Group #31011148 — subgroups

Subgroups — C₂×#22562454 #31011148

Order 17152.

582 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 203 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×#16863798	C₂	1	1	yes	maximal, characteristic
2	#22562454	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863200	C₂	1	4	yes	maximal
4	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
5	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524035	—	2	1	no	—
7	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
8	#16863798	—	2	2	no	—
9	#16863798	C₂²	1	4	yes	—
10	C₂×#14523977	C₂²	1	2	yes	—
11	C₂×#14523975	—	2	1	no	—
12	#16863200	C₂²	1	16	yes	—
13	#16863172	C₂²	1	4	yes	—
14	C₂²×Q₁₆×C₆₇	—	2	1	no	—
15	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
17	#494×C₆₇	—	2	2	no	—
18	#494×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#14524035	—	2	4	no	—
21	#14524035	—	4	2	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	C₂×Dic₆₇⋊C₄	—	4	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	4	no	—
25	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
26	#14523977	—	2	2	no	—
27	#14523977	C₂³	1	4	yes	—
28	#14523975	—	2	4	no	—
29	C₂²×Q₈×C₆₇	—	2	2	no	—
30	C₂×Q₁₆×C₆₇	—	2	2	no	—
31	C₂×Q₁₆×C₆₇	—	2	4	no	—
32	C₂×Q₁₆×C₆₇	—	2	8	no	—
33	C₂×Q₁₆×C₆₇	—	4	4	no	—
34	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₄⋊Q₈×C₆₇	—	2	4	no	—
38	C₄⋊Q₈×C₆₇	—	4	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	D₄	1	2	yes	—
42	C₄×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×Dic₆₇	—	4	2	no	—
44	C₄⋊Dic₆₇	—	2	4	no	—
45	C₄⋊Dic₆₇	—	4	2	no	—
46	Dic₆₇⋊C₄	—	4	8	no	—
47	C₆₇⋊C₁₆	—	2	4	no	—
48	C₂×Q₈×C₆₇	—	2	4	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	4	4	no	—
51	C₂×Q₈×C₆₇	—	4	4	no	—
52	C₂²×C₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	D₈	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₆₇	—	2	8	no	—
56	Q₁₆×C₆₇	—	4	4	no	—
57	Q₁₆×C₆₇	—	4	8	no	—
58	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
59	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₇	#41	1	4	yes	—
62	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₆₇	—	4	8	no	—
64	C₄²×C₆₇	D₈	1	2	yes	—
65	C₄²×C₆₇	C₂×D₄	1	2	yes	—
66	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₆₇	—	4	4	no	—
68	Q₈×C₆₇	—	4	4	no	—
69	Q₈×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	4	4	no	—
72	C₂×C₄×C₆₇	—	8	4	no	—
73	C₂×C₄×C₆₇	#124	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₆₇	—	4	4	no	—
79	C₂×Dic₆₇	—	8	4	no	—
80	C₈×C₆₇	#124	1	2	yes	—
81	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
82	C₈×C₆₇	C₂×#41	1	2	yes	—
83	C₂²×C₆₇	#486	1	2	yes	—
84	C₂²×C₆₇	#510	1	4	yes	—
85	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
86	C₄×C₆₇	—	2	4	no	—
87	C₄×C₆₇	—	8	4	no	—
88	C₄×C₆₇	#486	1	2	yes	—
89	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
90	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
91	Dic₆₇	—	8	4	no	—
92	C₂×#2245	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	#2245	1	4	yes	complemented
94	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
95	C₂×C₆₇	C₂×#510	1	2	yes	—
96	C₂×#499	—	67	1	no	—
97	C₂×#494	D₆₇	1	1	yes	characteristic
98	#2245	—	67	8	no	—
99	C₂×#358	—	67	4	no	—
100	C₂×#346	—	67	1	no	—
101	C₆₇	C₂×#2245	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₁₆	—	2	1	no	—
103	C₂²×Q₁₆	C₂×D₆₇	1	2	yes	—
104	C₂×C₄⋊Q₈	—	2	1	no	—
105	C₂×#125	—	134	1	no	—
106	C₂²×C₁₆	—	67	2	no	—
107	#499	—	67	4	no	—
108	#499	—	134	2	no	—
109	#494	—	2	2	no	—
110	#494	C₂×D₆₇	1	4	yes	—
111	C₂×#107	—	67	2	no	—
112	C₂×#106	—	134	1	no	—
113	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
114	#358	—	67	16	no	—
115	#346	—	67	4	no	—
116	C₂²×Q₈	—	2	2	no	—
117	C₂×Q₁₆	—	2	4	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	2	8	no	—
120	C₂×Q₁₆	—	4	4	no	—
121	C₂×Q₁₆	C₂²×D₆₇	1	4	yes	—
122	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	2	no	—
126	#125	—	268	2	no	—
127	#125	—	134	4	no	—
128	C₂×C₄⋊C₄	—	4	2	no	—
129	C₂×C₄⋊C₄	—	268	2	no	—
130	C₂×C₄⋊C₄	—	134	2	no	—
131	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
132	C₂×C₁₆	—	67	4	no	—
133	C₂×C₁₆	—	134	4	no	—
134	#107	—	67	4	no	—
135	#107	—	134	2	no	—
136	#106	—	134	4	no	—
137	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
138	C₄×C₈	C₂²×D₆₇	1	2	yes	—
139	C₂×Q₈	—	2	4	no	—
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	—	4	4	no	—
142	C₂×Q₈	—	4	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	268	2	no	—
145	C₂²×C₄	#11761613	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
147	Q₁₆	—	2	8	no	—
148	Q₁₆	—	4	4	no	—
149	Q₁₆	—	4	8	no	—
150	C₂×C₈	#11761616	1	4	yes	—
151	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
153	C₂×C₈	#11761627	1	4	yes	—
154	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	268	8	no	—
157	C₄⋊C₄	—	268	2	no	—
158	C₄⋊C₄	—	134	4	no	—
159	C₄²	#11761613	1	2	yes	—
160	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
161	C₁₆	—	134	4	no	—
162	C₂³	C₂×#11761613	1	1	yes	characteristic
163	Q₈	—	4	4	no	—
164	Q₈	—	4	4	no	—
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	268	4	no	—
170	C₂×C₄	—	536	4	no	—
171	C₂×C₄	#14524030	1	2	yes	—
172	C₂×C₄	C₂×#11761613	1	4	yes	—
173	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
174	C₂×C₄	C₂×#11761616	1	2	yes	—
175	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
176	C₈	#14524030	1	2	yes	—
177	C₈	C₂×#11761616	1	4	yes	—
178	C₈	C₂×#11761627	1	2	yes	—
179	C₂²	#16863742	1	2	yes	—
180	C₂²	#16863837	1	4	yes	—
181	C₂²	C₂²×#11761613	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	536	4	no	—
185	C₄	#16863742	1	2	yes	—
186	C₄	C₂×#14524030	1	1	yes	characteristic
187	C₄	C₂²×#11761616	1	1	yes	characteristic
188	C₂	#22562454	1	4	yes	direct product factor
189	C₂	C₂×#16863742	1	1	yes	characteristic
190	C₂	C₂×#16863837	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863798

C₂

yes

maximal, characteristic

#22562454

C₂

yes

maximal, direct product factor

C₂×#16863200

C₂

yes

maximal

C₂×#16863172

C₂

yes

maximal, characteristic

C₂×#494×C₆₇

C₂

yes

maximal, characteristic

C₂×#14524035

—

C₂²×C₆₇⋊C₁₆

C₂²

yes

—

#16863798

—

#16863798

C₂²

yes

—

C₂×#14523977

C₂²

yes

—

C₂×#14523975

—

#16863200

C₂²

yes

—

#16863172

C₂²

yes

—

C₂²×Q₁₆×C₆₇

—

C₂²×Q₁₆×C₆₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₆₇

—

#494×C₆₇

—

#494×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#14524035

—

#14524035

—

C₂×C₄⋊Dic₆₇

—

C₂×Dic₆₇⋊C₄

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂³

yes

—

#14523977

—

#14523977

C₂³

yes

—

#14523975

—

C₂²×Q₈×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

C₂³

yes

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₈

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₈

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

#124

yes

—

C₈×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

#486

yes

—

C₂²×C₆₇

#510

yes

—

C₂²×C₆₇

C₂²×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#486

yes

—

C₄×C₆₇

C₂×#124

yes

characteristic

C₄×C₆₇

C₂²×SD₁₆

yes

characteristic

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×#499

—

C₂×#494

D₆₇

yes

characteristic

#2245

—

C₂×#358

—

100

C₂×#346

—

101

C₆₇

C₂×#2245

yes

characteristic, complemented, Sylow

102

C₂²×Q₁₆

—

103

C₂²×Q₁₆

C₂×D₆₇

yes

—

104

C₂×C₄⋊Q₈

—

105

C₂×#125

—

134

—

106

C₂²×C₁₆

—

107

#499

—

108

#499

—

134

—

109

#494

—

110

#494

C₂×D₆₇

yes

—

111

C₂×#107

—

112

C₂×#106

—

134

—

113

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

114

#358

—

115

#346

—

116

C₂²×Q₈

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

C₂²×D₆₇

yes

—

122

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

123

C₂²×C₈

C₂²×D₆₇

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

#125

—

268

—

127

#125

—

134

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

268

—

130

C₂×C₄⋊C₄

—

134

—

131

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

—

134

#107

—

135

#107

—

134

—

136

#106

—

134

—

137

C₄×C₈

C₂²⋊D₆₇

yes

—

138

C₄×C₈

C₂²×D₆₇

yes

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

—

268

—

145

C₂²×C₄

#11761613

yes

—

146

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

147

Q₁₆

—

148

Q₁₆

—

149

Q₁₆

—

150

C₂×C₈

#11761616

yes

—

151

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

152

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

153

C₂×C₈

#11761627

yes

—

154

C₂×C₈

C₂³×D₆₇

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

268

—

157

C₄⋊C₄

—

268

—

158

C₄⋊C₄

—

134

—

159

C₄²

#11761613

yes

—

160

C₄²

C₂×C₂²⋊D₆₇

yes

—

161

C₁₆

—

134

—

162

C₂³

C₂×#11761613

yes

characteristic

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

268

—

170

C₂×C₄

—

536

—

171

C₂×C₄

#14524030

yes

—

172

C₂×C₄

C₂×#11761613

yes

—

173

C₂×C₄

C₂×#11761613

yes

characteristic

174

C₂×C₄

C₂×#11761616

yes

—

175

C₂×C₄

C₂²×C₂²⋊D₆₇

yes

characteristic

176

C₈

#14524030

yes

—

177

C₈

C₂×#11761616

yes

—

178

C₈

C₂×#11761627

yes

—

179

C₂²

#16863742

yes

—

180

C₂²

#16863837

yes

—

181

C₂²

C₂²×#11761613

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

536

—

185

C₄

#16863742

yes

—

186

C₄

C₂×#14524030

yes

characteristic

187

C₄

C₂²×#11761616

yes

characteristic

188

C₂

#22562454

yes

direct product factor

189

C₂

C₂×#16863742

yes

characteristic

190

C₂

C₂×#16863837

yes

—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	C₂×#2245
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×#11761613

commutator, cosocle

C₂×C₈×C₆₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₆₇

Fitting

C₂×#494×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
2. #22562454	C₂	direct (N × H)
93. C₂×C₆₇	#2245	semidirect
101. C₆₇	C₂×#2245	semidirect
188. C₂	#22562454	direct (N × H)

H ≅ G/N

type

2. #22562454

C₂

direct (N × H)

93. C₂×C₆₇

#2245

semidirect

101. C₆₇

C₂×#2245

semidirect

188. C₂

#22562454

direct (N × H)

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

subgroup	index
C₂×#16863798	2
#22562454	2
C₂×#16863200	2
C₂×#16863172	2
C₂×#494×C₆₇	2
C₂×#2245	67

subgroup

index

C₂×#494×C₆₇

Subgroups — C₂×#22562454 #31011148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22562454 #31011148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562454 #31011148