Group #20286102 — subgroups

Subgroups — #20286102

Order 6784.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811180	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810915	C₂	1	2	yes	maximal, complemented
3	#16810903	C₂	1	2	yes	maximal, complemented
4	#16810867	C₂	1	4	yes	maximal, complemented
5	#16810828	C₂	1	2	yes	maximal, complemented
6	#16810820	C₂	1	2	yes	maximal, complemented
7	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
9	C₂×#11719923	C₂²	1	4	yes	—
10	C₂×#11719917	C₂²	1	2	yes	—
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133217	C₂²	1	1	yes	characteristic
13	#13133209	—	2	1	no	—
14	#13133196	—	2	1	no	—
15	#13133196	C₂²	1	2	yes	—
16	C₄×Dic₁₀₆	—	2	1	no	—
17	C₄×Dic₁₀₆	C₂²	1	2	yes	—
18	#13133176	—	2	1	no	—
19	#13133150	C₂²	1	4	yes	—
20	C₄⋊#88322	C₂²	1	2	yes	—
21	C₄⋊#88322	C₂²	1	2	yes	—
22	#13133140	C₂²	1	4	yes	—
23	#13133138	C₂²	1	4	yes	—
24	C₂×#41×C₅₃	—	2	1	no	—
25	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
26	#127×C₅₃	C₂²	1	1	yes	characteristic
27	#124×C₅₃	—	2	1	no	—
28	#121×C₅₃	—	2	1	no	—
29	#121×C₅₃	C₂²	1	2	yes	—
30	C₄×D₄×C₅₃	—	2	1	no	—
31	C₄×D₄×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	#41×C₅₃	—	4	4	no	—
34	C₂×Q₈×C₅₃	—	2	1	no	—
35	C₂×D₄×C₅₃	—	2	2	no	—
36	C₂×D₄×C₅₃	—	2	1	no	—
37	C₂×D₄×C₅₃	—	2	1	no	—
38	C₂×D₄×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₀₆	—	2	1	no	—
44	C₂×Dic₁₀₆	C₂³	1	2	yes	—
45	C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₅₃	—	2	2	no	—
48	C₂²⋊C₄×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²⋊Dic₅₃	—	2	2	no	—
54	#11719923	—	4	4	no	—
55	C₄⋊Dic₅₃	—	2	1	no	—
56	C₄⋊Dic₅₃	—	2	1	no	—
57	C₄⋊Dic₅₃	—	2	1	no	—
58	C₄⋊Dic₅₃	C₂³	1	2	yes	—
59	#11719919	—	2	2	no	—
60	#11719919	—	2	2	no	—
61	C₄×Dic₅₃	—	2	2	no	—
62	#11719917	—	4	2	no	—
63	C₂×#88322	C₂³	1	4	yes	—
64	C₂³×C₅₃	—	2	2	no	—
65	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₅₃	—	4	2	no	—
67	D₄×C₅₃	—	2	2	no	—
68	D₄×C₅₃	—	2	2	no	—
69	D₄×C₅₃	—	4	2	no	—
70	D₄×C₅₃	—	4	4	no	—
71	D₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	4	4	no	—
78	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
84	C₂×Dic₅₃	—	2	2	no	—
85	C₂×Dic₅₃	—	2	2	no	—
86	Dic₁₀₆	—	2	2	no	—
87	Dic₁₀₆	—	2	2	no	—
88	Dic₁₀₆	—	4	2	no	—
89	#88322	—	2	4	no	—
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	4	2	no	—
94	C₂²×C₅₃	—	4	2	no	—
95	C₂²×C₅₃	—	4	2	no	—
96	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	1	no	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	4	2	no	—
101	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₄×C₅₃	#137	1	2	yes	—
103	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
104	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
105	Dic₅₃	—	4	2	no	—
106	Dic₅₃	—	4	2	no	—
107	#3254	—	53	1	no	maximal, Sylow
108	C₂×C₅₃	—	4	1	no	—
109	C₂×C₅₃	—	8	2	no	—
110	C₂×C₅₃	#537	1	1	yes	characteristic
111	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
113	#533	—	53	1	no	—
114	#532	D₅₃	1	1	yes	characteristic
115	#477	—	53	2	no	—
116	#474	—	53	2	no	—
117	#468	—	53	4	no	—
118	#461	—	53	2	no	—
119	#460	—	53	2	no	—
120	#423	—	53	1	no	—
121	C₅₃	#3254	1	1	yes	characteristic, complemented, Sylow
122	C₂×#41	—	2	1	no	—
123	C₂×SD₁₆	—	53	4	no	—
124	C₂×M₁₆	—	53	2	no	—
125	C₄⋊Q₈	—	53	1	no	—
126	C₄⋊Q₈	—	53	1	no	—
127	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
128	#127	C₂×D₅₃	1	1	yes	characteristic
129	#125	—	106	1	no	—
130	#124	—	2	1	no	—
131	C₂²⋊Q₈	—	53	2	no	—
132	C₂²⋊Q₈	—	106	1	no	—
133	#121	—	2	1	no	—
134	#121	C₂×D₅₃	1	2	yes	—
135	C₄×Q₈	—	53	2	no	—
136	C₄×Q₈	—	106	1	no	—
137	C₄×D₄	—	2	1	no	—
138	C₄×D₄	C₂×D₅₃	1	2	yes	—
139	#117	—	106	1	no	—
140	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
141	#107	—	53	4	no	—
142	C₄⋊C₈	—	53	2	no	—
143	C₄⋊C₈	—	53	2	no	—
144	#103	—	53	4	no	—
145	#102	—	53	4	no	—
146	#41	—	4	4	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	53	2	no	—
149	C₂×Q₈	—	106	1	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	C₂²×D₅₃	1	2	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
158	SD₁₆	—	212	4	no	—
159	M₁₆	—	212	2	no	—
160	C₂×C₈	—	53	4	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	53	2	no	—
163	C₄⋊C₄	—	106	1	no	—
164	C₄⋊C₄	—	106	1	no	—
165	C₄⋊C₄	—	106	2	no	—
166	C₄⋊C₄	—	106	2	no	—
167	C₄⋊C₄	—	106	1	no	—
168	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	106	2	no	—
172	C₄²	—	106	2	no	—
173	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₄²	C₂²×D₅₃	1	1	yes	characteristic
176	C₄²	C₂²×D₅₃	1	1	yes	characteristic
177	C₂³	—	2	2	no	—
178	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
179	Q₈	—	4	2	no	—
180	Q₈	—	106	2	no	—
181	Q₈	—	106	2	no	—
182	Q₈	—	212	2	no	—
183	D₄	—	2	2	no	—
184	D₄	—	2	2	no	—
185	D₄	—	4	2	no	—
186	D₄	—	4	4	no	—
187	D₄	—	4	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	106	2	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
202	C₈	—	106	4	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	4	2	no	—
214	C₄	—	212	2	no	—
215	C₄	—	212	2	no	—
216	C₄	#13133243	1	2	yes	—
217	C₄	#13133247	1	2	yes	—
218	C₄	#13133263	1	1	yes	characteristic
219	C₄	#13133264	1	1	yes	characteristic
220	C₂	—	4	1	no	—
221	C₂	—	8	2	no	—
222	C₂	#16811201	1	1	yes	characteristic
223	C₂	C₂×#13133243	1	1	yes	characteristic
224	C₂	C₂×#13133247	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic, complemented

#16810915

C₂

yes

maximal, complemented

#16810903

C₂

yes

maximal, complemented

#16810867

C₂

yes

maximal, complemented

#16810828

C₂

yes

maximal, complemented

#16810820

C₂

yes

maximal, complemented

#16810647

C₂

yes

maximal, characteristic, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

C₂×#11719923

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

—

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133176

—

#13133150

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133140

C₂²

yes

—

#13133138

C₂²

yes

—

C₂×#41×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#124×C₅₃

—

#121×C₅₃

—

#121×C₅₃

C₂²

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

100

C₄×C₅₃

—

101

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

102

C₄×C₅₃

#137

yes

—

103

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

104

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

105

Dic₅₃

—

106

Dic₅₃

—

107

#3254

—

maximal, Sylow

108

C₂×C₅₃

—

109

C₂×C₅₃

—

110

C₂×C₅₃

#537

yes

characteristic

111

C₂×C₅₃

C₂×AGL(1,ℤ/8)

yes

characteristic

112

C₂×C₅₃

C₂×#137

yes

characteristic

113

#533

—

114

#532

D₅₃

yes

characteristic

115

#477

—

116

#474

—

117

#468

—

118

#461

—

119

#460

—

120

#423

—

121

C₅₃

#3254

yes

characteristic, complemented, Sylow

122

C₂×#41

—

123

C₂×SD₁₆

—

124

C₂×M₁₆

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

128

#127

C₂×D₅₃

yes

characteristic

129

#125

—

106

—

130

#124

—

131

C₂²⋊Q₈

—

132

C₂²⋊Q₈

—

106

—

133

#121

—

134

#121

C₂×D₅₃

yes

—

135

C₄×Q₈

—

136

C₄×Q₈

—

106

—

137

C₄×D₄

—

138

C₄×D₄

C₂×D₅₃

yes

—

139

#117

—

106

—

140

C₂×C₄²

C₂×D₅₃

yes

characteristic

141

#107

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

#103

—

145

#102

—

146

#41

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

106

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

C₂²×D₅₃

yes

—

154

C₂²×C₄

—

155

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

156

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

157

C₂²×C₄

C₂²×D₅₃

yes

characteristic

158

SD₁₆

—

212

—

159

M₁₆

—

212

—

160

C₂×C₈

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

106

—

164

C₄⋊C₄

—

106

—

165

C₄⋊C₄

—

106

—

166

C₄⋊C₄

—

106

—

167

C₄⋊C₄

—

106

—

168

C₄⋊C₄

C₂²×D₅₃

yes

—

169

C₂²⋊C₄

—

170

C₂²⋊C₄

—

171

C₂²⋊C₄

—

106

—

172

C₄²

—

106

—

173

C₄²

C₂²⋊D₅₃

yes

characteristic

174

C₄²

C₂²⋊D₅₃

yes

characteristic

175

C₄²

C₂²×D₅₃

yes

characteristic

176

C₄²

C₂²×D₅₃

yes

characteristic

177

C₂³

—

178

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

179

Q₈

—

180

Q₈

—

106

—

181

Q₈

—

106

—

182

Q₈

—

212

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

187

D₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

106

—

195

C₂×C₄

—

106

—

196

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

197

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

198

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

199

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

200

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

201

C₂×C₄

C₂³×D₅₃

yes

characteristic

202

C₈

—

106

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

212

—

215

C₄

—

212

—

216

C₄

#13133243

yes

—

217

C₄

#13133247

yes

—

218

C₄

#13133263

yes

characteristic

219

C₄

#13133264

yes

characteristic

220

C₂

—

221

C₂

—

222

C₂

#16811201

yes

characteristic

223

C₂

C₂×#13133243

yes

characteristic

224

C₂

C₂×#13133247

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#532	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#532	D₅₃
hypocenter	C₅₃	#3254
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#532

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16811180	C₂	semidirect
2. #16810915	C₂	semidirect
3. #16810903	C₂	semidirect
4. #16810867	C₂	semidirect
5. #16810828	C₂	semidirect
6. #16810820	C₂	semidirect
7. #16810647	C₂	semidirect
121. C₅₃	#3254	semidirect

H ≅ G/N

type

1. #16811180

C₂

semidirect

2. #16810915

C₂

semidirect

3. #16810903

C₂

semidirect

4. #16810867

C₂

semidirect

5. #16810828

C₂

semidirect

6. #16810820

C₂

semidirect

7. #16810647

C₂

semidirect

121. C₅₃

#3254

semidirect

Subgroups — #20286102

Conjugacy classes of subgroups

Complements