Group #31913791 — subgroups

Subgroups — #31913791

Order 18176.

366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22596951	C₂	1	1	yes	maximal, characteristic
2	#22588386	C₂	1	1	yes	maximal, characteristic
3	#22586455	C₂	1	2	yes	maximal, complemented
4	#22586436	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2113	C₂	1	1	yes	maximal, characteristic
6	#17986766	—	2	1	no	—
7	#17986048	—	2	1	no	—
8	#17986048	C₄	1	4	yes	complemented
9	C₂×#14556064	—	2	2	no	—
10	C₂×#14556064	C₂²	1	2	yes	—
11	#17986037	C₂²	1	2	yes	—
12	#17985973	—	2	1	no	—
13	#17985756	—	2	1	no	—
14	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²⋊C₁₆×C₇₁	—	2	2	no	—
18	#14556135	—	4	4	no	—
19	C₂×#11765849	—	2	4	no	—
20	C₂×#11765849	—	2	2	no	—
21	#14556087	—	2	2	no	—
22	#14556064	—	2	4	no	—
23	#14556064	—	2	2	no	—
24	#14556064	C₂×C₄	1	4	yes	—
25	#14556062	—	2	2	no	—
26	#14556060	—	2	2	no	—
27	#14556043	—	4	2	no	—
28	C₂×#41×C₇₁	D₄	1	1	yes	characteristic
29	#131×C₇₁	—	2	2	no	—
30	#131×C₇₁	—	2	2	no	—
31	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	D₄	1	2	yes	—
34	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	M₃₂×C₇₁	—	2	2	no	—
36	C₂×C₁₆×C₇₁	—	2	1	no	—
37	C₂×C₁₆×C₇₁	—	2	1	no	—
38	C₂×C₁₆×C₇₁	—	2	2	no	—
39	#11765849	—	2	4	no	—
40	#11765849	—	4	4	no	—
41	#11765849	—	4	2	no	—
42	C₂×C₇₁⋊C₈	—	4	4	no	—
43	C₂×C₇₁⋊C₈	—	4	2	no	—
44	C₄⋊Dic₇₁	—	4	2	no	—
45	C₂²⋊Dic₇₁	—	4	2	no	—
46	C₄×Dic₇₁	—	4	2	no	—
47	#41×C₇₁	—	2	2	no	—
48	#41×C₇₁	—	2	2	no	—
49	C₂×Q₈×C₇₁	Q₁₆	1	1	yes	characteristic
50	C₂×D₄×C₇₁	SD₁₆	1	2	yes	—
51	C₂×D₄×C₇₁	Q₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	M₁₆×C₇₁	—	2	2	no	—
55	M₁₆×C₇₁	—	2	4	no	—
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	#41	1	2	yes	—
62	C₂×C₈×C₇₁	#41	1	2	yes	—
63	C₁₆×C₇₁	—	4	1	no	—
64	C₁₆×C₇₁	—	4	1	no	—
65	C₂³×C₇₁	#103	1	2	yes	—
66	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₇₁	—	2	2	no	—
68	D₄×C₇₁	—	2	4	no	—
69	D₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	#103	1	2	yes	—
73	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#123	1	2	yes	—
75	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×Dic₇₁	—	8	2	no	—
78	C₈×C₇₁	—	2	2	no	—
79	C₈×C₇₁	#117	1	2	yes	—
80	C₈×C₇₁	#123	1	1	yes	characteristic
81	C₈×C₇₁	#123	1	1	yes	characteristic
82	C₇₁⋊C₈	—	4	4	no	—
83	C₂²×C₇₁	—	2	2	no	—
84	C₂²×C₇₁	—	4	1	no	—
85	C₂²×C₇₁	—	4	2	no	—
86	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
87	C₂²×C₇₁	#481	1	2	yes	—
88	C₄×C₇₁	—	2	2	no	—
89	C₄×C₇₁	#381	1	1	yes	characteristic
90	C₄×C₇₁	#418	1	1	yes	characteristic
91	C₄×C₇₁	#484	1	2	yes	—
92	Dic₇₁	—	16	2	no	—
93	#12877	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	—	2	2	no	—
95	C₂×C₇₁	—	2	1	no	—
96	C₂×C₇₁	—	4	2	no	—
97	C₂×C₇₁	#1824	1	1	yes	characteristic
98	#2916	—	71	1	no	—
99	#2113	D₇₁	1	1	yes	characteristic
100	#1812	—	71	1	no	—
101	#1389	—	71	2	no	—
102	#1382	—	71	2	no	—
103	C₇₁	#12877	1	1	yes	characteristic, complemented, Sylow
104	#568	—	142	1	no	—
105	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
106	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
107	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
108	#430	—	71	4	no	—
109	#430	—	142	1	no	—
110	C₂×#108	—	71	2	no	—
111	C₂×#108	—	142	2	no	—
112	#427	—	71	2	no	—
113	#410	—	142	1	no	—
114	C₂²⋊C₁₆	—	2	2	no	—
115	H(3,ℤ/4)	—	142	1	no	—
116	C₂×#41	D₂₈₄	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	#131	—	284	4	no	—
120	C₂×M₁₆	—	142	4	no	—
121	C₂×M₁₆	—	142	2	no	—
122	C₂×M₁₆	C₄×D₇₁	1	2	yes	—
123	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
125	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
126	#117	—	142	2	no	—
127	M₃₂	—	2	2	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	2	1	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	71	4	no	—
132	#108	—	142	4	no	—
133	#108	—	142	2	no	—
134	#107	—	142	2	no	—
135	#106	—	142	2	no	—
136	#99	—	284	2	no	—
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	Dic₂₈₄	1	1	yes	characteristic
140	C₂×D₄	#11765854	1	2	yes	—
141	C₂×D₄	Dic₂₈₄	1	1	yes	characteristic
142	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
143	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	284	4	no	—
147	M₁₆	—	284	2	no	—
148	M₁₆	—	142	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	284	4	no	—
152	C₂×C₈	—	284	2	no	—
153	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
155	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
156	C₂×C₈	#11765864	1	2	yes	—
157	C₂×C₈	#11765864	1	2	yes	—
158	C₄⋊C₄	—	284	2	no	—
159	C₂²⋊C₄	—	284	2	no	—
160	C₄²	—	284	2	no	—
161	C₁₆	—	4	1	no	—
162	C₁₆	—	4	1	no	—
163	C₂³	#14556053	1	2	yes	—
164	C₂³	C₂×#11765854	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	568	2	no	—
171	C₂×C₄	#14556053	1	2	yes	—
172	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#14556112	1	2	yes	—
174	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
175	C₂×C₄	#14556156	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	284	4	no	—
178	C₈	#14556084	1	2	yes	—
179	C₈	#14556109	1	1	yes	characteristic
180	C₈	#14556109	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂×#14556053	1	1	yes	characteristic
185	C₂²	#17986318	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	1136	2	no	—
188	C₄	#17985860	1	1	yes	characteristic
189	C₄	#17986005	1	1	yes	characteristic
190	C₄	#17986335	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#22588431	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22596951

C₂

yes

maximal, characteristic

#22588386

C₂

yes

maximal, characteristic

#22586455

C₂

yes

maximal, complemented

#22586436

C₂

yes

maximal, complemented

C₇₁×#2113

C₂

yes

maximal, characteristic

#17986766

—

#17986048

—

#17986048

C₄

yes

complemented

C₂×#14556064

—

C₂×#14556064

C₂²

yes

—

#17986037

C₂²

yes

—

#17985973

—

#17985756

—

C₂×#131×C₇₁

C₂²

yes

characteristic

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₁

—

—

—

—

—

—

—

C₂×C₄

yes

—

#14556062

—

#14556060

—

#14556043

—

C₂×#41×C₇₁

D₄

yes

characteristic

#131×C₇₁

—

#131×C₇₁

—

C₂×M₁₆×C₇₁

C₂×C₄

yes

—

C₂×M₁₆×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

D₄

yes

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

#11765849

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

Q₁₆

yes

characteristic

C₂×D₄×C₇₁

SD₁₆

yes

—

C₂×D₄×C₇₁

Q₁₆

yes

characteristic

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

#103

yes

—

C₂³×C₇₁

C₂×SD₁₆

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#103

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#123

yes

—

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#117

yes

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#103

yes

characteristic

C₂²×C₇₁

#481

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

#12877

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

100

#1812

—

101

#1389

—

102

#1382

—

103

C₇₁

#12877

yes

characteristic, complemented, Sylow

104

#568

—

142

—

105

C₂×#131

C₂×D₇₁

yes

characteristic

106

C₂×M₃₂

C₂×D₇₁

yes

characteristic

107

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

108

#430

—

109

#430

—

142

—

110

C₂×#108

—

111

C₂×#108

—

142

—

112

#427

—

113

#410

—

142

—

114

C₂²⋊C₁₆

—

115

H(3,ℤ/4)

—

142

—

116

C₂×#41

D₂₈₄

yes

characteristic

117

#131

—

118

#131

—

119

#131

—

284

—

120

C₂×M₁₆

—

142

—

121

C₂×M₁₆

—

142

—

122

C₂×M₁₆

C₄×D₇₁

yes

—

123

C₂×M₁₆

D₂₈₄

yes

characteristic

124

C₂²×C₈

C₂²⋊D₇₁

yes

—

125

C₂²×C₈

C₂²×D₇₁

yes

characteristic

126

#117

—

142

—

127

M₃₂

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

142

—

133

#108

—

142

—

134

#107

—

142

—

135

#106

—

142

—

136

#99

—

284

—

137

#41

—

138

#41

—

139

C₂×Q₈

Dic₂₈₄

yes

characteristic

140

C₂×D₄

#11765854

yes

—

141

C₂×D₄

Dic₂₈₄

yes

characteristic

142

C₂²×C₄

(C₂×D₇₁)⋊C₄

yes

—

143

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

284

—

147

M₁₆

—

284

—

148

M₁₆

—

142

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

284

—

152

C₂×C₈

—

284

—

153

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

154

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

155

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

156

C₂×C₈

#11765864

yes

—

157

C₂×C₈

#11765864

yes

—

158

C₄⋊C₄

—

284

—

159

C₂²⋊C₄

—

284

—

160

C₄²

—

284

—

161

C₁₆

—

162

C₁₆

—

163

C₂³

#14556053

yes

—

164

C₂³

C₂×#11765854

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

568

—

171

C₂×C₄

#14556053

yes

—

172

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

173

C₂×C₄

#14556112

yes

—

174

C₂×C₄

C₂×Dic₂₈₄

yes

characteristic

175

C₂×C₄

#14556156

yes

—

176

C₈

—

177

C₈

—

284

—

178

C₈

#14556084

yes

—

179

C₈

#14556109

yes

characteristic

180

C₈

#14556109

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

C₂×#14556053

yes

characteristic

185

C₂²

#17986318

yes

—

186

C₄

—

187

C₄

—

1136

—

188

C₄

#17985860

yes

characteristic

189

C₄

#17986005

yes

characteristic

190

C₄

#17986335

yes

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#22588431

yes

characteristic

	subgroup	G/H
center	C₄	#17986005
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2113	C₂
socle	C₂×C₇₁	#1824
prime radical	C₂×#41×C₇₁	D₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2113	D₇₁
hypocenter	C₇₁	#12877
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

subgroup

G/H

center

C₄

#17986005

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#2113

C₂

socle

C₂×C₇₁

#1824

prime radical

C₂×#41×C₇₁

D₄

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

N	H ≅ G/N	type
3. #22586455	C₂	semidirect
4. #22586436	C₂	semidirect
8. #17986048	C₄	semidirect
103. C₇₁	#12877	semidirect

H ≅ G/N

type

3. #22586455

C₂

semidirect

4. #22586436

C₂

semidirect

8. #17986048

C₄

semidirect

103. C₇₁

#12877

semidirect

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

subgroup	index
#22596951	2
#22588386	2
#22586455	2
#22586436	2
C₇₁×#2113	2
#12877	71

subgroup

index

C₇₁×#2113

Subgroups — #31913791

Conjugacy classes of subgroups

Complements

Maximal subgroups