Group #16855481 — subgroups

Subgroups — C₂×#14521343 #16855481

Order 3968.

416 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 157 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×#11758465	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758464	C₂	1	1	yes	maximal, characteristic
3	#14521343	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11758395	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758404	C₂	1	1	yes	maximal, characteristic
7	C₂×#11758398	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₆₂	—	2	1	no	—
10	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁⋊C₄	—	2	1	no	—
16	C₂×Dic₃₁⋊C₄	—	2	1	no	—
17	C₂×C₄×Dic₃₁	—	2	1	no	—
18	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
19	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
20	#11758465	—	2	2	no	—
21	#11758465	C₂²	1	4	yes	—
22	#11758464	—	2	2	no	—
23	#11758464	C₂²	1	4	yes	—
24	C₃₁×#107	C₂²	1	4	yes	—
25	#11758404	C₂²	1	4	yes	—
26	Dic₃₁⋊C₈	C₂²	1	4	yes	—
27	#11758398	C₂²	1	4	yes	—
28	#11758395	C₂²	1	4	yes	—
29	C₂²×C₄×C₃₁	—	2	1	no	—
30	C₂²×C₄×C₃₁	—	2	1	no	—
31	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×Dic₃₁	—	2	1	no	—
33	C₂²×Dic₃₁	—	2	1	no	—
34	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
35	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
36	C₂×Dic₆₂	—	2	1	no	—
37	C₂×Dic₆₂	—	2	2	no	—
38	C₂×Dic₆₂	—	2	4	no	—
39	C₂×Dic₆₂	—	4	4	no	—
40	C₂×Dic₆₂	Q₈	1	4	yes	complemented
41	C₂×Dic₆₂	C₂³	1	2	yes	—
42	C₂×C₈×C₃₁	—	2	2	no	—
43	C₂×C₈×C₃₁	C₂³	1	2	yes	—
44	C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₄⋊C₄×C₃₁	—	2	1	no	—
46	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₄⋊Dic₃₁	—	2	1	no	—
49	C₄⋊Dic₃₁	C₂³	1	2	yes	—
50	Dic₃₁⋊C₄	—	2	4	no	—
51	Dic₃₁⋊C₄	—	2	4	no	—
52	C₄×Dic₃₁	—	2	4	no	—
53	C₄×Dic₃₁	D₄	1	2	yes	—
54	C₄×Dic₃₁	C₂³	1	2	yes	—
55	C₂×C₃₁⋊C₈	—	2	2	no	—
56	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
57	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	2	2	no	—
60	C₂×C₄×C₃₁	—	4	2	no	—
61	C₂×C₄×C₃₁	—	4	2	no	—
62	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₃₁	#41	1	2	yes	—
65	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₃₁	—	2	4	no	—
67	C₂×Dic₃₁	—	2	2	no	—
68	C₂×Dic₃₁	—	2	2	no	—
69	C₂×Dic₃₁	—	2	2	no	—
70	C₂×Dic₃₁	—	4	2	no	—
71	C₂×Dic₃₁	Q₁₆	1	4	yes	—
72	C₂×Dic₃₁	C₂×D₄	1	2	yes	—
73	C₂×Dic₃₁	C₂×D₄	1	2	yes	—
74	Dic₆₂	—	2	4	no	—
75	Dic₆₂	—	2	2	no	—
76	Dic₆₂	—	4	8	no	—
77	Dic₆₂	C₂×Q₈	1	4	yes	complemented
78	C₈×C₃₁	—	2	2	no	—
79	C₃₁⋊C₈	—	2	2	no	—
80	C₂×#477	—	31	1	no	maximal, Sylow
81	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
82	C₂²×C₃₁	C₂×Q₁₆	1	2	yes	—
83	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
84	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
85	C₄×C₃₁	—	4	2	no	—
86	C₄×C₃₁	—	4	2	no	—
87	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
88	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
89	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
90	Dic₃₁	—	2	4	no	—
91	Dic₃₁	—	2	2	no	—
92	Dic₃₁	—	4	2	no	—
93	Dic₃₁	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄⋊Q₈	—	31	1	no	—
95	C₂×C₄×Q₈	—	31	1	no	—
96	#477	—	31	8	no	—
97	C₂×#107	—	31	1	no	—
98	C₂×#107	D₃₁	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	31	1	no	—
100	C₂×#103	—	31	1	no	—
101	C₂×#103	—	31	1	no	—
102	C₂×C₃₁	#477	1	4	yes	complemented
103	C₂×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
105	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂²×Q₈	—	31	1	no	—
107	C₂²×Q₈	—	62	1	no	—
108	C₂²×C₈	—	31	1	no	—
109	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
110	C₄⋊Q₈	—	31	4	no	—
111	C₄⋊Q₈	—	62	2	no	—
112	C₄×Q₈	—	31	4	no	—
113	C₄×Q₈	—	62	2	no	—
114	C₂×C₄⋊C₄	—	31	1	no	—
115	C₂×C₄⋊C₄	—	62	1	no	—
116	C₂×C₄⋊C₄	—	62	1	no	—
117	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
119	C₂×C₄²	—	31	1	no	—
120	C₂×C₄²	—	62	1	no	—
121	#107	—	31	4	no	—
122	#107	C₂×D₃₁	1	4	yes	—
123	C₄⋊C₈	—	31	4	no	—
124	#103	—	31	4	no	—
125	#103	—	31	4	no	—
126	C₃₁	C₂×#477	1	1	yes	characteristic, complemented, Sylow
127	C₂×Q₈	—	31	4	no	—
128	C₂×Q₈	—	31	2	no	—
129	C₂×Q₈	—	62	4	no	—
130	C₂×Q₈	—	62	2	no	—
131	C₂×Q₈	—	62	1	no	—
132	C₂×Q₈	—	124	4	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	31	1	no	—
136	C₂²×C₄	—	31	1	no	—
137	C₂²×C₄	—	62	1	no	—
138	C₂²×C₄	—	62	1	no	—
139	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	31	2	no	—
142	C₂×C₈	—	62	2	no	—
143	C₂×C₈	C₂²×D₃₁	1	2	yes	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	31	2	no	—
147	C₄⋊C₄	—	62	4	no	—
148	C₄⋊C₄	—	62	4	no	—
149	C₄⋊C₄	—	62	1	no	—
150	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
151	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
152	C₄²	—	31	2	no	—
153	C₄²	—	31	2	no	—
154	C₄²	—	62	4	no	—
155	C₂³	D₄×D₃₁	1	1	yes	characteristic
156	Q₈	—	31	4	no	—
157	Q₈	—	62	4	no	—
158	Q₈	—	62	2	no	—
159	Q₈	—	124	8	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	31	4	no	—
165	C₂×C₄	—	31	2	no	—
166	C₂×C₄	—	31	2	no	—
167	C₂×C₄	—	62	4	no	—
168	C₂×C₄	—	62	2	no	—
169	C₂×C₄	—	62	2	no	—
170	C₂×C₄	—	62	2	no	—
171	C₂×C₄	—	124	2	no	—
172	C₂×C₄	#92495	1	2	yes	—
173	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
174	C₂×C₄	Q₈×D₃₁	1	2	yes	—
175	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	62	2	no	—
178	C₂²	#11758473	1	2	yes	—
179	C₂²	#11758488	1	2	yes	—
180	C₂²	Q₁₆×D₃₁	1	2	yes	—
181	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	31	4	no	—
185	C₄	—	62	4	no	—
186	C₄	—	62	2	no	—
187	C₄	—	124	2	no	—
188	C₄	#11758473	1	2	yes	—
189	C₄	C₂×#92495	1	1	yes	characteristic
190	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
191	C₂	#14521343	1	4	yes	direct product factor
192	C₂	C₂×#11758473	1	1	yes	characteristic
193	C₂	C₂×Q₁₆×D₃₁	1	1	yes	characteristic
194	C₂	C₂×#11758488	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758465

C₂

yes

maximal, characteristic

C₂×#11758464

C₂

yes

maximal, characteristic

#14521343

C₂

yes

maximal, direct product factor

C₂×#11758395

C₂

yes

maximal, characteristic

C₂×Dic₃₁⋊C₈

C₂

yes

maximal, characteristic

C₂×#11758404

C₂

yes

maximal, characteristic

C₂×#11758398

C₂

yes

maximal, characteristic

C₂×C₃₁×#107

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

—

C₂²×Dic₆₂

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

characteristic

C₂×Dic₃₁⋊C₄

—

C₂×Dic₃₁⋊C₄

—

C₂×C₄×Dic₃₁

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

#11758465

—

#11758465

C₂²

yes

—

#11758464

—

#11758464

C₂²

yes

—

C₃₁×#107

C₂²

yes

—

#11758404

C₂²

yes

—

Dic₃₁⋊C₈

C₂²

yes

—

#11758398

C₂²

yes

—

#11758395

C₂²

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂²×Dic₃₁

D₄

yes

characteristic

C₂²×Dic₃₁

D₄

yes

characteristic

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

Q₈

yes

complemented

C₂×Dic₆₂

C₂³

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

Q₁₆

yes

—

C₂×Dic₃₁

C₂×D₄

yes

—

C₂×Dic₃₁

C₂×D₄

yes

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

C₂×Q₈

yes

complemented

C₈×C₃₁

—

C₃₁⋊C₈

—

C₂×#477

—

maximal, Sylow

C₂²×C₃₁

C₂²⋊Q₈

yes

—

C₂²×C₃₁

C₂×Q₁₆

yes

—

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

C₂²×Q₈

yes

characteristic

C₄×C₃₁

C₂×#41

yes

characteristic

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

C₂×Q₁₆

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

#477

—

C₂×#107

—

C₂×#107

D₃₁

yes

characteristic

C₂×C₄⋊C₈

—

100

C₂×#103

—

101

C₂×#103

—

102

C₂×C₃₁

#477

yes

complemented

103

C₂×C₃₁

C₂×C₂²⋊Q₈

yes

characteristic

104

C₂×C₃₁

C₂²×Q₁₆

yes

characteristic

105

C₂×C₃₁

C₂×AGL(1,ℤ/8)

yes

characteristic

106

C₂²×Q₈

—

107

C₂²×Q₈

—

108

C₂²×C₈

—

109

C₂²×C₈

C₂×D₃₁

yes

characteristic

110

C₄⋊Q₈

—

111

C₄⋊Q₈

—

112

C₄×Q₈

—

113

C₄×Q₈

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

118

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

119

C₂×C₄²

—

120

C₂×C₄²

—

121

#107

—

122

#107

C₂×D₃₁

yes

—

123

C₄⋊C₈

—

124

#103

—

125

#103

—

126

C₃₁

C₂×#477

yes

characteristic, complemented, Sylow

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

124

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂²×D₃₁

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

C₂²×D₃₁

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

C₂²×D₃₁

yes

—

151

C₄⋊C₄

C₂²×D₃₁

yes

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₂³

D₄×D₃₁

yes

characteristic

156

Q₈

—

157

Q₈

—

158

Q₈

—

159

Q₈

—

124

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

124

—

172

C₂×C₄

#92495

yes

—

173

C₂×C₄

D₄×D₃₁

yes

characteristic

174

C₂×C₄

Q₈×D₃₁

yes

—

175

C₂×C₄

C₂³×D₃₁

yes

characteristic

176

C₈

—

177

C₈

—

178

C₂²

#11758473

yes

—

179

C₂²

#11758488

yes

—

180

C₂²

Q₁₆×D₃₁

yes

—

181

C₂²

C₂×D₄×D₃₁

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

124

—

188

C₄

#11758473

yes

—

189

C₄

C₂×#92495

yes

characteristic

190

C₄

C₂×Q₈×D₃₁

yes

characteristic

191

C₂

#14521343

yes

direct product factor

192

C₂

C₂×#11758473

yes

characteristic

193

C₂

C₂×Q₁₆×D₃₁

yes

characteristic

194

C₂

C₂×#11758488

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×#107	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₂×#107	D₃₁
hypocenter	C₃₁	C₂×#477
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₃₁×#107

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #14521343	C₂	direct (N × H)
40. C₂×Dic₆₂	Q₈	semidirect
77. Dic₆₂	C₂×Q₈	semidirect
102. C₂×C₃₁	#477	semidirect
126. C₃₁	C₂×#477	semidirect
191. C₂	#14521343	direct (N × H)

H ≅ G/N

type

3. #14521343

C₂

direct (N × H)

40. C₂×Dic₆₂

Q₈

semidirect

77. Dic₆₂

C₂×Q₈

semidirect

102. C₂×C₃₁

#477

semidirect

126. C₃₁

C₂×#477

semidirect

191. C₂

#14521343

direct (N × H)

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

subgroup	index
C₂×#11758465	2
C₂×#11758464	2
#14521343	2
C₂×#11758395	2
C₂×Dic₃₁⋊C₈	2
C₂×#11758404	2
C₂×#11758398	2
C₂×C₃₁×#107	2
C₂×#477	31

subgroup

index

C₂×Dic₃₁⋊C₈

C₂×C₃₁×#107

Subgroups — C₂×#14521343 #16855481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521343 #16855481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521343 #16855481