Group #27205639 — subgroups

Subgroups — #27205639

Order 13184.

532 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15693051	C₂	1	2	yes	maximal, complemented
2	C₂²×Dic₄₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20268362	C₂	1	2	yes	maximal, complemented
4	#20268182	C₂	1	4	yes	maximal, complemented
5	#20268174	C₂	1	2	yes	maximal, complemented
6	#20268169	C₂	1	2	yes	maximal, complemented
7	#20267746	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₀₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₀₆	C₂²	1	2	yes	—
10	#15693051	—	2	8	no	—
11	C₂×Dic₄₁₂	—	2	4	no	—
12	C₂×Dic₄₁₂	C₂²	1	2	yes	—
13	C₂×Dic₄₁₂	C₂²	1	2	yes	—
14	C₂×#13110095	C₂²	1	4	yes	—
15	C₂×#13110093	C₂²	1	4	yes	—
16	C₂×#13110089	C₂²	1	1	yes	characteristic
17	#15693019	C₂²	1	4	yes	—
18	#15693008	C₂²	1	4	yes	—
19	#15693002	—	2	4	no	—
20	#15693000	—	2	4	no	—
21	#15692980	C₂²	1	1	yes	characteristic
22	#15692931	C₂²	1	2	yes	—
23	C₈×Dic₁₀₃	C₂²	1	2	yes	—
24	C₂×#41×C₁₀₃	C₂²	1	2	yes	—
25	#135×C₁₀₃	—	2	4	no	—
26	C₂×Q₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
27	C₂×SD₁₆×C₁₀₃	C₂²	1	2	yes	—
28	C₂×D₈×C₁₀₃	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
30	C₂×Dic₂₀₆	—	2	4	no	—
31	C₂×Dic₂₀₆	—	2	4	no	—
32	C₂×Dic₂₀₆	—	2	2	no	—
33	C₂×Dic₂₀₆	C₂³	1	4	yes	—
34	C₂²×Dic₁₀₃	—	2	2	no	—
35	Dic₄₁₂	—	2	4	no	—
36	Dic₄₁₂	—	2	4	no	—
37	#13110095	—	2	8	no	—
38	#13110093	—	2	8	no	—
39	#13110089	D₄	1	4	yes	—
40	C₂×C₁₀₃⋊C₈	C₂³	1	2	yes	—
41	C₄⋊Dic₁₀₃	D₄	1	2	yes	—
42	Dic₁₀₃⋊C₄	—	2	8	no	—
43	C₂²⋊Dic₁₀₃	—	2	8	no	—
44	C₂²⋊Dic₁₀₃	D₄	1	2	yes	—
45	C₄×Dic₁₀₃	C₂³	1	2	yes	—
46	#41×C₁₀₃	—	2	4	no	—
47	#41×C₁₀₃	—	2	4	no	—
48	C₂×Q₈×C₁₀₃	C₂³	1	2	yes	—
49	C₂×D₄×C₁₀₃	—	2	2	no	—
50	C₂×D₄×C₁₀₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₀₃	—	2	2	no	—
52	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
53	Q₁₆×C₁₀₃	—	2	2	no	—
54	SD₁₆×C₁₀₃	—	2	4	no	—
55	D₈×C₁₀₃	—	2	2	no	—
56	C₂×C₈×C₁₀₃	D₄	1	4	yes	—
57	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
59	C₂³×C₁₀₃	—	2	2	no	—
60	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
61	Q₈×C₁₀₃	—	2	2	no	—
62	Q₈×C₁₀₃	—	2	2	no	—
63	D₄×C₁₀₃	—	2	2	no	—
64	D₄×C₁₀₃	—	2	2	no	—
65	D₄×C₁₀₃	—	4	4	no	—
66	C₂×C₄×C₁₀₃	—	2	2	no	—
67	C₂×C₄×C₁₀₃	—	4	4	no	—
68	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₁₀₃	—	2	4	no	—
73	C₂×Dic₁₀₃	—	4	4	no	—
74	C₂×Dic₁₀₃	C₂×D₄	1	4	yes	—
75	Dic₂₀₆	—	2	4	no	—
76	Dic₂₀₆	—	2	4	no	—
77	Dic₂₀₆	—	4	4	no	—
78	C₈×C₁₀₃	C₂×D₄	1	2	yes	—
79	C₈×C₁₀₃	C₂×D₄	1	2	yes	—
80	C₁₀₃⋊C₈	C₂×D₄	1	4	yes	—
81	C₂²×C₁₀₃	—	2	2	no	—
82	C₂²×C₁₀₃	—	4	2	no	—
83	C₂²×C₁₀₃	—	4	4	no	—
84	C₂²×C₁₀₃	#127	1	2	yes	—
85	C₂²×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₁₀₃	—	4	2	no	—
87	C₄×C₁₀₃	#127	1	2	yes	—
88	C₄×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
90	Dic₁₀₃	—	2	4	no	—
91	Dic₁₀₃	—	4	4	no	—
92	C₂×C₁₀₃	—	2	2	no	—
93	C₂×C₁₀₃	—	8	2	no	—
94	C₂×C₁₀₃	C₂×#127	1	1	yes	characteristic
95	C₂×C₁₀₃	#578	1	2	yes	—
96	#3156	—	103	1	no	maximal, Sylow
97	C₁₀₃	#3156	1	1	yes	characteristic, complemented, Sylow
98	C₂×#137	—	103	2	no	—
99	C₂×#135	D₁₀₃	1	1	yes	characteristic
100	C₂²×Q₁₆	—	103	1	no	—
101	#536	—	103	2	no	—
102	#497	—	103	4	no	—
103	#495	—	103	2	no	—
104	#494	—	103	2	no	—
105	#405	—	103	1	no	—
106	C₂×#41	C₂×D₁₀₃	1	2	yes	—
107	C₂²×Q₈	—	103	2	no	—
108	#137	—	206	8	no	—
109	#135	—	2	4	no	—
110	C₂×Q₁₆	—	103	4	no	—
111	C₂×Q₁₆	—	103	2	no	—
112	C₂×Q₁₆	—	103	2	no	—
113	C₂×Q₁₆	—	206	4	no	—
114	C₂×Q₁₆	C₂×D₁₀₃	1	1	yes	characteristic
115	C₂×SD₁₆	—	103	4	no	—
116	C₂×SD₁₆	C₂×D₁₀₃	1	2	yes	—
117	C₂×D₈	C₂×D₁₀₃	1	1	yes	characteristic
118	C₂×M₁₆	—	103	1	no	—
119	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
120	C₄⋊Q₈	—	103	4	no	—
121	#124	—	103	4	no	—
122	#123	—	206	4	no	—
123	C₂²⋊Q₈	—	206	4	no	—
124	#117	—	103	1	no	—
125	#97	—	103	2	no	—
126	C₄×C₈	—	103	2	no	—
127	#41	—	2	4	no	—
128	#41	—	2	4	no	—
129	C₂×Q₈	—	103	4	no	—
130	C₂×Q₈	—	206	4	no	—
131	C₂×Q₈	—	206	4	no	—
132	C₂×Q₈	—	206	2	no	—
133	C₂×Q₈	C₂²×D₁₀₃	1	2	yes	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	C₂²×D₁₀₃	1	2	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	206	2	no	—
138	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
139	Q₁₆	—	2	2	no	—
140	Q₁₆	—	206	8	no	—
141	Q₁₆	—	206	4	no	—
142	Q₁₆	—	206	4	no	—
143	SD₁₆	—	2	4	no	—
144	SD₁₆	—	206	8	no	—
145	D₈	—	2	2	no	—
146	M₁₆	—	103	4	no	—
147	C₂×C₈	—	103	2	no	—
148	C₂×C₈	C₂²⋊D₁₀₃	1	4	yes	—
149	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
151	C₄⋊C₄	—	103	2	no	—
152	C₄⋊C₄	—	206	8	no	—
153	C₂²⋊C₄	—	103	2	no	—
154	C₂²⋊C₄	—	206	8	no	—
155	C₄²	—	103	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	D₄×D₁₀₃	1	1	yes	characteristic
158	Q₈	—	2	2	no	—
159	Q₈	—	2	2	no	—
160	Q₈	—	412	4	no	—
161	Q₈	—	206	4	no	—
162	Q₈	—	206	4	no	—
163	D₄	—	2	2	no	—
164	D₄	—	2	2	no	—
165	D₄	—	4	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	103	4	no	—
169	C₂×C₄	—	412	4	no	—
170	C₂×C₄	—	206	4	no	—
171	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
172	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
174	C₂×C₄	C₂³×D₁₀₃	1	1	yes	characteristic
175	C₈	—	103	4	no	—
176	C₈	C₂×C₂²⋊D₁₀₃	1	2	yes	—
177	C₈	C₂×C₂²⋊D₁₀₃	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	#15693017	1	2	yes	—
182	C₂²	C₂×D₄×D₁₀₃	1	1	yes	characteristic
183	C₄	—	4	2	no	—
184	C₄	—	412	4	no	—
185	C₄	—	206	4	no	—
186	C₄	#15693017	1	2	yes	—
187	C₄	C₂×D₄×D₁₀₃	1	1	yes	characteristic
188	C₄	C₂²×C₂²⋊D₁₀₃	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	8	2	no	—
191	C₂	C₂×#15693017	1	1	yes	characteristic
192	C₂	#20268605	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15693051

C₂

yes

maximal, complemented

C₂²×Dic₄₁₂

C₂

yes

maximal, characteristic, complemented

#20268362

C₂

yes

maximal, complemented

#20268182

C₂

yes

maximal, complemented

#20268174

C₂

yes

maximal, complemented

#20268169

C₂

yes

maximal, complemented

#20267746

C₂

yes

maximal, characteristic, complemented

C₂×#135×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₆

C₂²

yes

—

#15693051

—

C₂×Dic₄₁₂

—

C₂×Dic₄₁₂

C₂²

yes

—

C₂×Dic₄₁₂

C₂²

yes

—

C₂×#13110095

C₂²

yes

—

C₂×#13110093

C₂²

yes

—

C₂×#13110089

C₂²

yes

characteristic

#15693019

C₂²

yes

—

#15693008

C₂²

yes

—

#15693002

—

#15693000

—

#15692980

C₂²

yes

characteristic

#15692931

C₂²

yes

—

C₈×Dic₁₀₃

C₂²

yes

—

C₂×#41×C₁₀₃

C₂²

yes

—

#135×C₁₀₃

—

C₂×Q₁₆×C₁₀₃

C₂²

yes

characteristic

C₂×SD₁₆×C₁₀₃

C₂²

yes

—

C₂×D₈×C₁₀₃

C₂²

yes

characteristic

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

C₂³

yes

—

C₂²×Dic₁₀₃

—

Dic₄₁₂

—

Dic₄₁₂

—

#13110095

—

#13110093

—

#13110089

D₄

yes

—

C₂×C₁₀₃⋊C₈

C₂³

yes

—

C₄⋊Dic₁₀₃

D₄

yes

—

Dic₁₀₃⋊C₄

—

C₂²⋊Dic₁₀₃

—

C₂²⋊Dic₁₀₃

D₄

yes

—

C₄×Dic₁₀₃

C₂³

yes

—

#41×C₁₀₃

—

#41×C₁₀₃

—

C₂×Q₈×C₁₀₃

C₂³

yes

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

C₂³

yes

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

Q₁₆×C₁₀₃

—

SD₁₆×C₁₀₃

—

D₈×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₂×C₈×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

C₂³

yes

characteristic

C₂³×C₁₀₃

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂⁴

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

C₂×D₄

yes

—

Dic₂₀₆

—

Dic₂₀₆

—

Dic₂₀₆

—

C₈×C₁₀₃

C₂×D₄

yes

—

C₈×C₁₀₃

C₂×D₄

yes

—

C₁₀₃⋊C₈

C₂×D₄

yes

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#127

yes

—

C₂²×C₁₀₃

C₂²×D₄

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

#127

yes

—

C₄×C₁₀₃

C₂²×D₄

yes

characteristic

C₄×C₁₀₃

C₂²×D₄

yes

characteristic

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

C₂×#127

yes

characteristic

C₂×C₁₀₃

#578

yes

—

#3156

—

103

maximal, Sylow

C₁₀₃

#3156

yes

characteristic, complemented, Sylow

C₂×#137

—

103

—

C₂×#135

D₁₀₃

yes

characteristic

100

C₂²×Q₁₆

—

103

—

101

#536

—

103

—

102

#497

—

103

—

103

#495

—

103

—

104

#494

—

103

—

105

#405

—

103

—

106

C₂×#41

C₂×D₁₀₃

yes

—

107

C₂²×Q₈

—

103

—

108

#137

—

206

—

109

#135

—

110

C₂×Q₁₆

—

103

—

111

C₂×Q₁₆

—

103

—

112

C₂×Q₁₆

—

103

—

113

C₂×Q₁₆

—

206

—

114

C₂×Q₁₆

C₂×D₁₀₃

yes

characteristic

115

C₂×SD₁₆

—

103

—

116

C₂×SD₁₆

C₂×D₁₀₃

yes

—

117

C₂×D₈

C₂×D₁₀₃

yes

characteristic

118

C₂×M₁₆

—

103

—

119

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

120

C₄⋊Q₈

—

103

—

121

#124

—

103

—

122

#123

—

206

—

123

C₂²⋊Q₈

—

206

—

124

#117

—

103

—

125

#97

—

103

—

126

C₄×C₈

—

103

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

103

—

130

C₂×Q₈

—

206

—

131

C₂×Q₈

—

206

—

132

C₂×Q₈

—

206

—

133

C₂×Q₈

C₂²×D₁₀₃

yes

—

134

C₂×D₄

—

135

C₂×D₄

C₂²×D₁₀₃

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

206

—

138

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

139

Q₁₆

—

140

Q₁₆

—

206

—

141

Q₁₆

—

206

—

142

Q₁₆

—

206

—

143

SD₁₆

—

144

SD₁₆

—

206

—

145

D₈

—

146

M₁₆

—

103

—

147

C₂×C₈

—

103

—

148

C₂×C₈

C₂²⋊D₁₀₃

yes

—

149

C₂×C₈

C₂²×D₁₀₃

yes

characteristic

150

C₂×C₈

C₂²×D₁₀₃

yes

characteristic

151

C₄⋊C₄

—

103

—

152

C₄⋊C₄

—

206

—

153

C₂²⋊C₄

—

103

—

154

C₂²⋊C₄

—

206

—

155

C₄²

—

103

—

156

C₂³

—

157

C₂³

D₄×D₁₀₃

yes

characteristic

158

Q₈

—

159

Q₈

—

160

Q₈

—

412

—

161

Q₈

—

206

—

162

Q₈

—

206

—

163

D₄

—

164

D₄

—

165

D₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

103

—

169

C₂×C₄

—

412

—

170

C₂×C₄

—

206

—

171

C₂×C₄

D₄×D₁₀₃

yes

—

172

C₂×C₄

D₄×D₁₀₃

yes

characteristic

173

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

—

174

C₂×C₄

C₂³×D₁₀₃

yes

characteristic

175

C₈

—

103

—

176

C₈

C₂×C₂²⋊D₁₀₃

yes

—

177

C₈

C₂×C₂²⋊D₁₀₃

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

#15693017

yes

—

182

C₂²

C₂×D₄×D₁₀₃

yes

characteristic

183

C₄

—

184

C₄

—

412

—

185

C₄

—

206

—

186

C₄

#15693017

yes

—

187

C₄

C₂×D₄×D₁₀₃

yes

characteristic

188

C₄

C₂²×C₂²⋊D₁₀₃

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

C₂×#15693017

yes

characteristic

192

C₂

#20268605

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₃
commutator, cosocle	C₂×C₄×C₁₀₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting, prime radical	C₂×#135×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂²×D₄
hypercenter	C₂×#135	D₁₀₃
hypocenter	C₁₀₃	#3156
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

subgroup

G/H

center

C₂²

C₂×D₄×D₁₀₃

commutator, cosocle

C₂×C₄×C₁₀₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₃

Fitting, prime radical

C₂×#135×C₁₀₃

C₂

socle

C₂²×C₁₀₃

C₂²×D₄

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂×#15693051	C₂	semidirect
2. C₂²×Dic₄₁₂	C₂	semidirect
3. #20268362	C₂	semidirect
4. #20268182	C₂	semidirect
5. #20268174	C₂	semidirect
6. #20268169	C₂	semidirect
7. #20267746	C₂	semidirect
97. C₁₀₃	#3156	semidirect

H ≅ G/N

type

1. C₂×#15693051

C₂

semidirect

2. C₂²×Dic₄₁₂

C₂

semidirect

3. #20268362

C₂

semidirect

4. #20268182

C₂

semidirect

5. #20268174

C₂

semidirect

6. #20268169

C₂

semidirect

7. #20267746

C₂

semidirect

97. C₁₀₃

#3156

semidirect

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

subgroup	index
C₂×#15693051	2
C₂²×Dic₄₁₂	2
#20268362	2
#20268182	2
#20268174	2
#20268169	2
#20267746	2
C₂×#135×C₁₀₃	2
#3156	103

subgroup

index

C₂²×Dic₄₁₂

C₂×#135×C₁₀₃

103

Subgroups — #27205639

Conjugacy classes of subgroups

Complements

Maximal subgroups