Group #20274986 — subgroups

Subgroups — #20274986

Order 6784.

298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 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×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133149	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133142	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133142	C₂	1	1	yes	maximal, characteristic
5	#16810521	C₂	1	1	yes	maximal, characteristic
6	#16810386	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₄×Dic₁₀₆	—	2	2	no	—
10	C₄×Dic₁₀₆	—	2	2	no	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×#11719919	—	2	1	no	—
15	C₂×#11719919	—	2	1	no	—
16	C₂×C₄×Dic₅₃	—	2	1	no	—
17	#13133149	—	2	2	no	—
18	#13133142	—	2	2	no	—
19	#13133142	C₄	1	4	yes	complemented
20	#13133124	—	2	1	no	—
21	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₃	—	2	1	no	—
28	C₂²×Dic₅₃	—	2	1	no	—
29	C₂²×Dic₅₃	—	2	1	no	—
30	C₂²×Dic₅₃	—	2	1	no	—
31	C₂×Dic₁₀₆	—	2	1	no	—
32	C₂×Dic₁₀₆	—	2	1	no	—
33	C₂×Dic₁₀₆	—	2	1	no	—
34	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
35	C₂×Dic₁₀₆	D₄	1	2	yes	—
36	C₂×Dic₁₀₆	Q₈	1	2	yes	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄²×C₅₃	—	2	1	no	—
44	C₄²×C₅₃	—	2	1	no	—
45	C₄⋊Dic₅₃	—	2	1	no	—
46	C₄⋊Dic₅₃	—	2	1	no	—
47	C₄⋊Dic₅₃	—	2	1	no	—
48	C₄⋊Dic₅₃	—	2	1	no	—
49	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₅₃	D₄	1	2	yes	—
51	#11719919	—	2	4	no	—
52	#11719919	—	4	2	no	—
53	C₄×Dic₅₃	—	2	4	no	—
54	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
61	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	—	2	2	no	—
72	C₂×Dic₅₃	—	4	1	no	—
73	C₂×Dic₅₃	—	4	1	no	—
74	C₂×Dic₅₃	—	4	1	no	—
75	C₂×Dic₅₃	—	4	1	no	—
76	C₂×Dic₅₃	—	4	1	no	—
77	C₂×Dic₅₃	—	4	1	no	—
78	C₂×Dic₅₃	—	4	1	no	—
79	Dic₁₀₆	—	2	2	no	—
80	Dic₁₀₆	—	2	2	no	—
81	Dic₁₀₆	—	2	2	no	—
82	Dic₁₀₆	C₄⋊C₄	1	4	yes	complemented
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	4	1	no	—
85	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₅₃	#120	1	1	yes	characteristic
87	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂²×C₅₃	#135	1	1	yes	characteristic
90	C₂²×C₅₃	#137	1	1	yes	characteristic
91	C₂²×C₅₃	#137	1	1	yes	characteristic
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	4	1	no	—
94	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₄×C₅₃	#123	1	1	yes	characteristic
98	Dic₅₃	—	2	4	no	—
99	Dic₅₃	—	4	2	no	—
100	Dic₅₃	—	4	2	no	—
101	Dic₅₃	—	8	1	no	—
102	#1865	—	53	1	no	maximal, Sylow
103	C₂×C₅₃	#385	1	1	yes	characteristic
104	C₂×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
105	C₂×C₅₃	#441	1	1	yes	characteristic
106	C₂×C₅₃	#448	1	1	yes	characteristic
107	C₂×C₅₃	#452	1	1	yes	characteristic
108	C₂×C₅₃	#475	1	1	yes	characteristic
109	C₂×C₅₃	#479	1	1	yes	characteristic
110	C₂×C₄×Q₈	—	53	1	no	—
111	C₂×#106	—	53	1	no	—
112	C₂×#103	—	53	1	no	—
113	C₂×#103	—	53	1	no	—
114	#389	—	53	1	no	—
115	#340	—	53	1	no	—
116	#336	D₅₃	1	1	yes	characteristic
117	C₅₃	#1865	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	53	1	no	—
119	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
121	C₄×Q₈	—	106	2	no	—
122	C₄×Q₈	—	106	2	no	—
123	C₂×C₄⋊C₄	—	53	1	no	—
124	C₂×C₄⋊C₄	—	53	1	no	—
125	C₂×C₄⋊C₄	—	53	1	no	—
126	C₂×C₄⋊C₄	—	106	1	no	—
127	C₂×C₄⋊C₄	—	106	1	no	—
128	C₂×C₄²	—	106	1	no	—
129	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
130	#106	—	106	2	no	—
131	#103	—	53	4	no	—
132	#103	—	106	2	no	—
133	#95	—	106	1	no	—
134	C₂×Q₈	—	53	2	no	—
135	C₂×Q₈	—	53	2	no	—
136	C₂×Q₈	—	53	2	no	—
137	C₂×Q₈	—	106	1	no	—
138	C₂×Q₈	—	106	1	no	—
139	C₂×Q₈	—	106	1	no	—
140	C₂²×C₄	—	106	1	no	—
141	C₂²×C₄	—	106	1	no	—
142	C₂²×C₄	—	106	1	no	—
143	C₂²×C₄	—	106	1	no	—
144	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
145	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
146	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	C₄×D₅₃	1	2	yes	—
153	C₄⋊C₄	—	53	2	no	—
154	C₄⋊C₄	—	53	2	no	—
155	C₄⋊C₄	—	106	4	no	—
156	C₄⋊C₄	—	106	1	no	—
157	C₄⋊C₄	—	106	1	no	—
158	C₄⋊C₄	—	106	1	no	—
159	C₄⋊C₄	—	106	1	no	—
160	C₄⋊C₄	—	212	2	no	—
161	C₄²	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	106	4	no	—
164	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
165	Q₈	—	53	4	no	—
166	Q₈	—	106	2	no	—
167	Q₈	—	106	2	no	—
168	Q₈	—	106	2	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	106	2	no	—
175	C₂×C₄	—	106	2	no	—
176	C₂×C₄	—	106	2	no	—
177	C₂×C₄	—	106	2	no	—
178	C₂×C₄	—	106	2	no	—
179	C₂×C₄	—	212	1	no	—
180	C₂×C₄	—	212	1	no	—
181	C₂×C₄	—	212	1	no	—
182	C₂×C₄	—	212	1	no	—
183	C₂×C₄	—	212	1	no	—
184	C₂×C₄	—	212	1	no	—
185	C₂×C₄	—	212	1	no	—
186	C₂×C₄	#11719913	1	2	yes	—
187	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₂×C₄	#11719945	1	1	yes	characteristic
189	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
190	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
191	C₂×C₄	#11719947	1	1	yes	characteristic
192	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	4	1	no	—
195	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
196	C₂²	#13133187	1	1	yes	characteristic
197	C₂²	#13133197	1	1	yes	characteristic
198	C₂²	C₂×#11719913	1	1	yes	characteristic
199	C₂²	#13133239	1	1	yes	characteristic
200	C₂²	#13133250	1	1	yes	characteristic
201	C₂²	#13133250	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	4	1	no	—
204	C₄	—	106	4	no	—
205	C₄	—	424	1	no	—
206	C₄	—	212	2	no	—
207	C₄	—	212	2	no	—
208	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
209	C₄	#13133177	1	1	yes	characteristic
210	C₄	#13133195	1	1	yes	characteristic
211	C₄	#13133199	1	1	yes	characteristic
212	C₂	#16810497	1	1	yes	characteristic
213	C₂	C₄×#11719913	1	1	yes	characteristic
214	C₂	#16810731	1	1	yes	characteristic
215	C₂	#16810764	1	1	yes	characteristic
216	C₂	#16810787	1	1	yes	characteristic
217	C₂	#16810909	1	1	yes	characteristic
218	C₂	#16810931	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

C₂×#13133149

C₂

yes

maximal, characteristic

C₂×#13133142

C₂

yes

maximal, characteristic

C₂×#13133142

C₂

yes

maximal, characteristic

#16810521

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal, characteristic

C₅₃×#336

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719919

—

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

#13133149

—

#13133142

—

#13133142

C₄

yes

complemented

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×Dic₁₀₆

D₄

yes

—

C₂×Dic₁₀₆

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₄⋊C₄

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#123

yes

characteristic

Dic₅₃

—

Dic₅₃

—

100

Dic₅₃

—

101

Dic₅₃

—

102

#1865

—

maximal, Sylow

103

C₂×C₅₃

#385

yes

characteristic

104

C₂×C₅₃

C₄×SD₁₆

yes

characteristic

105

C₂×C₅₃

#441

yes

characteristic

106

C₂×C₅₃

#448

yes

characteristic

107

C₂×C₅₃

#452

yes

characteristic

108

C₂×C₅₃

#475

yes

characteristic

109

C₂×C₅₃

#479

yes

characteristic

110

C₂×C₄×Q₈

—

111

C₂×#106

—

112

C₂×#103

—

113

C₂×#103

—

114

#389

—

115

#340

—

116

#336

D₅₃

yes

characteristic

117

C₅₃

#1865

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

—

119

C₂²×C₈

C₂×D₅₃

yes

characteristic

120

C₂²×C₈

C₂×D₅₃

yes

characteristic

121

C₄×Q₈

—

106

—

122

C₄×Q₈

—

106

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

106

—

127

C₂×C₄⋊C₄

—

106

—

128

C₂×C₄²

—

106

—

129

C₂×C₄²

C₂×D₅₃

yes

characteristic

130

#106

—

106

—

131

#103

—

132

#103

—

106

—

133

#95

—

106

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

106

—

138

C₂×Q₈

—

106

—

139

C₂×Q₈

—

106

—

140

C₂²×C₄

—

106

—

141

C₂²×C₄

—

106

—

142

C₂²×C₄

—

106

—

143

C₂²×C₄

—

106

—

144

C₂²×C₄

D₂₁₂

yes

characteristic

145

C₂²×C₄

D₂₁₂

yes

characteristic

146

C₂²×C₄

C₂²×D₅₃

yes

characteristic

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

C₄×D₅₃

yes

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

106

—

156

C₄⋊C₄

—

106

—

157

C₄⋊C₄

—

106

—

158

C₄⋊C₄

—

106

—

159

C₄⋊C₄

—

106

—

160

C₄⋊C₄

—

212

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

106

—

164

C₂³

C₂×D₂₁₂

yes

characteristic

165

Q₈

—

166

Q₈

—

106

—

167

Q₈

—

106

—

168

Q₈

—

106

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

106

—

175

C₂×C₄

—

106

—

176

C₂×C₄

—

106

—

177

C₂×C₄

—

106

—

178

C₂×C₄

—

106

—

179

C₂×C₄

—

212

—

180

C₂×C₄

—

212

—

181

C₂×C₄

—

212

—

182

C₂×C₄

—

212

—

183

C₂×C₄

—

212

—

184

C₂×C₄

—

212

—

185

C₂×C₄

—

212

—

186

C₂×C₄

#11719913

yes

—

187

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

188

C₂×C₄

#11719945

yes

characteristic

189

C₂×C₄

D₄×D₅₃

yes

characteristic

190

C₂×C₄

D₄×D₅₃

yes

characteristic

191

C₂×C₄

#11719947

yes

characteristic

192

C₂×C₄

Q₈×D₅₃

yes

characteristic

193

C₈

—

194

C₈

—

195

C₂²

C₄×D₂₁₂

yes

characteristic

196

C₂²

#13133187

yes

characteristic

197

C₂²

#13133197

yes

characteristic

198

C₂²

C₂×#11719913

yes

characteristic

199

C₂²

#13133239

yes

characteristic

200

C₂²

#13133250

yes

characteristic

201

C₂²

#13133250

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

106

—

205

C₄

—

424

—

206

C₄

—

212

—

207

C₄

—

212

—

208

C₄

C₄⋊C₄×D₅₃

yes

characteristic

209

C₄

#13133177

yes

characteristic

210

C₄

#13133195

yes

characteristic

211

C₄

#13133199

yes

characteristic

212

C₂

#16810497

yes

characteristic

213

C₂

C₄×#11719913

yes

characteristic

214

C₂

#16810731

yes

characteristic

215

C₂

#16810764

yes

characteristic

216

C₂

#16810787

yes

characteristic

217

C₂

#16810909

yes

characteristic

218

C₂

#16810931

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#336	D₅₃
hypocenter	C₅₃	#1865
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#336

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
19. #13133142	C₄	semidirect
82. Dic₁₀₆	C₄⋊C₄	semidirect
117. C₅₃	#1865	semidirect

H ≅ G/N

type

19. #13133142

C₄

semidirect

82. Dic₁₀₆

C₄⋊C₄

semidirect

117. C₅₃

#1865

semidirect

Subgroups — #20274986

Conjugacy classes of subgroups

Complements