Group #31911604 — subgroups

Subgroups — #31911604

Order 18176.

218 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22590493	C₂	1	1	yes	maximal, characteristic
2	#22590391	C₂	1	1	yes	maximal, characteristic
3	#22587937	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic
5	#22586397	C₂	1	1	yes	maximal, characteristic
6	#22586390	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1367	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₆₈	—	2	1	no	—
9	#17986404	C₂²	1	1	yes	characteristic
10	#17986397	—	2	1	no	—
11	#17986379	C₂²	1	1	yes	characteristic
12	#17986350	—	2	1	no	—
13	C₈×Dic₁₄₂	—	2	1	no	—
14	C₄×Dic₂₈₄	C₂²	1	1	yes	characteristic
15	#17985825	—	2	1	no	—
16	#17985804	—	2	1	no	—
17	#17985804	—	2	1	no	—
18	#17985752	C₂²	1	1	yes	characteristic
19	#17985731	—	2	1	no	—
20	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	C₈²×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₈₄	—	2	1	no	—
24	C₂×Dic₂₈₄	—	2	1	no	—
25	C₂×Dic₂₈₄	D₄	1	1	yes	characteristic
26	#14556128	—	2	1	no	—
27	#14556128	—	2	1	no	—
28	C₄×Dic₁₄₂	—	2	1	no	—
29	Dic₅₆₈	—	4	2	no	—
30	#14556062	—	2	1	no	—
31	#14556062	—	2	1	no	—
32	#14556062	D₄	1	1	yes	characteristic
33	#14556060	—	4	1	no	—
34	#14556060	—	4	1	no	—
35	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
36	Dic₇₁⋊C₈	—	4	1	no	—
37	#14556053	—	2	1	no	—
38	#14556053	—	4	2	no	—
39	C₈×Dic₇₁	—	4	1	no	—
40	C₂×C₁₆×C₇₁	—	2	1	no	—
41	C₂×C₁₆×C₇₁	—	2	1	no	—
42	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₄₂	—	2	1	no	—
46	C₂×Dic₁₄₂	—	4	1	no	—
47	C₂×Dic₁₄₂	—	4	1	no	—
48	Dic₂₈₄	—	2	1	no	—
49	Dic₂₈₄	—	4	1	no	—
50	Dic₂₈₄	—	4	1	no	—
51	Dic₂₈₄	—	8	1	no	—
52	Dic₂₈₄	SD₁₆	1	2	yes	—
53	C₂×C₇₁⋊C₈	—	4	1	no	—
54	C₄⋊Dic₇₁	—	2	1	no	—
55	C₄⋊Dic₇₁	—	4	1	no	—
56	C₄⋊Dic₇₁	—	4	1	no	—
57	C₄⋊Dic₇₁	—	4	1	no	—
58	C₄⋊Dic₇₁	—	4	1	no	—
59	Dic₇₁⋊C₄	—	4	1	no	—
60	C₄×Dic₇₁	—	4	1	no	—
61	C₂×C₈×C₇₁	—	2	1	no	—
62	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
63	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
64	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
66	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₁₆×C₇₁	—	4	1	no	—
68	C₁₆×C₇₁	—	4	1	no	—
69	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×Dic₇₁	—	4	1	no	—
73	C₂×Dic₇₁	—	4	1	no	—
74	C₂×Dic₇₁	—	8	1	no	—
75	C₂×Dic₇₁	—	8	1	no	—
76	Dic₁₄₂	—	2	2	no	—
77	Dic₁₄₂	—	4	1	no	—
78	Dic₁₄₂	—	8	2	no	—
79	Dic₁₄₂	—	8	1	no	—
80	Dic₁₄₂	—	8	1	no	—
81	C₈×C₇₁	—	2	1	no	—
82	C₈×C₇₁	—	2	1	no	—
83	C₈×C₇₁	—	4	1	no	—
84	C₈×C₇₁	Q₃₂	1	2	yes	complemented
85	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
86	C₈×C₇₁	#135	1	1	yes	characteristic
87	C₇₁⋊C₈	—	8	1	no	—
88	C₂²×C₇₁	#459	1	1	yes	characteristic
89	C₄×C₇₁	—	2	1	no	—
90	C₄×C₇₁	#465	1	1	yes	characteristic
91	C₄×C₇₁	#471	1	1	yes	characteristic
92	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
93	C₄×C₇₁	#510	1	1	yes	characteristic
94	Dic₇₁	—	4	2	no	—
95	Dic₇₁	—	8	1	no	—
96	Dic₇₁	—	16	1	no	—
97	Dic₇₁	—	16	1	no	—
98	#12421	—	71	1	no	maximal, Sylow
99	C₂×C₇₁	#1665	1	1	yes	characteristic
100	C₂×C₇₁	#2209	1	1	yes	characteristic
101	C₂×C₇₁	#2212	1	1	yes	characteristic
102	#2227	—	71	1	no	—
103	#2209	—	71	1	no	—
104	#1707	—	71	1	no	—
105	C₈×Q₁₆	—	71	1	no	—
106	#1367	D₇₁	1	1	yes	characteristic
107	#1365	—	71	1	no	—
108	#1362	—	71	1	no	—
109	C₇₁	#12421	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₃₂	—	142	1	no	—
111	#500	—	71	1	no	—
112	#498	—	142	1	no	—
113	#494	—	71	1	no	—
114	#487	—	142	1	no	—
115	C₈×Q₈	—	142	1	no	—
116	C₄×Q₁₆	—	71	1	no	—
117	#366	—	142	1	no	—
118	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
119	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
120	#358	—	142	1	no	—
121	#358	—	142	1	no	—
122	#335	—	71	1	no	—
123	Q₈⋊C₈	—	142	1	no	—
124	C₈²	C₂×D₇₁	1	1	yes	characteristic
125	C₂×Q₁₆	—	71	1	no	—
126	C₂×Q₁₆	—	142	1	no	—
127	C₂×Q₁₆	—	142	1	no	—
128	C₄⋊Q₈	—	142	1	no	—
129	C₄⋊Q₈	—	142	1	no	—
130	C₄×Q₈	—	142	1	no	—
131	Q₃₂	—	284	2	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	2	1	no	—
134	#107	—	71	1	no	—
135	#107	—	142	1	no	—
136	#107	—	142	1	no	—
137	#106	—	284	1	no	—
138	#106	—	284	1	no	—
139	C₄⋊C₈	—	284	1	no	—
140	C₄⋊C₈	—	142	1	no	—
141	#103	—	284	2	no	—
142	#103	—	142	1	no	—
143	C₄×C₈	—	284	1	no	—
144	C₄×C₈	D₂₈₄	1	1	yes	characteristic
145	C₄×C₈	D₂₈₄	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
147	C₂×Q₈	—	284	1	no	—
148	C₂×Q₈	—	284	1	no	—
149	C₂×Q₈	—	142	1	no	—
150	Q₁₆	—	284	1	no	—
151	Q₁₆	—	284	1	no	—
152	Q₁₆	—	568	1	no	—
153	Q₁₆	—	71	2	no	—
154	Q₁₆	—	142	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	284	1	no	—
157	C₂×C₈	D₅₆₈	1	1	yes	characteristic
158	C₂×C₈	D₅₆₈	1	1	yes	characteristic
159	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
160	C₂×C₈	#11765866	1	1	yes	characteristic
161	C₄⋊C₄	—	284	1	no	—
162	C₄⋊C₄	—	284	1	no	—
163	C₄⋊C₄	—	284	1	no	—
164	C₄⋊C₄	—	284	1	no	—
165	C₄⋊C₄	—	284	1	no	—
166	C₄⋊C₄	—	142	1	no	—
167	C₄²	—	284	1	no	—
168	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
169	C₁₆	—	4	1	no	—
170	C₁₆	—	4	1	no	—
171	Q₈	—	284	1	no	—
172	Q₈	—	568	2	no	—
173	Q₈	—	568	1	no	—
174	Q₈	—	568	1	no	—
175	Q₈	—	142	2	no	—
176	C₂×C₄	—	284	1	no	—
177	C₂×C₄	—	284	1	no	—
178	C₂×C₄	—	568	1	no	—
179	C₂×C₄	—	568	1	no	—
180	C₂×C₄	#14556103	1	1	yes	characteristic
181	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
182	C₂×C₄	#14556156	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	1	no	—
186	C₈	—	568	1	no	—
187	C₈	Dic₅₆₈	1	2	yes	complemented
188	C₈	SD₁₆×D₇₁	1	1	yes	characteristic
189	C₈	#14556149	1	1	yes	characteristic
190	C₂²	#17986195	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	284	2	no	—
193	C₄	—	568	1	no	—
194	C₄	—	1136	1	no	—
195	C₄	—	1136	1	no	—
196	C₄	#17986229	1	1	yes	characteristic
197	C₄	#17986263	1	1	yes	characteristic
198	C₄	C₂×Dic₅₆₈	1	1	yes	characteristic
199	C₄	#17986442	1	1	yes	characteristic
200	C₂	#22587682	1	1	yes	characteristic
201	C₂	#22590391	1	1	yes	characteristic
202	C₂	#22590410	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590493

C₂

yes

maximal, characteristic

#22590391

C₂

yes

maximal, characteristic

#22587937

C₂

yes

maximal, characteristic

C₈×Dic₂₈₄

C₂

yes

maximal, characteristic

#22586397

C₂

yes

maximal, characteristic

#22586390

C₂

yes

maximal, characteristic

C₇₁×#1367

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

—

#17986404

C₂²

yes

characteristic

#17986397

—

#17986379

C₂²

yes

characteristic

#17986350

—

C₈×Dic₁₄₂

—

C₄×Dic₂₈₄

C₂²

yes

—

—

—

C₂²

yes

characteristic

#17985731

—

C₄⋊C₁₆×C₇₁

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₁

C₂²

yes

characteristic

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

D₄

yes

characteristic

#14556128

—

#14556128

—

C₄×Dic₁₄₂

—

Dic₅₆₈

—

#14556062

—

#14556062

—

#14556062

D₄

yes

characteristic

#14556060

—

#14556060

—

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

#14556053

—

#14556053

—

C₈×Dic₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

SD₁₆

yes

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

Q₃₂

yes

complemented

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₈×C₇₁

#135

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

#459

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×Q₃₂

yes

characteristic

C₄×C₇₁

#510

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

100

C₂×C₇₁

#2209

yes

characteristic

101

C₂×C₇₁

#2212

yes

characteristic

102

#2227

—

103

#2209

—

104

#1707

—

105

C₈×Q₁₆

—

106

#1367

D₇₁

yes

characteristic

107

#1365

—

108

#1362

—

109

C₇₁

#12421

yes

characteristic, complemented, Sylow

110

C₂×Q₃₂

—

142

—

111

#500

—

112

#498

—

142

—

113

#494

—

114

#487

—

142

—

115

C₈×Q₈

—

142

—

116

C₄×Q₁₆

—

117

#366

—

142

—

118

C₄⋊C₁₆

C₂×D₇₁

yes

characteristic

119

C₄⋊C₁₆

C₂×D₇₁

yes

characteristic

120

#358

—

142

—

121

#358

—

142

—

122

#335

—

123

Q₈⋊C₈

—

142

—

124

C₈²

C₂×D₇₁

yes

characteristic

125

C₂×Q₁₆

—

126

C₂×Q₁₆

—

142

—

127

C₂×Q₁₆

—

142

—

128

C₄⋊Q₈

—

142

—

129

C₄⋊Q₈

—

142

—

130

C₄×Q₈

—

142

—

131

Q₃₂

—

284

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

#107

—

135

#107

—

142

—

136

#107

—

142

—

137

#106

—

284

—

138

#106

—

284

—

139

C₄⋊C₈

—

284

—

140

C₄⋊C₈

—

142

—

141

#103

—

284

—

142

#103

—

142

—

143

C₄×C₈

—

284

—

144

C₄×C₈

D₂₈₄

yes

characteristic

145

C₄×C₈

D₂₈₄

yes

characteristic

146

C₄×C₈

C₂²×D₇₁

yes

characteristic

147

C₂×Q₈

—

284

—

148

C₂×Q₈

—

284

—

149

C₂×Q₈

—

142

—

150

Q₁₆

—

284

—

151

Q₁₆

—

284

—

152

Q₁₆

—

568

—

153

Q₁₆

—

154

Q₁₆

—

142

—

155

C₂×C₈

—

156

C₂×C₈

—

284

—

157

C₂×C₈

D₅₆₈

yes

characteristic

158

C₂×C₈

D₅₆₈

yes

characteristic

159

C₂×C₈

D₄×D₇₁

yes

characteristic

160

C₂×C₈

#11765866

yes

characteristic

161

C₄⋊C₄

—

284

—

162

C₄⋊C₄

—

284

—

163

C₄⋊C₄

—

284

—

164

C₄⋊C₄

—

284

—

165

C₄⋊C₄

—

284

—

166

C₄⋊C₄

—

142

—

167

C₄²

—

284

—

168

C₄²

C₂×D₂₈₄

yes

characteristic

169

C₁₆

—

170

C₁₆

—

171

Q₈

—

284

—

172

Q₈

—

568

—

173

Q₈

—

568

—

174

Q₈

—

568

—

175

Q₈

—

142

—

176

C₂×C₄

—

284

—

177

C₂×C₄

—

284

—

178

C₂×C₄

—

568

—

179

C₂×C₄

—

568

—

180

C₂×C₄

#14556103

yes

characteristic

181

C₂×C₄

C₂×D₅₆₈

yes

characteristic

182

C₂×C₄

#14556156

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

568

—

187

C₈

Dic₅₆₈

yes

complemented

188

C₈

SD₁₆×D₇₁

yes

characteristic

189

C₈

#14556149

yes

characteristic

190

C₂²

#17986195

yes

characteristic

191

C₄

—

192

C₄

—

284

—

193

C₄

—

568

—

194

C₄

—

1136

—

195

C₄

—

1136

—

196

C₄

#17986229

yes

characteristic

197

C₄

#17986263

yes

characteristic

198

C₄

C₂×Dic₅₆₈

yes

characteristic

199

C₄

#17986442

yes

characteristic

200

C₂

#22587682

yes

characteristic

201

C₂

#22590391

yes

characteristic

202

C₂

#22590410

yes

characteristic

	subgroup	G/H
center	C₂²	#17986195
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1367	C₂
socle, prime radical	C₂²×C₇₁	#459
hypercenter	#1367	D₇₁
hypocenter	C₇₁	#12421
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#17986195

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1367

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
84. C₈×C₇₁	Q₃₂	semidirect
109. C₇₁	#12421	semidirect
187. C₈	Dic₅₆₈	semidirect

H ≅ G/N

type

84. C₈×C₇₁

Q₃₂

semidirect

109. C₇₁

#12421

semidirect

187. C₈

Dic₅₆₈

semidirect

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

subgroup	index
#22590493	2
#22590391	2
#22587937	2
C₈×Dic₂₈₄	2
#22586397	2
#22586390	2
C₇₁×#1367	2
#12421	71

subgroup

index

C₈×Dic₂₈₄

C₇₁×#1367

Subgroups — #31911604

Conjugacy classes of subgroups

Complements

Maximal subgroups