Group #20276261 — subgroups

Subgroups — #20276261

Order 6784.

238 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 85 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×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133148	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133141	C₂	1	1	yes	maximal, characteristic
4	#16810530	C₂	1	1	yes	maximal, characteristic
5	#16810488	C₂	1	1	yes	maximal, characteristic
6	#16810377	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	#13133151	—	2	2	no	—
13	#13133148	—	2	2	no	—
14	#13133141	—	2	2	no	—
15	#13133124	—	2	2	no	—
16	#13133124	—	2	2	no	—
17	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	#103×C₅₃	—	2	2	no	—
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×Q₈×C₅₃	—	2	1	no	—
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×Q₈×C₅₃	—	2	1	no	—
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×Q₈×C₅₃	Q₈	1	2	yes	—
27	C₂²×C₄×C₅₃	—	2	1	no	—
28	C₂²×C₄×C₅₃	—	2	1	no	—
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₃	—	2	1	no	—
31	C₂²×Dic₅₃	—	2	1	no	—
32	C₂²×Dic₅₃	D₄	1	2	yes	—
33	C₂×C₈×C₅₃	—	2	1	no	—
34	C₂×C₈×C₅₃	—	2	1	no	—
35	C₂×C₈×C₅₃	D₄	1	2	yes	—
36	C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₄⋊Dic₅₃	—	2	1	no	—
39	C₄⋊Dic₅₃	—	2	1	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	—	2	1	no	—
42	C₄×Dic₅₃	—	2	1	no	—
43	C₄×Dic₅₃	—	2	1	no	—
44	C₂×#88322	—	2	1	no	—
45	C₂×#88322	—	2	1	no	—
46	C₂×#88322	D₄	1	2	yes	—
47	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₅₃	—	2	2	no	—
49	Q₈×C₅₃	—	2	2	no	—
50	Q₈×C₅₃	—	4	1	no	—
51	Q₈×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	—	4	1	no	—
56	C₂×C₄×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂×Dic₅₃	—	2	2	no	—
64	C₂×Dic₅₃	—	2	2	no	—
65	C₂×Dic₅₃	—	2	2	no	—
66	C₂×Dic₅₃	—	4	1	no	—
67	C₂×Dic₅₃	—	4	1	no	—
68	C₂×Dic₅₃	—	4	2	no	—
69	C₂×Dic₅₃	—	4	2	no	—
70	C₈×C₅₃	—	2	2	no	—
71	#88322	—	2	2	no	—
72	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂²×C₅₃	#123	1	1	yes	characteristic
74	C₂²×C₅₃	#127	1	1	yes	characteristic
75	C₂²×C₅₃	#135	1	2	yes	—
76	C₂²×C₅₃	#137	1	2	yes	—
77	C₄×C₅₃	—	4	2	no	—
78	C₄×C₅₃	—	8	1	no	—
79	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₄×C₅₃	#123	1	1	yes	characteristic
82	C₄×C₅₃	#123	1	1	yes	characteristic
83	Dic₅₃	—	4	2	no	—
84	Dic₅₃	—	8	1	no	—
85	Dic₅₃	—	8	1	no	—
86	#2097	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	#399	1	1	yes	characteristic
88	C₂×C₅₃	#479	1	2	yes	—
89	C₂×C₅₃	#485	1	2	yes	—
90	C₂×C₅₃	#495	1	1	yes	characteristic
91	C₂×C₅₃	#497	1	1	yes	characteristic
92	C₂×#107	—	53	1	no	—
93	C₂×#106	—	53	1	no	—
94	C₂×#103	—	53	1	no	—
95	C₂×#103	D₅₃	1	1	yes	characteristic
96	#391	—	53	1	no	—
97	#382	—	53	1	no	—
98	#336	—	53	1	no	—
99	C₅₃	#2097	1	1	yes	characteristic, complemented, Sylow
100	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
101	C₂²×C₈	—	53	1	no	—
102	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	53	1	no	—
104	C₂×C₄⋊C₄	—	53	1	no	—
105	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
106	C₂×C₄²	—	53	1	no	—
107	#107	—	106	2	no	—
108	#106	—	106	2	no	—
109	#103	—	2	2	no	—
110	#103	—	106	2	no	—
111	#95	—	106	2	no	—
112	#95	—	106	2	no	—
113	C₂×Q₈	—	2	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	Dic₁₀₆	1	2	yes	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	53	2	no	—
122	C₂²×C₄	—	106	1	no	—
123	C₂²×C₄	—	106	1	no	—
124	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	53	2	no	—
128	C₂×C₈	—	106	1	no	—
129	C₂×C₈	—	106	1	no	—
130	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	106	1	no	—
134	C₄⋊C₄	—	106	1	no	—
135	C₄⋊C₄	—	106	1	no	—
136	C₄⋊C₄	—	106	1	no	—
137	C₄²	—	106	1	no	—
138	C₄²	—	106	1	no	—
139	C₂³	D₄×D₅₃	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	Q₈	—	2	2	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	4	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	106	2	no	—
150	C₂×C₄	—	106	2	no	—
151	C₂×C₄	—	106	2	no	—
152	C₂×C₄	—	212	1	no	—
153	C₂×C₄	—	212	1	no	—
154	C₂×C₄	—	212	2	no	—
155	C₂×C₄	—	212	2	no	—
156	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
157	C₂×C₄	#11719945	1	1	yes	characteristic
158	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
159	C₂×C₄	#11719947	1	1	yes	characteristic
160	C₂×C₄	#11719947	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	106	2	no	—
164	C₂²	#13133194	1	1	yes	characteristic
165	C₂²	#13133199	1	1	yes	characteristic
166	C₂²	#13133213	1	1	yes	characteristic
167	C₂²	#13133240	1	2	yes	—
168	C₂²	#13133246	1	2	yes	—
169	C₄	—	4	2	no	—
170	C₄	—	8	1	no	—
171	C₄	—	424	1	no	—
172	C₄	—	424	1	no	—
173	C₄	—	212	2	no	—
174	C₄	#13133194	1	1	yes	characteristic
175	C₄	#13133196	1	1	yes	characteristic
176	C₄	#13133198	1	1	yes	characteristic
177	C₄	#13133199	1	1	yes	characteristic
178	C₂	#16810559	1	1	yes	characteristic
179	C₂	#16810932	1	2	yes	—
180	C₂	#16810963	1	2	yes	—
181	C₂	#16811008	1	1	yes	characteristic
182	C₂	#16811014	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×#13133148

C₂

yes

maximal, characteristic

C₂×#13133141

C₂

yes

maximal, characteristic

#16810530

C₂

yes

maximal, characteristic

#16810488

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal, characteristic

C₂×#103×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×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×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#485

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

D₅₃

yes

—

—

—

C₅₃

#2097

yes

characteristic, complemented, Sylow

100

C₂²×Q₈

C₂×D₅₃

yes

characteristic

101

C₂²×C₈

—

102

C₂²×C₈

C₂×D₅₃

yes

characteristic

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

106

C₂×C₄²

—

107

#107

—

106

—

108

#106

—

106

—

109

#103

—

110

#103

—

106

—

111

#95

—

106

—

112

#95

—

106

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

Dic₁₀₆

yes

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

—

106

—

124

C₂²×C₄

C₂²×D₅₃

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

106

—

129

C₂×C₈

—

106

—

130

C₂×C₈

C₂²⋊D₅₃

yes

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

—

106

—

135

C₄⋊C₄

—

106

—

136

C₄⋊C₄

—

106

—

137

C₄²

—

106

—

138

C₄²

—

106

—

139

C₂³

D₄×D₅₃

yes

characteristic

140

Q₈

—

141

Q₈

—

142

Q₈

—

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

106

—

151

C₂×C₄

—

106

—

152

C₂×C₄

—

212

—

153

C₂×C₄

—

212

—

154

C₂×C₄

—

212

—

155

C₂×C₄

—

212

—

156

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

157

C₂×C₄

#11719945

yes

characteristic

158

C₂×C₄

D₄×D₅₃

yes

characteristic

159

C₂×C₄

#11719947

yes

characteristic

160

C₂×C₄

#11719947

yes

characteristic

161

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

162

C₈

—

163

C₈

—

106

—

164

C₂²

#13133194

yes

characteristic

165

C₂²

#13133199

yes

characteristic

166

C₂²

#13133213

yes

characteristic

167

C₂²

#13133240

yes

—

168

C₂²

#13133246

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

424

—

172

C₄

—

424

—

173

C₄

—

212

—

174

C₄

#13133194

yes

characteristic

175

C₄

#13133196

yes

characteristic

176

C₄

#13133198

yes

characteristic

177

C₄

#13133199

yes

characteristic

178

C₂

#16810559

yes

characteristic

179

C₂

#16810932

yes

—

180

C₂

#16810963

yes

—

181

C₂

#16811008

yes

characteristic

182

C₂

#16811014

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#103×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#103	D₅₃
hypocenter	C₅₃	#2097
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#103×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
99. C₅₃	#2097	semidirect

H ≅ G/N

type

99. C₅₃

#2097

semidirect

Subgroups — #20276261

Conjugacy classes of subgroups

Complements