Group #27539410 — subgroups

Subgroups — C₂×#20276961 #27539410

Order 13568.

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×#16811025	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
3	#20276961	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810428	C₂	1	4	yes	maximal
5	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	—	2	1	no	—
7	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
8	C₂×#13133217	—	2	1	no	—
9	C₂×#13133209	—	2	1	no	—
10	#16811025	—	2	2	no	—
11	#16811025	C₂²	1	4	yes	—
12	#16811003	—	2	2	no	—
13	#16811003	C₂²	1	4	yes	—
14	C₂×#13133151	C₂²	1	2	yes	—
15	C₂×#13133149	—	2	1	no	—
16	#16810428	C₂²	1	16	yes	—
17	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#346	C₂²	1	4	yes	—
20	C₂²×Dic₁₀₆	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	4	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	8	no	—
24	C₂×Dic₂₁₂	—	4	4	no	—
25	C₂×Dic₂₁₂	C₂³	1	4	yes	—
26	#13133217	—	2	4	no	—
27	#13133217	—	4	2	no	—
28	#13133209	—	2	4	no	—
29	#13133209	—	4	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	—	4	2	no	—
32	C₂×#11719919	—	4	2	no	—
33	#13133151	—	2	2	no	—
34	#13133151	C₂³	1	4	yes	—
35	#13133149	—	2	4	no	—
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₃	—	2	4	no	—
40	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
41	C₄×C₈×C₅₃	D₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₅₃	D₈	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂²×Dic₅₃	—	4	2	no	—
47	C₂×Dic₁₀₆	—	2	4	no	—
48	C₂×Dic₁₀₆	—	4	4	no	—
49	C₂×Dic₁₀₆	—	4	4	no	—
50	C₂×Dic₁₀₆	—	4	2	no	—
51	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
52	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₅₃	#41	1	4	yes	—
55	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₄²×C₅₃	D₈	1	2	yes	—
57	C₄²×C₅₃	C₂×D₄	1	2	yes	—
58	C₄⋊Dic₅₃	—	2	4	no	—
59	C₄⋊Dic₅₃	—	4	2	no	—
60	C₄⋊Dic₅₃	—	4	8	no	—
61	#11719919	—	4	8	no	—
62	Dic₂₁₂	—	2	8	no	—
63	Dic₂₁₂	—	4	4	no	—
64	Dic₂₁₂	—	4	8	no	—
65	C₁₆×C₅₃	—	2	4	no	—
66	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	#124	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
70	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×Dic₅₃	—	4	4	no	—
74	C₂×Dic₅₃	—	4	4	no	—
75	C₂×Dic₅₃	—	8	4	no	—
76	C₂×Dic₅₃	—	8	4	no	—
77	Dic₁₀₆	—	4	4	no	—
78	Dic₁₀₆	—	4	4	no	—
79	Dic₁₀₆	—	8	4	no	—
80	C₈×C₅₃	#124	1	2	yes	—
81	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
82	C₈×C₅₃	C₂×#41	1	2	yes	—
83	C₂×#2245	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	#486	1	2	yes	—
85	C₂²×C₅₃	#510	1	4	yes	—
86	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
87	C₄×C₅₃	—	2	4	no	—
88	C₄×C₅₃	#486	1	2	yes	—
89	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
90	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
91	Dic₅₃	—	8	4	no	—
92	Dic₅₃	—	8	4	no	—
93	C₂×#499	—	53	1	no	—
94	C₂×#494	—	53	1	no	—
95	#2245	—	53	8	no	—
96	C₂×#358	—	53	4	no	—
97	C₂×#346	D₅₃	1	1	yes	characteristic
98	C₂×C₅₃	#2245	1	4	yes	complemented
99	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
100	C₂×C₅₃	C₂×#510	1	2	yes	—
101	C₂²×Q₁₆	—	53	2	no	—
102	C₂²×Q₁₆	—	106	1	no	—
103	C₂×C₄⋊Q₈	—	106	1	no	—
104	C₂×#125	—	106	1	no	—
105	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
106	#499	—	53	4	no	—
107	#499	—	106	2	no	—
108	#494	—	53	4	no	—
109	#494	—	106	2	no	—
110	C₂×#107	—	53	2	no	—
111	C₂×#106	—	106	1	no	—
112	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
113	#358	—	53	16	no	—
114	#346	C₂×D₅₃	1	4	yes	—
115	C₅₃	C₂×#2245	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	106	2	no	—
117	C₂×Q₁₆	—	53	4	no	—
118	C₂×Q₁₆	—	106	2	no	—
119	C₂×Q₁₆	—	106	4	no	—
120	C₂×Q₁₆	—	106	8	no	—
121	C₂×Q₁₆	—	212	4	no	—
122	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	106	4	no	—
125	C₄⋊Q₈	—	212	2	no	—
126	#125	—	106	4	no	—
127	#125	—	212	2	no	—
128	C₂×C₄⋊C₄	—	106	2	no	—
129	C₂×C₄⋊C₄	—	212	2	no	—
130	C₂×C₄⋊C₄	—	212	2	no	—
131	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
132	C₂×C₁₆	—	2	4	no	—
133	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
134	#107	—	53	4	no	—
135	#107	—	106	2	no	—
136	#106	—	106	4	no	—
137	C₄×C₈	D₂₁₂	1	2	yes	—
138	C₄×C₈	C₂²×D₅₃	1	2	yes	—
139	C₂×Q₈	—	106	4	no	—
140	C₂×Q₈	—	212	4	no	—
141	C₂×Q₈	—	212	4	no	—
142	C₂×Q₈	—	212	2	no	—
143	C₂²×C₄	—	212	2	no	—
144	C₂²×C₄	—	212	2	no	—
145	C₂²×C₄	D₄₂₄	1	2	yes	—
146	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
147	Q₁₆	—	106	8	no	—
148	Q₁₆	—	212	4	no	—
149	Q₁₆	—	212	8	no	—
150	C₂×C₈	#11719913	1	4	yes	—
151	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
152	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
153	C₂×C₈	#11719945	1	4	yes	—
154	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
155	C₄⋊C₄	—	106	4	no	—
156	C₄⋊C₄	—	212	2	no	—
157	C₄⋊C₄	—	212	8	no	—
158	C₄⋊C₄	—	212	8	no	—
159	C₄²	D₄₂₄	1	2	yes	—
160	C₄²	C₂×D₂₁₂	1	2	yes	—
161	C₁₆	—	2	4	no	—
162	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
163	Q₈	—	424	4	no	—
164	Q₈	—	212	4	no	—
165	Q₈	—	212	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	424	4	no	—
168	C₂×C₄	—	424	4	no	—
169	C₂×C₄	—	212	4	no	—
170	C₂×C₄	—	212	4	no	—
171	C₂×C₄	#13133205	1	2	yes	—
172	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
173	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
174	C₂×C₄	C₂×#11719913	1	2	yes	—
175	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
176	C₈	#13133205	1	2	yes	—
177	C₈	C₂×#11719913	1	4	yes	—
178	C₈	C₂×#11719945	1	2	yes	—
179	C₂²	#16810970	1	2	yes	—
180	C₂²	#16811066	1	4	yes	—
181	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	424	4	no	—
184	C₄	—	424	4	no	—
185	C₄	#16810970	1	2	yes	—
186	C₄	C₂×#13133205	1	1	yes	characteristic
187	C₄	C₂²×#11719913	1	1	yes	characteristic
188	C₂	#20276961	1	4	yes	direct product factor
189	C₂	C₂×#16810970	1	1	yes	characteristic
190	C₂	C₂×#16811066	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811025

C₂

yes

maximal, characteristic

C₂×#16811003

C₂

yes

maximal, characteristic

#20276961

C₂

yes

maximal, direct product factor

C₂×#16810428

C₂

yes

maximal

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

—

C₂²×Dic₂₁₂

C₂²

yes

—

—

—

—

C₂²

yes

—

#16811003

—

#16811003

C₂²

yes

—

C₂×#13133151

C₂²

yes

—

C₂×#13133149

—

#16810428

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

C₂²

yes

—

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂³

yes

—

—

—

—

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×#11719919

—

#13133151

—

#13133151

C₂³

yes

—

#13133149

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂³

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

C₂⁴

yes

characteristic

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

#124

yes

—

C₈×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

C₂×#41

yes

—

C₂×#2245

—

maximal, Sylow

C₂²×C₅₃

#486

yes

—

C₂²×C₅₃

#510

yes

—

C₂²×C₅₃

C₂²×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#486

yes

—

C₄×C₅₃

C₂×#124

yes

characteristic

C₄×C₅₃

C₂²×SD₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

—

—

—

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

100

C₂×C₅₃

C₂×#510

yes

—

101

C₂²×Q₁₆

—

102

C₂²×Q₁₆

—

106

—

103

C₂×C₄⋊Q₈

—

106

—

104

C₂×#125

—

106

—

105

C₂²×C₁₆

C₂×D₅₃

yes

—

106

#499

—

107

#499

—

106

—

108

#494

—

109

#494

—

106

—

110

C₂×#107

—

111

C₂×#106

—

106

—

112

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

113

#358

—

114

#346

C₂×D₅₃

yes

—

115

C₅₃

C₂×#2245

yes

characteristic, complemented, Sylow

116

C₂²×Q₈

—

106

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

106

—

119

C₂×Q₁₆

—

106

—

120

C₂×Q₁₆

—

106

—

121

C₂×Q₁₆

—

212

—

122

C₂²×C₈

D₂₁₂

yes

characteristic

123

C₂²×C₈

C₂²×D₅₃

yes

characteristic

124

C₄⋊Q₈

—

106

—

125

C₄⋊Q₈

—

212

—

126

#125

—

106

—

127

#125

—

212

—

128

C₂×C₄⋊C₄

—

106

—

129

C₂×C₄⋊C₄

—

212

—

130

C₂×C₄⋊C₄

—

212

—

131

C₂×C₄²

D₂₁₂

yes

characteristic

132

C₂×C₁₆

—

133

C₂×C₁₆

C₂²×D₅₃

yes

—

134

#107

—

135

#107

—

106

—

136

#106

—

106

—

137

C₄×C₈

D₂₁₂

yes

—

138

C₄×C₈

C₂²×D₅₃

yes

—

139

C₂×Q₈

—

106

—

140

C₂×Q₈

—

212

—

141

C₂×Q₈

—

212

—

142

C₂×Q₈

—

212

—

143

C₂²×C₄

—

212

—

144

C₂²×C₄

—

212

—

145

C₂²×C₄

D₄₂₄

yes

—

146

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

147

Q₁₆

—

106

—

148

Q₁₆

—

212

—

149

Q₁₆

—

212

—

150

C₂×C₈

#11719913

yes

—

151

C₂×C₈

C₂×D₂₁₂

yes

characteristic

152

C₂×C₈

C₂×D₂₁₂

yes

—

153

C₂×C₈

#11719945

yes

—

154

C₂×C₈

C₂³×D₅₃

yes

characteristic

155

C₄⋊C₄

—

106

—

156

C₄⋊C₄

—

212

—

157

C₄⋊C₄

—

212

—

158

C₄⋊C₄

—

212

—

159

C₄²

D₄₂₄

yes

—

160

C₄²

C₂×D₂₁₂

yes

—

161

C₁₆

—

162

C₂³

C₂×D₄₂₄

yes

characteristic

163

Q₈

—

424

—

164

Q₈

—

212

—

165

Q₈

—

212

—

166

C₂×C₄

—

167

C₂×C₄

—

424

—

168

C₂×C₄

—

424

—

169

C₂×C₄

—

212

—

170

C₂×C₄

—

212

—

171

C₂×C₄

#13133205

yes

—

172

C₂×C₄

C₂×D₄₂₄

yes

—

173

C₂×C₄

C₂×D₄₂₄

yes

characteristic

174

C₂×C₄

C₂×#11719913

yes

—

175

C₂×C₄

C₂²×D₂₁₂

yes

characteristic

176

C₈

#13133205

yes

—

177

C₈

C₂×#11719913

yes

—

178

C₈

C₂×#11719945

yes

—

179

C₂²

#16810970

yes

—

180

C₂²

#16811066

yes

—

181

C₂²

C₂²×D₄₂₄

yes

characteristic

182

C₄

—

183

C₄

—

424

—

184

C₄

—

424

—

185

C₄

#16810970

yes

—

186

C₄

C₂×#13133205

yes

characteristic

187

C₄

C₂²×#11719913

yes

characteristic

188

C₂

#20276961

yes

direct product factor

189

C₂

C₂×#16810970

yes

characteristic

190

C₂

C₂×#16811066

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	C₂×#2245
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₄₂₄

commutator, cosocle

C₂×C₈×C₅₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₃

Fitting

C₂×C₅₃×#346

C₂

socle, prime radical

C₂³×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
3. #20276961	C₂	direct (N × H)
98. C₂×C₅₃	#2245	semidirect
115. C₅₃	C₂×#2245	semidirect
188. C₂	#20276961	direct (N × H)

H ≅ G/N

type

3. #20276961

C₂

direct (N × H)

98. C₂×C₅₃

#2245

semidirect

115. C₅₃

C₂×#2245

semidirect

188. C₂

#20276961

direct (N × H)

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

subgroup	index
C₂×#16811025	2
C₂×#16811003	2
#20276961	2
C₂×#16810428	2
C₂×C₅₃×#346	2
C₂×#2245	53

subgroup

index

C₂×C₅₃×#346

Subgroups — C₂×#20276961 #27539410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20276961 #27539410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276961 #27539410