Group #31916027 — subgroups

Subgroups — #31916027

Order 18176.

398 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 159 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×#17986404	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
3	#22586432	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556128	—	2	1	no	—
6	#17986404	—	2	2	no	—
7	#17986404	C₄	1	4	yes	complemented
8	C₂×#14556062	—	2	1	no	—
9	C₂×#14556062	C₂²	1	2	yes	—
10	C₂×#14556062	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₇₁	—	2	1	no	—
12	#17985762	—	2	1	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₁	—	2	2	no	—
16	C₂²×Dic₁₄₂	—	4	1	no	—
17	#14556128	—	2	4	no	—
18	#14556128	—	4	2	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	4	1	no	—
22	C₂×C₄×Dic₇₁	—	4	1	no	—
23	#14556062	—	2	4	no	—
24	#14556062	—	2	2	no	—
25	#14556062	—	2	2	no	—
26	#14556062	C₂×C₄	1	4	yes	—
27	#14556062	Q₈	1	4	yes	complemented
28	C₇₁⋊#95	—	4	1	no	—
29	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₁	—	2	2	no	—
33	C₂×C₁₆×C₇₁	—	2	4	no	—
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₁	D₄	1	2	yes	—
36	C₂×Dic₁₄₂	—	4	4	no	—
37	C₂×Dic₁₄₂	—	8	4	no	—
38	C₂²×Dic₇₁	—	4	2	no	—
39	C₂²×Dic₇₁	—	4	2	no	—
40	C₄⋊Dic₇₁	—	2	4	no	—
41	C₄⋊Dic₇₁	—	2	4	no	—
42	C₄⋊Dic₇₁	—	4	4	no	—
43	C₄⋊Dic₇₁	—	4	2	no	—
44	C₄⋊Dic₇₁	—	4	2	no	—
45	C₄×Dic₇₁	—	8	2	no	—
46	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	—	2	2	no	—
50	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
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	2	yes	—
54	C₂×C₈×C₇₁	#41	1	2	yes	—
55	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₇₁	SD₁₆	1	2	yes	—
57	C₁₆×C₇₁	—	4	2	no	—
58	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	#102	1	2	yes	—
61	C₂×C₄×C₇₁	#102	1	2	yes	—
62	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
63	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
64	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
66	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#137	1	2	yes	—
68	C₂×Dic₇₁	—	4	4	no	—
69	C₂×Dic₇₁	—	4	4	no	—
70	C₂×Dic₇₁	—	8	2	no	—
71	C₂×Dic₇₁	—	8	2	no	—
72	C₂×Dic₇₁	—	8	4	no	—
73	Dic₁₄₂	—	8	8	no	—
74	C₈×C₇₁	—	2	2	no	—
75	C₈×C₇₁	C₄×Q₈	1	2	yes	—
76	C₈×C₇₁	#124	1	1	yes	characteristic
77	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
79	C₂²×C₇₁	#474	1	2	yes	—
80	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
81	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
82	C₂²×C₇₁	#509	1	2	yes	—
83	C₄×C₇₁	—	2	2	no	—
84	C₄×C₇₁	#358	1	4	yes	—
85	C₄×C₇₁	#391	1	1	yes	characteristic
86	C₄×C₇₁	#419	1	1	yes	characteristic
87	C₄×C₇₁	#475	1	2	yes	—
88	Dic₇₁	—	8	4	no	—
89	Dic₇₁	—	8	4	no	—
90	#13276	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	#1988	1	1	yes	characteristic
92	C₂×C₇₁	C₂×#358	1	1	yes	characteristic
93	C₂×C₇₁	#2140	1	1	yes	characteristic
94	C₂×C₇₁	#2226	1	2	yes	—
95	C₂×C₇₁	#2227	1	2	yes	—
96	C₂×#500	—	71	1	no	—
97	C₂×C₄⋊C₁₆	D₇₁	1	1	yes	characteristic
98	C₄×#107	—	71	1	no	—
99	#1380	—	71	4	no	—
100	C₇₁	#13276	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	—	142	1	no	—
102	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
103	#500	—	71	4	no	—
104	#500	—	142	2	no	—
105	C₂×#107	—	71	2	no	—
106	C₂×#107	—	71	2	no	—
107	C₂×#107	—	142	1	no	—
108	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	142	1	no	—
110	C₄⋊C₁₆	—	2	2	no	—
111	#340	—	142	1	no	—
112	C₂²×Q₈	—	284	1	no	—
113	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
115	C₄⋊Q₈	—	284	2	no	—
116	C₄⋊Q₈	—	142	4	no	—
117	C₂×C₄⋊C₄	—	284	1	no	—
118	C₂×C₄⋊C₄	—	142	2	no	—
119	C₂×C₄⋊C₄	—	142	2	no	—
120	C₂×C₄²	—	284	1	no	—
121	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	4	no	—
124	#107	—	71	4	no	—
125	#107	—	71	4	no	—
126	#107	—	142	4	no	—
127	#107	—	142	2	no	—
128	#107	—	142	2	no	—
129	C₄×C₈	C₄×D₇₁	1	2	yes	—
130	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
131	#95	—	284	1	no	—
132	C₂×Q₈	—	284	4	no	—
133	C₂×Q₈	—	568	4	no	—
134	C₂²×C₄	—	284	2	no	—
135	C₂²×C₄	—	284	2	no	—
136	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
137	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
141	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₇₁	1	2	yes	—
144	C₂×C₈	#11765864	1	2	yes	—
145	C₄⋊C₄	—	284	2	no	—
146	C₄⋊C₄	—	284	4	no	—
147	C₄⋊C₄	—	284	2	no	—
148	C₄⋊C₄	—	142	4	no	—
149	C₄⋊C₄	—	142	4	no	—
150	C₄²	—	568	2	no	—
151	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
152	C₄²	#11765854	1	2	yes	—
153	C₁₆	—	4	2	no	—
154	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
155	Q₈	—	568	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	284	4	no	—
158	C₂×C₄	—	284	4	no	—
159	C₂×C₄	—	568	2	no	—
160	C₂×C₄	—	568	4	no	—
161	C₂×C₄	—	568	2	no	—
162	C₂×C₄	#14556050	1	2	yes	—
163	C₂×C₄	#14556050	1	2	yes	—
164	C₂×C₄	#14556073	1	2	yes	—
165	C₂×C₄	Dic₅₆₈	1	2	yes	—
166	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
167	C₂×C₄	#14556108	1	2	yes	—
168	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
169	C₂×C₄	#14556161	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	#14556093	1	2	yes	—
172	C₈	#14556115	1	1	yes	characteristic
173	C₈	#14556126	1	1	yes	characteristic
174	C₂²	C₂×#14556050	1	1	yes	characteristic
175	C₂²	#17986280	1	2	yes	—
176	C₂²	C₂×#14556073	1	1	yes	characteristic
177	C₂²	C₂×Dic₅₆₈	1	1	yes	characteristic
178	C₂²	#17986437	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	568	4	no	—
181	C₄	—	568	4	no	—
182	C₄	#17985804	1	4	yes	—
183	C₄	#17985904	1	1	yes	characteristic
184	C₄	#17986009	1	1	yes	characteristic
185	C₄	#17986285	1	2	yes	—
186	C₂	#22589291	1	1	yes	characteristic
187	C₂	C₂×#17985804	1	1	yes	characteristic
188	C₂	#22590057	1	1	yes	characteristic
189	C₂	#22590488	1	2	yes	—
190	C₂	#22590493	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986404

C₂

yes

maximal, characteristic

C₄×#14556062

C₂

yes

maximal, characteristic

#22586432

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556128

—

#17986404

—

#17986404

C₄

yes

complemented

C₂×#14556062

—

C₂×#14556062

C₂²

yes

—

C₂×#14556062

C₂²

yes

—

C₄×C₄⋊Dic₇₁

—

#17985762

—

C₂²×C₁₆×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₁

—

C₂²×Dic₁₄₂

—

#14556128

—

#14556128

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄×Dic₇₁

—

—

—

—

C₂×C₄

yes

—

#14556062

Q₈

yes

complemented

C₇₁⋊#95

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

D₈

yes

characteristic

C₂²×C₄×C₇₁

D₈

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

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

C₂×Q₈

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

SD₁₆

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

SD₃₂

yes

—

C₂×C₄×C₇₁

Q₃₂

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

#137

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₄×Q₈

yes

—

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

C₂×SD₃₂

yes

characteristic

C₂²×C₇₁

C₂×Q₃₂

yes

characteristic

C₂²×C₇₁

#509

yes

—

C₄×C₇₁

—

C₄×C₇₁

#358

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2227

yes

—

C₂×#500

—

C₂×C₄⋊C₁₆

D₇₁

yes

characteristic

C₄×#107

—

#1380

—

100

C₇₁

#13276

yes

characteristic, complemented, Sylow

101

C₂×C₄⋊Q₈

—

142

—

102

C₂²×C₁₆

C₂×D₇₁

yes

—

103

#500

—

104

#500

—

142

—

105

C₂×#107

—

106

C₂×#107

—

107

C₂×#107

—

142

—

108

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

109

C₄×C₄⋊C₄

—

142

—

110

C₄⋊C₁₆

—

111

#340

—

142

—

112

C₂²×Q₈

—

284

—

113

C₂²×C₈

D₂₈₄

yes

characteristic

114

C₂²×C₈

C₂²×D₇₁

yes

characteristic

115

C₄⋊Q₈

—

284

—

116

C₄⋊Q₈

—

142

—

117

C₂×C₄⋊C₄

—

284

—

118

C₂×C₄⋊C₄

—

142

—

119

C₂×C₄⋊C₄

—

142

—

120

C₂×C₄²

—

284

—

121

C₂×C₄²

D₂₈₄

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

#107

—

125

#107

—

126

#107

—

142

—

127

#107

—

142

—

128

#107

—

142

—

129

C₄×C₈

C₄×D₇₁

yes

—

130

C₄×C₈

C₂²⋊D₇₁

yes

—

131

#95

—

284

—

132

C₂×Q₈

—

284

—

133

C₂×Q₈

—

568

—

134

C₂²×C₄

—

284

—

135

C₂²×C₄

—

284

—

136

C₂²×C₄

D₅₆₈

yes

characteristic

137

C₂²×C₄

D₅₆₈

yes

characteristic

138

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

141

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

142

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

143

C₂×C₈

Q₈×D₇₁

yes

—

144

C₂×C₈

#11765864

yes

—

145

C₄⋊C₄

—

284

—

146

C₄⋊C₄

—

284

—

147

C₄⋊C₄

—

284

—

148

C₄⋊C₄

—

142

—

149

C₄⋊C₄

—

142

—

150

C₄²

—

568

—

151

C₄²

(C₂×D₇₁)⋊C₄

yes

—

152

C₄²

#11765854

yes

—

153

C₁₆

—

154

C₂³

C₂×D₅₆₈

yes

characteristic

155

Q₈

—

568

—

156

C₂×C₄

—

157

C₂×C₄

—

284

—

158

C₂×C₄

—

284

—

159

C₂×C₄

—

568

—

160

C₂×C₄

—

568

—

161

C₂×C₄

—

568

—

162

C₂×C₄

#14556050

yes

—

163

C₂×C₄

#14556050

yes

—

164

C₂×C₄

#14556073

yes

—

165

C₂×C₄

Dic₅₆₈

yes

—

166

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

167

C₂×C₄

#14556108

yes

—

168

C₂×C₄

C₂×#11765854

yes

characteristic

169

C₂×C₄

#14556161

yes

—

170

C₈

—

171

C₈

#14556093

yes

—

172

C₈

#14556115

yes

characteristic

173

C₈

#14556126

yes

characteristic

174

C₂²

C₂×#14556050

yes

characteristic

175

C₂²

#17986280

yes

—

176

C₂²

C₂×#14556073

yes

characteristic

177

C₂²

C₂×Dic₅₆₈

yes

characteristic

178

C₂²

#17986437

yes

—

179

C₄

—

180

C₄

—

568

—

181

C₄

—

568

—

182

C₄

#17985804

yes

—

183

C₄

#17985904

yes

characteristic

184

C₄

#17986009

yes

characteristic

185

C₄

#17986285

yes

—

186

C₂

#22589291

yes

characteristic

187

C₂

C₂×#17985804

yes

characteristic

188

C₂

#22590057

yes

characteristic

189

C₂

#22590488

yes

—

190

C₂

#22590493

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄⋊C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#13276
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂³

C₂×D₅₆₈

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×C₄⋊C₁₆×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂×D₈

cosocle

C₂²×C₈×C₇₁

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₇₁

hypocenter

C₇₁

#13276

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
7. #17986404	C₄	semidirect
27. #14556062	Q₈	semidirect
100. C₇₁	#13276	semidirect

H ≅ G/N

type

7. #17986404

C₄

semidirect

27. #14556062

Q₈

semidirect

100. C₇₁

#13276

semidirect

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

subgroup	index
C₂×#17986404	2
C₄×#14556062	2
#22586432	2
C₂×C₄⋊C₁₆×C₇₁	2
#13276	71

subgroup

index

C₂×C₄⋊C₁₆×C₇₁

Subgroups — #31916027

Conjugacy classes of subgroups

Complements

Maximal subgroups