Group #27277048 — subgroups

Subgroups — #27277048

Order 13568.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
2	#20273476	C₂	1	2	yes	maximal
3	#20273059	C₂	1	1	yes	maximal, characteristic
4	#20272584	C₂	1	2	yes	maximal
5	C₅₃×#1467	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88322	—	2	2	no	—
7	Q₈×#88322	—	2	2	no	—
8	#16810680	C₂²	1	2	yes	—
9	C₂×C₄⋊#88322	C₄	1	2	yes	—
10	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810375	—	2	2	no	—
14	#16810373	—	2	2	no	—
15	#16810349	—	2	2	no	—
16	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#432	C₂²	1	2	yes	—
18	C₅₃×#324	—	2	2	no	—
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂×C₄	1	2	yes	—
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₂²⋊#88322	—	2	2	no	—
26	#13133128	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	—	2	2	no	—
29	C₄×#88322	D₄	1	2	yes	—
30	C₄×#88322	D₄	1	2	yes	—
31	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₄×Q₈×C₅₃	—	2	2	no	—
33	C₄×Q₈×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₂²⋊C₈×C₅₃	—	2	2	no	—
40	#97×C₅₃	—	2	2	no	—
41	C₂×Q₈×C₅₃	—	2	4	no	—
42	C₂×Q₈×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₄⋊C₄×C₅₃	—	2	4	no	—
50	C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄²×C₅₃	—	2	2	no	—
53	C₄²×C₅₃	C₂×D₄	1	2	yes	—
54	C₄²×C₅₃	#41	1	2	yes	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	4	1	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×#88322	C₂²⋊C₄	1	2	yes	—
62	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
63	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	Q₈×C₅₃	—	2	8	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	4	1	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	#108	1	4	yes	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#121	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₅₃	—	8	2	no	—
79	#88322	—	2	1	no	—
80	#88322	—	2	2	no	—
81	#88322	—	4	2	no	—
82	#88322	—	8	2	no	—
83	#88322	#101	1	2	yes	—
84	#8623	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	#380	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×#108	1	2	yes	—
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	—	2	4	no	—
91	C₄×C₅₃	—	2	2	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	4	2	no	—
94	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
95	C₄×C₅₃	#413	1	1	yes	characteristic
96	C₂×C₈×Q₈	—	53	1	no	—
97	#1568	—	53	2	no	—
98	#1467	—	53	1	no	—
99	#1467	D₅₃	1	1	yes	characteristic
100	#1290	—	53	2	no	—
101	C₂×C₅₃	—	2	2	no	—
102	C₂×C₅₃	#1801	1	1	yes	characteristic
103	C₂×C₅₃	#1810	1	1	yes	characteristic
104	C₂×C₅₃	#1812	1	1	yes	characteristic
105	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
106	C₈×Q₈	—	106	2	no	—
107	C₈×Q₈	—	106	2	no	—
108	#432	—	53	2	no	—
109	#432	C₂×D₅₃	1	2	yes	—
110	C₂×C₄⋊C₈	—	53	2	no	—
111	C₂×C₄⋊C₈	—	53	1	no	—
112	C₂×C₄×C₈	—	53	1	no	—
113	C₂×C₄×C₈	—	53	2	no	—
114	#335	—	106	2	no	—
115	#334	—	106	2	no	—
116	#324	—	2	2	no	—
117	#324	—	106	2	no	—
118	C₅₃	#8623	1	1	yes	characteristic, complemented, Sylow
119	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
120	C₂²×C₈	—	53	1	no	—
121	C₂²×C₈	—	53	1	no	—
122	C₂²×C₈	—	53	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
127	C₂×C₄²	C₄×D₅₃	1	2	yes	—
128	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	106	4	no	—
131	C₄⋊C₈	—	106	2	no	—
132	C₄⋊C₈	—	106	2	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₂²⋊C₈	—	106	2	no	—
135	#97	—	2	2	no	—
136	#97	—	106	2	no	—
137	C₄×C₈	—	53	2	no	—
138	C₄×C₈	—	53	2	no	—
139	C₄×C₈	—	106	2	no	—
140	C₄×C₈	—	106	2	no	—
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂²×C₄	#11719919	1	2	yes	—
144	C₂²×C₄	#11719919	1	2	yes	—
145	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	53	1	no	—
150	C₂×C₈	—	53	1	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	106	2	no	—
153	C₂×C₈	—	106	2	no	—
154	C₂×C₈	—	212	1	no	—
155	C₂×C₈	—	212	2	no	—
156	C₂×C₈	—	212	2	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	D₄×D₅₃	1	2	yes	—
162	C₄²	#11719947	1	2	yes	—
163	C₂³	C₂×#11719919	1	1	yes	characteristic
164	Q₈	—	2	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	#13133152	1	4	yes	—
174	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₄	#13133173	1	1	yes	characteristic
176	C₂×C₄	#13133189	1	2	yes	—
177	C₂×C₄	#13133194	1	2	yes	—
178	C₈	—	8	2	no	—
179	C₈	—	53	2	no	—
180	C₈	—	106	1	no	—
181	C₈	—	106	2	no	—
182	C₈	—	424	2	no	—
183	C₈	—	212	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	#16810481	1	1	yes	characteristic
187	C₂²	C₂×#13133152	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	#101×D₅₃	1	1	yes	characteristic
194	C₄	#16810606	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	#20274709	1	1	yes	characteristic
197	C₂	#20274751	1	1	yes	characteristic
198	C₂	#20274759	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×#88322

C₂

yes

maximal, characteristic

#20273476

C₂

yes

maximal

#20273059

C₂

yes

maximal, characteristic

#20272584

C₂

yes

maximal

C₅₃×#1467

C₂

yes

maximal, characteristic

Q₈×#88322

—

Q₈×#88322

—

#16810680

C₂²

yes

—

C₂×C₄⋊#88322

C₄

yes

—

C₂×C₄⋊#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

—

—

—

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×#432

C₂²

yes

—

C₅₃×#324

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

—

—

—

—

—

—

D₄

yes

—

C₄×#88322

D₄

yes

—

C₂²×Q₈×C₅₃

Q₈

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂²⋊C₄

yes

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

Q₈×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

#108

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

—

—

—

—

—

yes

—

#8623

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#380

yes

characteristic

C₂²×C₅₃

C₂×#108

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₈×Q₈

—

#1568

—

#1467

—

#1467

D₅₃

yes

characteristic

100

#1290

—

101

C₂×C₅₃

—

102

C₂×C₅₃

#1801

yes

characteristic

103

C₂×C₅₃

#1810

yes

characteristic

104

C₂×C₅₃

#1812

yes

characteristic

105

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

106

C₈×Q₈

—

106

—

107

C₈×Q₈

—

106

—

108

#432

—

109

#432

C₂×D₅₃

yes

—

110

C₂×C₄⋊C₈

—

111

C₂×C₄⋊C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

#335

—

106

—

115

#334

—

106

—

116

#324

—

117

#324

—

106

—

118

C₅₃

#8623

yes

characteristic, complemented, Sylow

119

C₂²×Q₈

Dic₁₀₆

yes

characteristic

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

C₂×C₄⋊C₄

Dic₁₀₆

yes

characteristic

126

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

127

C₂×C₄²

C₄×D₅₃

yes

—

128

C₂×C₄²

C₂²×D₅₃

yes

characteristic

129

C₄⋊C₈

—

130

C₄⋊C₈

—

106

—

131

C₄⋊C₈

—

106

—

132

C₄⋊C₈

—

106

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

106

—

135

#97

—

136

#97

—

106

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

106

—

140

C₄×C₈

—

106

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

#11719919

yes

—

144

C₂²×C₄

#11719919

yes

—

145

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

146

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

147

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

106

—

153

C₂×C₈

—

106

—

154

C₂×C₈

—

212

—

155

C₂×C₈

—

212

—

156

C₂×C₈

—

212

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

D₄×D₅₃

yes

—

162

C₄²

#11719947

yes

—

163

C₂³

C₂×#11719919

yes

characteristic

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#13133152

yes

—

174

C₂×C₄

C₂²⋊C₄×D₅₃

yes

characteristic

175

C₂×C₄

#13133173

yes

characteristic

176

C₂×C₄

#13133189

yes

—

177

C₂×C₄

#13133194

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

106

—

181

C₈

—

106

—

182

C₈

—

424

—

183

C₈

—

212

—

184

C₂²

—

185

C₂²

—

186

C₂²

#16810481

yes

characteristic

187

C₂²

C₂×#13133152

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#101×D₅₃

yes

characteristic

194

C₄

#16810606

yes

characteristic

195

C₂

—

196

C₂

#20274709

yes

characteristic

197

C₂

#20274751

yes

characteristic

198

C₂

#20274759

yes

characteristic

	subgroup	G/H
center	C₂²	#16810481
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1467	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1467	D₅₃
hypocenter	C₅₃	#8623
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

subgroup

G/H

center

C₂²

#16810481

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1467

C₂

socle

C₂²×C₅₃

#380

prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

N	H ≅ G/N	type
118. C₅₃	#8623	semidirect

H ≅ G/N

type

118. C₅₃

#8623

semidirect

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

subgroup	index
C₂×Q₈×#88322	2
#20273476	2
#20273059	2
#20272584	2
C₅₃×#1467	2
#8623	53

subgroup

index

C₂×Q₈×#88322

C₅₃×#1467

Subgroups — #27277048

Conjugacy classes of subgroups

Complements

Maximal subgroups