	142	1	no	—
114	C₂×M₁₆	—	142	1	no	—
115	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	#117	—	142	1	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
120	M₃₂	—	71	4	no	—
121	M₃₂	—	142	2	no	—
122	C₂×C₁₆	—	142	2	no	—
123	#108	—	71	2	no	—
124	#108	—	142	1	no	—
125	#107	—	4	2	no	—
126	#107	—	71	2	no	—
127	#106	—	2	2	no	—
128	#106	—	2	2	no	—
129	#106	—	142	1	no	—
130	#97	—	284	1	no	—
131	C₄×C₈	—	284	1	no	—
132	C₄×C₈	Dic₁₄₂	1	2	yes	—
133	C₄×C₈	C₄×D₇₁	1	2	yes	—
134	#95	—	4	2	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
138	M₁₆	—	284	1	no	—
139	M₁₆	—	142	2	no	—
140	C₂×C₈	—	284	1	no	—
141	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
142	C₂×C₈	#11765852	1	1	yes	characteristic
143	C₂×C₈	#11765852	1	1	yes	characteristic
144	C₂×C₈	#11765853	1	2	yes	—
145	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
146	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
148	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
149	C₂×C₈	#11765864	1	1	yes	characteristic
150	C₂×C₈	#11765866	1	1	yes	characteristic
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	142	2	no	—
154	C₂²⋊C₄	—	284	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	284	1	no	—
157	C₄²	Dic₇₁⋊C₄	1	2	yes	—
158	C₁₆	—	142	4	no	—
159	C₂³	#14556154	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	284	2	no	—
165	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
166	C₂×C₄	#14556099	1	1	yes	characteristic
167	C₂×C₄	#14556104	1	1	yes	characteristic
168	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
169	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
170	C₂×C₄	#14556159	1	1	yes	characteristic
171	C₈	—	284	2	no	—
172	C₈	#14556059	1	2	yes	—
173	C₈	#14556059	1	2	yes	—
174	C₈	#14556090	1	1	yes	characteristic
175	C₈	#14556093	1	1	yes	characteristic
176	C₈	#14556119	1	1	yes	characteristic
177	C₈	#14556127	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#17986039	1	1	yes	characteristic
180	C₂²	#17986196	1	1	yes	characteristic
181	C₂²	#17986286	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	16	2	no	—
184	C₄	—	568	2	no	—
185	C₄	#17985895	1	1	yes	characteristic
186	C₄	C₂×#14556059	1	1	yes	characteristic
187	C₄	#17986201	1	1	yes	characteristic
188	C₄	#17986281	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#22589101	1	1	yes	characteristic
192	C₂	#22590513	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985821

C₂

yes

maximal

C₂×#17985820

C₂

yes

maximal, characteristic

#22588219

C₂

yes

maximal, characteristic

#22586447

C₂

yes

maximal

C₇₁×#1846

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

—

C₂×#14556064

C₂²

yes

characteristic

#17986037

C₂²

yes

—

—

C₄

yes

complemented

#17985759

—

C₂×#107×C₇₁

—

C₂×#106×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#382×C₇₁

—

—

—

—

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

—

#14556064

—

#14556064

D₄

yes

—

#14556062

Q₈

yes

—

#14556060

—

#14556038

—

C₈×Dic₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

#107×C₇₁

—

#106×C₇₁

—

#106×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

#95×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

characteristic

C₂×C₈×C₇₁

SD₁₆

yes

characteristic

C₂×C₈×C₇₁

SD₁₆

yes

—

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

—

C₄²×C₇₁

—

C₄²×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

#137

yes

characteristic

C₂×Dic₇₁

—

C₈×C₇₁

#106

yes

—

C₈×C₇₁

#106

yes

—

C₈×C₇₁

C₄×D₄

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

characteristic

C₈×C₇₁

#125

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

#13187

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1948

yes

characteristic

C₂×C₇₁

#2231

yes

—

C₂×#365

—

C₂×#364

—

#1846

D₇₁

yes

characteristic

100

#1772

—

101

#1386

—

102

C₇₁

#13187

yes

characteristic, complemented, Sylow

103

C₂×M₃₂

—

104

C₂×#108

—

105

#427

—

106

C₂×#107

—

107

C₂×#106

C₂×D₇₁

yes

—

108

#405

—

142

—

109

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

110

#382

—

111

#365

—

142

—

112

#364

—

113

#339

—

142

—

114

C₂×M₁₆

—

142

—

115

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

116

C₂²×C₈

C₂²×D₇₁

yes

characteristic

117

#117

—

142

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

120

M₃₂

—

121

M₃₂

—

142

—

122

C₂×C₁₆

—

142

—

123

#108

—

124

#108

—

142

—

125

#107

—

126

#107

—

127

#106

—

128

#106

—

129

#106

—

142

—

130

#97

—

284

—

131

C₄×C₈

—

284

—

132

C₄×C₈

Dic₁₄₂

yes

—

133

C₄×C₈

C₄×D₇₁

yes

—

134

#95

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

138

M₁₆

—

284

—

139

M₁₆

—

142

—

140

C₂×C₈

—

284

—

141

C₂×C₈

Dic₇₁⋊C₄

yes

—

142

C₂×C₈

#11765852

yes

characteristic

143

C₂×C₈

#11765852

yes

characteristic

144

C₂×C₈

#11765853

yes

—

145

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

146

C₂×C₈

D₄×D₇₁

yes

characteristic

147

C₂×C₈

Q₈×D₇₁

yes

characteristic

148

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

149

C₂×C₈

#11765864

yes

characteristic

150

C₂×C₈

#11765866

yes

characteristic

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

142

—

154

C₂²⋊C₄

—

284

—

155

C₄²

—

156

C₄²

—

284

—

157

C₄²

Dic₇₁⋊C₄

yes

—

158

C₁₆

—

142

—

159

C₂³

#14556154

yes

characteristic

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

284

—

165

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

166

C₂×C₄

#14556099

yes

characteristic

167

C₂×C₄

#14556104

yes

characteristic

168

C₂×C₄

C₂×#11765852

yes

characteristic

169

C₂×C₄

C₂×#11765853

yes

characteristic

170

C₂×C₄

#14556159

yes

characteristic

171

C₈

—

284

—

172

C₈

#14556059

yes

—

173

C₈

#14556059

yes

—

174

C₈

#14556090

yes

characteristic

175

C₈

#14556093

yes

characteristic

176

C₈

#14556119

yes

characteristic

177

C₈

#14556127

yes

characteristic

178

C₂²

—

179

C₂²

#17986039

yes

characteristic

180

C₂²

#17986196

yes

characteristic

181

C₂²

#17986286

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

568

—

185

C₄

#17985895

yes

characteristic

186

C₄

C₂×#14556059

yes

characteristic

187

C₄

#17986201

yes

characteristic

188

C₄

#17986281

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#22589101

yes

characteristic

192

C₂

#22590513

yes

—

	subgroup	G/H
center	C₂²	#17986286
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1846	C₂
socle	C₂²×C₇₁	#476
prime radical	C₂³×C₇₁	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1846	D₇₁
hypocenter	C₇₁	#13187
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂²

#17986286

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1846

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 ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
11. #17985820	C₄	semidirect
102. C₇₁	#13187	semidirect

H ≅ G/N

type

11. #17985820

C₄

semidirect

102. C₇₁

#13187

semidirect

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

subgroup	index
C₂×#17985821	2
C₂×#17985820	2
#22588219	2
#22586447	2
C₇₁×#1846	2
#13187	71

subgroup

index

C₇₁×#1846

Subgroups — #31915596

Conjugacy classes of subgroups

Complements

Maximal subgroups