Group #20273128 — subgroups

Subgroups — #20273128

Order 6784.

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

normal subgroups only

This group has 234 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×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈⋊#88322	C₂	1	2	yes	maximal, complemented
4	D₄⋊#88322	C₂	1	2	yes	maximal, complemented
5	C₅₃×#529	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	—	2	1	no	—
7	C₂²×#88322	—	2	1	no	—
8	C₄⋊#88322	—	2	1	no	—
9	C₄⋊#88322	C₂²	1	2	yes	—
10	#13133128	—	2	1	no	—
11	C₄×#88322	C₂²	1	2	yes	—
12	#126×C₅₃	—	2	1	no	—
13	#125×C₅₃	C₄	1	1	yes	characteristic
14	#124×C₅₃	C₄	1	1	yes	characteristic
15	#123×C₅₃	—	2	1	no	—
16	C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
17	#121×C₅₃	C₄	1	1	yes	characteristic
18	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₅₃	—	2	1	no	—
20	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	#117×C₅₃	—	2	1	no	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	—	2	1	no	—
27	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²⋊C₄×C₅₃	—	2	1	no	—
37	C₂²⋊C₄×C₅₃	—	2	1	no	—
38	C₂²⋊C₄×C₅₃	—	2	1	no	—
39	C₂²⋊C₄×C₅₃	—	2	1	no	—
40	C₂²⋊C₄×C₅₃	—	2	1	no	—
41	C₄²×C₅₃	—	2	1	no	—
42	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	#11719917	—	2	4	no	—
47	C₂×#88322	—	2	2	no	—
48	C₂×#88322	—	2	2	no	—
49	C₂×#88322	—	4	2	no	—
50	C₂³×C₅₃	—	2	1	no	—
51	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	Q₈×C₅₃	—	2	2	no	—
53	D₄×C₅₃	—	2	2	no	—
54	D₄×C₅₃	—	4	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	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	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	4	1	no	—
63	C₂×C₄×C₅₃	—	4	1	no	—
64	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	#88322	—	2	4	no	—
71	#88322	—	4	2	no	—
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	—	4	1	no	—
75	C₂²×C₅₃	—	4	1	no	—
76	C₂²×C₅₃	—	4	1	no	—
77	C₂²×C₅₃	#104	1	2	yes	—
78	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	4	1	no	—
84	C₄×C₅₃	—	4	1	no	—
85	C₄×C₅₃	—	4	1	no	—
86	C₄×C₅₃	#131	1	2	yes	—
87	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₄×C₅₃	#137	1	1	yes	characteristic
89	#1483	—	53	1	no	maximal, Sylow
90	C₂×C₅₃	—	2	2	no	—
91	C₂×C₅₃	—	8	1	no	—
92	C₂×C₅₃	#408	1	1	yes	characteristic
93	C₂×C₅₃	#417	1	1	yes	characteristic
94	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
95	#529	D₅₃	1	1	yes	characteristic
96	C₂×C₄⋊C₈	—	53	1	no	—
97	C₄×M₁₆	—	53	1	no	—
98	Q₈⋊C₈	—	53	2	no	—
99	D₄⋊C₈	—	53	2	no	—
100	C₅₃	#1483	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	106	1	no	—
102	C₂²×C₈	—	106	1	no	—
103	#126	—	2	1	no	—
104	#125	Dic₅₃	1	1	yes	characteristic
105	#124	Dic₅₃	1	1	yes	characteristic
106	#123	—	2	1	no	—
107	C₂²⋊Q₈	Dic₅₃	1	1	yes	characteristic
108	#121	Dic₅₃	1	1	yes	characteristic
109	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
110	C₄×D₄	—	2	1	no	—
111	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
112	#117	—	2	1	no	—
113	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
114	C₄⋊C₈	—	53	2	no	—
115	C₄⋊C₈	—	106	1	no	—
116	#97	—	106	1	no	—
117	C₄×C₈	—	53	2	no	—
118	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
119	C₂×D₄	—	2	1	no	—
120	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
125	M₁₆	—	106	4	no	—
126	C₂×C₈	—	106	2	no	—
127	C₂×C₈	—	106	2	no	—
128	C₂×C₈	—	212	2	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
134	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
135	C₂²⋊C₄	—	2	1	no	—
136	C₂²⋊C₄	—	2	1	no	—
137	C₂²⋊C₄	—	2	1	no	—
138	C₂²⋊C₄	—	2	1	no	—
139	C₂²⋊C₄	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
142	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
143	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
144	C₄²	C₂²×D₅₃	1	1	yes	characteristic
145	C₂³	—	2	1	no	—
146	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
147	Q₈	—	2	2	no	—
148	D₄	—	2	2	no	—
149	D₄	—	4	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
160	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
161	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
162	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
165	C₈	—	106	4	no	—
166	C₈	—	212	2	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	#13133144	1	2	yes	—
173	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	4	1	no	—
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	#13133224	1	2	yes	—
182	C₄	#13133242	1	1	yes	characteristic
183	C₄	#13133247	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	—	8	1	no	—
186	C₂	#16810590	1	1	yes	characteristic
187	C₂	#16810624	1	1	yes	characteristic
188	C₂	C₂×#13133144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic, complemented

C₄×#11719917

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, complemented

D₄⋊#88322

C₂

yes

maximal, complemented

C₅₃×#529

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

#13133128

—

C₄×#88322

C₂²

yes

—

#126×C₅₃

—

#125×C₅₃

C₄

yes

characteristic

#124×C₅₃

C₄

yes

characteristic

#123×C₅₃

—

C₂²⋊Q₈×C₅₃

C₄

yes

characteristic

#121×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₂²

yes

characteristic

#117×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×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×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#104

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

C₄×M₁₆

—

Q₈⋊C₈

—

D₄⋊C₈

—

100

C₅₃

#1483

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

106

—

102

C₂²×C₈

—

106

—

103

#126

—

104

#125

Dic₅₃

yes

characteristic

105

#124

Dic₅₃

yes

characteristic

106

#123

—

107

C₂²⋊Q₈

Dic₅₃

yes

characteristic

108

#121

Dic₅₃

yes

characteristic

109

C₄×Q₈

C₂×D₅₃

yes

characteristic

110

C₄×D₄

—

111

C₄×D₄

C₂×D₅₃

yes

characteristic

112

#117

—

113

C₂×C₄²

C₂×D₅₃

yes

characteristic

114

C₄⋊C₈

—

115

C₄⋊C₈

—

106

—

116

#97

—

106

—

117

C₄×C₈

—

118

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

119

C₂×D₄

—

120

C₂×D₄

C₂×Dic₅₃

yes

characteristic

121

C₂²×C₄

—

122

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

123

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

124

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

125

M₁₆

—

106

—

126

C₂×C₈

—

106

—

127

C₂×C₈

—

106

—

128

C₂×C₈

—

212

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

C₂×Dic₅₃

yes

characteristic

134

C₄⋊C₄

C₂×Dic₅₃

yes

characteristic

135

C₂²⋊C₄

—

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

C₂×Dic₅₃

yes

characteristic

142

C₄²

C₂²⋊D₅₃

yes

characteristic

143

C₄²

C₂²⋊D₅₃

yes

characteristic

144

C₄²

C₂²×D₅₃

yes

characteristic

145

C₂³

—

146

C₂³

C₂²⋊Dic₅₃

yes

characteristic

147

Q₈

—

148

D₄

—

149

D₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

160

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

161

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

162

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

164

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

165

C₈

—

106

—

166

C₈

—

212

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#13133144

yes

—

173

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

#13133224

yes

—

182

C₄

#13133242

yes

characteristic

183

C₄

#13133247

yes

characteristic

184

C₂

—

185

C₂

—

186

C₂

#16810590

yes

characteristic

187

C₂

#16810624

yes

characteristic

188

C₂

C₂×#13133144

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#529	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#529	D₅₃
hypocenter	C₅₃	#1483
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#529

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

N	H ≅ G/N	type
1. C₂×C₄⋊#88322	C₂	semidirect
2. C₄×#11719917	C₂	semidirect
3. Q₈⋊#88322	C₂	semidirect
4. D₄⋊#88322	C₂	semidirect
100. C₅₃	#1483	semidirect

H ≅ G/N

type

1. C₂×C₄⋊#88322

C₂

semidirect

2. C₄×#11719917

C₂

semidirect

3. Q₈⋊#88322

C₂

semidirect

4. D₄⋊#88322

C₂

semidirect

100. C₅₃

#1483

semidirect

Subgroups — #20273128

Conjugacy classes of subgroups

Complements