Group #31913644 — subgroups

Subgroups — #31913644

Order 18176.

354 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 111 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×#17986417	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985828	C₂	1	2	yes	maximal, complemented
3	#22588387	C₂	1	2	yes	maximal
4	#22586444	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556134	C₂²	1	1	yes	characteristic
7	#17986417	—	2	4	no	—
8	C₂×#14556067	C₄	1	2	yes	—
9	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₇₁	C₄	1	2	yes	—
11	C₂×#14556064	C₂²	1	2	yes	—
12	#17985981	—	2	2	no	—
13	#17985828	—	2	4	no	—
14	C₇₁⋊#338	—	2	2	no	—
15	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#108×C₇₁	C₂²	1	2	yes	—
17	C₂×#11765865	D₄	1	1	yes	characteristic
18	#14556134	—	2	2	no	—
19	#14556134	—	2	2	no	—
20	C₂×#11765849	—	2	2	no	—
21	C₂×#11765849	D₄	1	1	yes	characteristic
22	C₂×#11765848	Q₈	1	2	yes	—
23	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
24	#14556067	—	2	4	no	—
25	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
28	C₁₆×D₇₁	—	2	4	no	—
29	#14556064	—	2	2	no	—
30	#14556064	—	2	2	no	—
31	#14556046	—	4	2	no	—
32	#14556044	—	4	2	no	—
33	C₂×M₁₆×C₇₁	—	2	2	no	—
34	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	M₃₂×C₇₁	—	2	2	no	—
36	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
37	#108×C₇₁	—	2	2	no	—
38	#108×C₇₁	—	2	2	no	—
39	#11765865	—	2	2	no	—
40	#11765865	—	2	2	no	—
41	C₂×Dic₁₄₂	D₈	1	1	yes	characteristic
42	C₂×D₂₈₄	D₈	1	1	yes	characteristic
43	C₂×C₂²⋊D₇₁	Q₁₆	1	2	yes	—
44	C₂×C₄×D₇₁	C₄⋊C₄	1	2	yes	—
45	#11765849	—	2	2	no	—
46	#11765849	—	4	2	no	—
47	#11765849	—	4	2	no	—
48	#11765848	—	2	4	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
51	C₈×D₇₁	—	2	2	no	—
52	C₈×D₇₁	—	2	2	no	—
53	C₇₁⋊C₁₆	C₂²⋊C₄	1	2	yes	—
54	C₇₁⋊C₁₆	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₇₁	—	4	2	no	—
57	M₁₆×C₇₁	—	4	2	no	—
58	C₂×C₈×C₇₁	—	4	2	no	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₁	#41	1	2	yes	—
63	C₁₆×C₇₁	—	2	2	no	—
64	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×D₇₁	#107	1	2	yes	—
66	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#121	1	2	yes	—
68	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#135	1	2	yes	—
70	C₂²⋊D₇₁	—	2	4	no	—
71	C₂×Dic₇₁	#107	1	2	yes	—
72	D₂₈₄	—	2	2	no	—
73	C₄×D₇₁	—	2	2	no	—
74	C₄×D₇₁	—	2	2	no	—
75	Dic₁₄₂	—	2	2	no	—
76	C₈×C₇₁	—	8	2	no	—
77	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₇₁	#117	1	1	yes	characteristic
79	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
80	C₇₁⋊C₈	—	2	2	no	—
81	C₇₁⋊C₈	—	8	2	no	—
82	C₂²×C₇₁	—	4	1	no	—
83	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
84	C₂²×C₇₁	#467	1	2	yes	—
85	C₂×D₇₁	—	2	2	no	—
86	C₂×D₇₁	—	4	2	no	—
87	C₄×C₇₁	#380	1	1	yes	characteristic
88	C₄×C₇₁	#427	1	1	yes	characteristic
89	C₄×C₇₁	#466	1	2	yes	—
90	Dic₇₁	—	2	2	no	—
91	#12840	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	—	2	2	no	—
94	C₂×C₇₁	#1808	1	1	yes	characteristic
95	D₇₁	—	4	2	no	—
96	C₂×#504	—	71	1	no	—
97	C₂×#368	—	71	2	no	—
98	#1813	—	71	2	no	—
99	#1385	—	71	1	no	—
100	#1385	D₇₁	1	1	yes	characteristic, complemented
101	C₇₁	#12840	1	1	yes	characteristic, complemented, Sylow
102	C₂×#131	—	71	1	no	—
103	#504	—	142	4	no	—
104	C₂×M₃₂	—	71	2	no	—
105	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
106	C₂²×C₁₆	—	71	1	no	—
107	C₂²×C₁₆	—	71	2	no	—
108	C₂×#108	—	71	2	no	—
109	C₂×#108	C₂×D₇₁	1	2	yes	—
110	#413	—	142	2	no	—
111	#368	—	142	4	no	—
112	#338	—	142	2	no	—
113	C₂×#41	—	71	1	no	—
114	#131	—	142	2	no	—
115	#131	—	142	2	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	71	1	no	—
118	C₂×M₁₆	—	71	2	no	—
119	C₂×M₁₆	—	142	2	no	—
120	C₂²×C₈	—	71	2	no	—
121	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	—	142	4	no	—
124	C₂×C₁₆	—	71	1	no	—
125	C₂×C₁₆	—	71	1	no	—
126	C₂×C₁₆	—	71	4	no	—
127	C₂×C₁₆	—	142	4	no	—
128	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
129	#108	—	2	2	no	—
130	#108	—	2	2	no	—
131	#108	—	142	2	no	—
132	#108	—	142	2	no	—
133	#101	—	284	2	no	—
134	#100	—	284	2	no	—
135	#41	—	142	2	no	—
136	#41	—	142	2	no	—
137	C₂×Q₈	—	71	1	no	—
138	C₂×D₄	—	71	2	no	—
139	C₂×D₄	—	71	1	no	—
140	C₂²×C₄	—	71	2	no	—
141	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	284	2	no	—
145	M₁₆	—	284	2	no	—
146	M₁₆	—	142	2	no	—
147	M₁₆	—	142	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	284	2	no	—
150	C₂×C₈	—	71	2	no	—
151	C₂×C₈	—	142	2	no	—
152	C₂×C₈	—	142	2	no	—
153	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
154	C₂×C₈	D₄×D₇₁	1	2	yes	—
155	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
156	C₂×C₈	#11765865	1	2	yes	—
157	C₁₆	—	2	2	no	—
158	C₁₆	—	71	2	no	—
159	C₁₆	—	71	2	no	—
160	C₂³	—	71	2	no	—
161	C₂³	Q₁₆×D₇₁	1	1	yes	characteristic
162	Q₈	—	142	2	no	—
163	D₄	—	142	4	no	—
164	D₄	—	142	2	no	—
165	C₂×C₄	—	71	2	no	—
166	C₂×C₄	—	142	2	no	—
167	C₂×C₄	—	142	2	no	—
168	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
169	C₂×C₄	#14556101	1	2	yes	—
170	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
171	C₂×C₄	#14556149	1	2	yes	—
172	C₈	—	8	2	no	—
173	C₈	—	568	2	no	—
174	C₈	—	142	2	no	—
175	C₈	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
176	C₈	#14556085	1	1	yes	characteristic
177	C₈	#14556106	1	2	yes	—
178	C₂²	—	4	1	no	—
179	C₂²	—	284	2	no	—
180	C₂²	—	142	2	no	—
181	C₂²	#107×D₇₁	1	1	yes	characteristic
182	C₂²	#17986239	1	2	yes	—
183	C₄	—	142	2	no	—
184	C₄	#17985858	1	1	yes	characteristic
185	C₄	#17986034	1	1	yes	characteristic
186	C₄	#17986234	1	2	yes	—
187	C₂	—	2	1	no	—
188	C₂	—	2	2	no	—
189	C₂	—	284	2	no	—
190	C₂	#22588368	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986417

C₂

yes

maximal, characteristic

C₂×#17985828

C₂

yes

maximal, complemented

#22588387

C₂

yes

maximal

#22586444

C₂

yes

maximal, characteristic, complemented

C₇₁×#1385

C₂

yes

maximal, characteristic, complemented

C₂×#14556134

C₂²

yes

characteristic

#17986417

—

C₂×#14556067

C₄

yes

—

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₆×D₇₁

C₄

yes

—

C₂×#14556064

C₂²

yes

—

#17985981

—

#17985828

—

C₇₁⋊#338

—

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

C₂×#108×C₇₁

C₂²

yes

—

C₂×#11765865

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11765848

Q₈

yes

—

C₂×C₈×D₇₁

C₂×C₄

yes

—

#14556067

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

D₄

yes

—

C₁₆×D₇₁

—

—

—

—

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

#108×C₇₁

—

#108×C₇₁

—

#11765865

—

#11765865

—

C₂×Dic₁₄₂

D₈

yes

characteristic

C₂×D₂₈₄

D₈

yes

characteristic

C₂×C₂²⋊D₇₁

Q₁₆

yes

—

C₂×C₄×D₇₁

C₄⋊C₄

yes

—

—

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₈×D₇₁

—

C₈×D₇₁

—

C₇₁⋊C₁₆

C₂²⋊C₄

yes

—

C₇₁⋊C₁₆

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×Q₁₆

yes

characteristic

C₂²×D₇₁

#107

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#121

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

—

C₂²⋊D₇₁

—

C₂×Dic₇₁

#107

yes

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

#467

yes

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

#12840

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1808

yes

characteristic

D₇₁

—

—

—

—

—

100

#1385

D₇₁

yes

characteristic, complemented

101

C₇₁

#12840

yes

characteristic, complemented, Sylow

102

C₂×#131

—

103

#504

—

142

—

104

C₂×M₃₂

—

105

C₂×M₃₂

C₂×D₇₁

yes

characteristic

106

C₂²×C₁₆

—

107

C₂²×C₁₆

—

108

C₂×#108

—

109

C₂×#108

C₂×D₇₁

yes

—

110

#413

—

142

—

111

#368

—

142

—

112

#338

—

142

—

113

C₂×#41

—

114

#131

—

142

—

115

#131

—

142

—

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

142

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂²×D₇₁

yes

characteristic

122

M₃₂

—

123

M₃₂

—

142

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

142

—

128

C₂×C₁₆

C₄×D₇₁

yes

—

129

#108

—

130

#108

—

131

#108

—

142

—

132

#108

—

142

—

133

#101

—

284

—

134

#100

—

284

—

135

#41

—

142

—

136

#41

—

142

—

137

C₂×Q₈

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₇₁

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

284

—

145

M₁₆

—

284

—

146

M₁₆

—

142

—

147

M₁₆

—

142

—

148

C₂×C₈

—

149

C₂×C₈

—

284

—

150

C₂×C₈

—

151

C₂×C₈

—

142

—

152

C₂×C₈

—

142

—

153

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

154

C₂×C₈

D₄×D₇₁

yes

—

155

C₂×C₈

Q₈×D₇₁

yes

characteristic

156

C₂×C₈

#11765865

yes

—

157

C₁₆

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

—

161

C₂³

Q₁₆×D₇₁

yes

characteristic

162

Q₈

—

142

—

163

D₄

—

142

—

164

D₄

—

142

—

165

C₂×C₄

—

166

C₂×C₄

—

142

—

167

C₂×C₄

—

142

—

168

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

169

C₂×C₄

#14556101

yes

—

170

C₂×C₄

D₈×D₇₁

yes

characteristic

171

C₂×C₄

#14556149

yes

—

172

C₈

—

173

C₈

—

568

—

174

C₈

—

142

—

175

C₈

C₂²⋊C₄×D₇₁

yes

characteristic

176

C₈

#14556085

yes

characteristic

177

C₈

#14556106

yes

—

178

C₂²

—

179

C₂²

—

284

—

180

C₂²

—

142

—

181

C₂²

#107×D₇₁

yes

characteristic

182

C₂²

#17986239

yes

—

183

C₄

—

142

—

184

C₄

#17985858

yes

characteristic

185

C₄

#17986034

yes

characteristic

186

C₄

#17986234

yes

—

187

C₂

—

188

C₂

—

189

C₂

—

284

—

190

C₂

#22588368

yes

characteristic

	subgroup	G/H
center	C₄	#17986034
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1385	C₂
socle	C₂×C₇₁	#1808
prime radical	C₂×#11765865	D₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1385	D₇₁
hypocenter	C₇₁	#12840
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#17986034

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1385

C₂

C₂×C₇₁

D₄

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
2. C₂×#17985828	C₂	semidirect
4. #22586444	C₂	semidirect
5. C₇₁×#1385	C₂	semidirect
100. #1385	D₇₁	semidirect
101. C₇₁	#12840	semidirect

H ≅ G/N

type

2. C₂×#17985828

C₂

semidirect

4. #22586444

C₂

semidirect

5. C₇₁×#1385

C₂

semidirect

100. #1385

D₇₁

semidirect

101. C₇₁

#12840

semidirect

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

subgroup	index
C₂×#17986417	2
C₂×#17985828	2
#22588387	2
#22586444	2
C₇₁×#1385	2
#12840	71

subgroup

index

C₇₁×#1385

Subgroups — #31913644

Conjugacy classes of subgroups

Complements

Maximal subgroups