Group #27171566 — subgroups

Subgroups — #27171566

Order 12928.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂²×#13108058	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692179	C₂	1	2	yes	maximal, complemented
3	#20265037	C₂	1	1	yes	maximal, characteristic
4	#20264939	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	—	2	4	no	—
7	C₂×#13108058	—	2	2	no	—
8	C₂²×#11717579	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
10	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
11	#15692179	—	2	4	no	—
12	#15692162	—	2	1	no	—
13	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²⋊Q₈×C₁₀₁	—	2	2	no	—
16	C₂²⋊Q₈×C₁₀₁	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₀₁	—	2	1	no	—
20	C₂²×Dic₁₀₁	—	2	2	no	—
21	#13108058	—	2	8	no	—
22	C₂×#11717579	—	2	2	no	—
23	C₂×#11717579	—	2	2	no	—
24	C₂×#11717579	D₄	1	4	yes	—
25	C₄⋊Dic₁₀₁	—	2	2	no	—
26	C₄⋊Dic₁₀₁	—	2	2	no	—
27	C₂²⋊Dic₁₀₁	—	4	2	no	—
28	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
29	C₂×Q₈×C₁₀₁	—	2	1	no	—
30	C₂×Q₈×C₁₀₁	—	2	2	no	—
31	C₂×Q₈×C₁₀₁	—	2	2	no	—
32	C₂×Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₀₁	—	2	2	no	—
34	C₂²×C₄×C₁₀₁	—	2	2	no	—
35	C₂²×C₄×C₁₀₁	—	2	1	no	—
36	C₂²×C₄×C₁₀₁	—	2	1	no	—
37	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
40	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₁	—	2	4	no	—
42	C₄⋊C₄×C₁₀₁	—	2	1	no	—
43	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
44	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
45	C₂³×C₁₀₁	—	2	2	no	—
46	C₂³×C₁₀₁	—	2	1	no	—
47	C₂³×C₁₀₁	—	2	1	no	—
48	C₂³×C₁₀₁	—	2	2	no	—
49	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₀₁	—	2	4	no	—
52	Q₈×C₁₀₁	—	4	1	no	—
53	Q₈×C₁₀₁	—	4	1	no	—
54	C₂×C₄×C₁₀₁	—	2	2	no	—
55	C₂×C₄×C₁₀₁	—	2	4	no	—
56	C₂×C₄×C₁₀₁	—	2	2	no	—
57	C₂×C₄×C₁₀₁	—	2	2	no	—
58	C₂×C₄×C₁₀₁	—	2	2	no	—
59	C₂×C₄×C₁₀₁	—	2	2	no	—
60	C₂×C₄×C₁₀₁	—	4	2	no	—
61	C₂×C₄×C₁₀₁	—	4	1	no	—
62	C₂×C₄×C₁₀₁	—	4	1	no	—
63	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₁	#41	1	2	yes	—
68	C₂×Dic₁₀₁	—	4	2	no	—
69	C₂×Dic₁₀₁	—	4	2	no	—
70	C₂×Dic₁₀₁	—	4	2	no	—
71	#11717579	—	2	4	no	—
72	C₂²×C₁₀₁	—	2	2	no	—
73	C₂²×C₁₀₁	—	2	2	no	—
74	C₂²×C₁₀₁	—	2	2	no	—
75	C₂²×C₁₀₁	—	4	2	no	—
76	C₂²×C₁₀₁	—	4	2	no	—
77	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₁₀₁	#121	1	2	yes	—
79	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂²×C₁₀₁	#137	1	1	yes	characteristic
81	C₂²×C₁₀₁	#137	1	2	yes	—
82	C₄×C₁₀₁	—	2	2	no	—
83	C₄×C₁₀₁	—	4	2	no	—
84	C₄×C₁₀₁	—	4	2	no	—
85	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
86	C₄×C₁₀₁	#120	1	1	yes	characteristic
87	C₄×C₁₀₁	#123	1	1	yes	characteristic
88	Dic₁₀₁	—	8	2	no	—
89	C₂×C₁₀₁	—	4	2	no	—
90	C₂×C₁₀₁	#386	1	1	yes	characteristic
91	C₂×C₁₀₁	#417	1	1	yes	characteristic
92	C₂×C₁₀₁	#419	1	1	yes	characteristic
93	C₂×C₁₀₁	#468	1	2	yes	—
94	C₂×C₁₀₁	#470	1	2	yes	—
95	#1894	—	101	1	no	maximal, Sylow
96	C₁₀₁	#1894	1	1	yes	characteristic, complemented, Sylow
97	C₂²×M₁₆	—	101	1	no	—
98	C₂×C₂²⋊Q₈	D₁₀₁	1	1	yes	characteristic
99	C₂×#103	—	101	2	no	—
100	#380	—	101	1	no	—
101	#340	—	101	2	no	—
102	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
104	C₂×M₁₆	—	202	4	no	—
105	C₂×M₁₆	—	202	2	no	—
106	C₂²×C₈	—	101	2	no	—
107	C₂²⋊Q₈	—	2	2	no	—
108	C₂²⋊Q₈	Dic₁₀₁	1	4	yes	complemented
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₄⋊C₄	—	101	2	no	—
111	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
112	C₂×C₂²⋊C₄	—	2	1	no	—
113	C₂×C₂²⋊C₄	—	202	1	no	—
114	#103	—	202	4	no	—
115	#95	—	202	1	no	—
116	C₂⁴	C₂²⋊D₁₀₁	1	1	yes	characteristic
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	C₂×Dic₁₀₁	1	2	yes	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	202	2	no	—
126	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
127	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
129	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
130	M₁₆	—	202	8	no	—
131	C₂×C₈	—	101	4	no	—
132	C₂×C₈	—	202	2	no	—
133	C₂×C₈	—	202	2	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	202	2	no	—
137	C₄⋊C₄	—	202	2	no	—
138	C₄⋊C₄	C₂×Dic₁₀₁	1	2	yes	—
139	C₂²⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	404	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	1	no	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	2	no	—
145	C₂³	C₂²⋊Dic₁₀₁	1	2	yes	—
146	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
147	Q₈	—	2	4	no	—
148	Q₈	—	4	1	no	—
149	Q₈	—	4	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	404	2	no	—
160	C₂×C₄	—	404	2	no	—
161	C₂×C₄	—	404	2	no	—
162	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
163	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
164	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
165	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
166	C₂×C₄	#13108073	1	2	yes	—
167	C₈	—	202	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
174	C₂²	#15692227	1	2	yes	—
175	C₂²	#15692280	1	1	yes	characteristic
176	C₂²	#15692285	1	1	yes	characteristic
177	C₂²	#15692286	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	808	2	no	—
182	C₄	D₄×Dic₁₀₁	1	2	yes	—
183	C₄	#15692223	1	1	yes	characteristic
184	C₄	#15692236	1	1	yes	characteristic
185	C₂	—	4	2	no	—
186	C₂	#20265057	1	1	yes	characteristic
187	C₂	#20265178	1	1	yes	characteristic
188	C₂	#20265186	1	1	yes	characteristic
189	C₂	#20265425	1	2	yes	—
190	C₂	#20265433	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13108058

C₂

yes

maximal, characteristic

C₂×#15692179

C₂

yes

maximal, complemented

#20265037

C₂

yes

maximal, characteristic

#20264939

C₂

yes

maximal, complemented

C₂×C₂²⋊Q₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

—

C₂×#13108058

—

C₂²×#11717579

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

—

#15692179

—

#15692162

—

C₂²×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²⋊Q₈×C₁₀₁

—

C₂²⋊Q₈×C₁₀₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₀₁

—

C₂²×Dic₁₀₁

—

—

—

—

D₄

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₂⁴×C₁₀₁

D₄

yes

characteristic

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

#121

yes

—

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#470

yes

—

#1894

—

101

maximal, Sylow

C₁₀₁

#1894

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

101

—

C₂×C₂²⋊Q₈

D₁₀₁

yes

characteristic

C₂×#103

—

101

—

100

#380

—

101

—

101

#340

—

101

—

102

C₂²×Q₈

C₂×D₁₀₁

yes

characteristic

103

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

104

C₂×M₁₆

—

202

—

105

C₂×M₁₆

—

202

—

106

C₂²×C₈

—

101

—

107

C₂²⋊Q₈

—

108

C₂²⋊Q₈

Dic₁₀₁

yes

complemented

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

101

—

111

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

characteristic

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

—

202

—

114

#103

—

202

—

115

#95

—

202

—

116

C₂⁴

C₂²⋊D₁₀₁

yes

characteristic

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

C₂×Dic₁₀₁

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

202

—

126

C₂²×C₄

C₂×Dic₁₀₁

yes

—

127

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

128

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

129

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

130

M₁₆

—

202

—

131

C₂×C₈

—

101

—

132

C₂×C₈

—

202

—

133

C₂×C₈

—

202

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

202

—

137

C₄⋊C₄

—

202

—

138

C₄⋊C₄

C₂×Dic₁₀₁

yes

—

139

C₂²⋊C₄

—

140

C₂²⋊C₄

—

404

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂²⋊Dic₁₀₁

yes

—

146

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

404

—

160

C₂×C₄

—

404

—

161

C₂×C₄

—

404

—

162

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

163

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

164

C₂×C₄

D₄×D₁₀₁

yes

—

165

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

166

C₂×C₄

#13108073

yes

—

167

C₈

—

202

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

174

C₂²

#15692227

yes

—

175

C₂²

#15692280

yes

characteristic

176

C₂²

#15692285

yes

characteristic

177

C₂²

#15692286

yes

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

808

—

182

C₄

D₄×Dic₁₀₁

yes

—

183

C₄

#15692223

yes

characteristic

184

C₄

#15692236

yes

characteristic

185

C₂

—

186

C₂

#20265057

yes

characteristic

187

C₂

#20265178

yes

characteristic

188

C₂

#20265186

yes

characteristic

189

C₂

#20265425

yes

—

190

C₂

#20265433

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×C₂²⋊Q₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂×D₄
prime radical	C₂⁴×C₁₀₁	D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₀₁
hypocenter	C₁₀₁	#1894
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×C₂²⋊Q₈×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂×D₄

prime radical

C₂⁴×C₁₀₁

D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂×C₂²⋊Q₈

D₁₀₁

hypocenter

C₁₀₁

#1894

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

N	H ≅ G/N	type
2. C₂×#15692179	C₂	semidirect
4. #20264939	C₂	semidirect
16. C₂²⋊Q₈×C₁₀₁	C₄	semidirect
96. C₁₀₁	#1894	semidirect
108. C₂²⋊Q₈	Dic₁₀₁	semidirect

H ≅ G/N

type

2. C₂×#15692179

C₂

semidirect

4. #20264939

C₂

semidirect

16. C₂²⋊Q₈×C₁₀₁

C₄

semidirect

96. C₁₀₁

#1894

semidirect

108. C₂²⋊Q₈

Dic₁₀₁

semidirect

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

subgroup	index
C₂²×#13108058	2
C₂×#15692179	2
#20265037	2
#20264939	2
C₂×C₂²⋊Q₈×C₁₀₁	2
#1894	101

subgroup

index

C₂×C₂²⋊Q₈×C₁₀₁

101

Subgroups — #27171566

Conjugacy classes of subgroups

Complements

Maximal subgroups