Group #27371799 — subgroups

Subgroups — #27371799

Order 13568.

706 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 187 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	#20285300	C₂	1	4	yes	maximal
2	#20285226	C₂	1	2	yes	maximal
3	#20274332	C₂	1	2	yes	maximal, complemented
4	#20274325	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274311	C₂	1	4	yes	maximal, complemented
6	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811178	C₂²	1	4	yes	—
9	#16811028	C₂²	1	2	yes	—
10	#16811026	C₂²	1	4	yes	—
11	#16811024	—	2	4	no	—
12	#16811019	C₂²	1	4	yes	—
13	#16811002	C₂²	1	4	yes	—
14	#16810974	C₂²	1	4	yes	—
15	#16810863	—	2	8	no	—
16	#16810715	—	2	8	no	—
17	#16810660	—	2	4	no	—
18	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₅₃	C₂²	1	2	yes	—
20	C₈×#88322	C₂²	1	2	yes	—
21	C₈×#88322	C₂²	1	1	yes	characteristic
22	C₅₃×#500	C₂²	1	2	yes	—
23	C₅₃×#487	C₂²	1	4	yes	—
24	C₈²×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#11719925	—	2	8	no	—
26	C₂×C₈×D₅₃	D₄	1	4	yes	—
27	C₂×C₈×D₅₃	D₄	1	2	yes	—
28	#13133217	C₂³	1	4	yes	—
29	#13133208	—	2	4	no	—
30	#13133197	—	2	4	no	—
31	#13133193	—	2	4	no	—
32	Q₈×Dic₅₃	—	2	4	no	—
33	#13133182	—	2	4	no	—
34	#13133175	—	2	4	no	—
35	C₄²×D₅₃	C₂³	1	1	yes	characteristic
36	#13133151	—	2	4	no	—
37	#13133150	—	2	8	no	—
38	#13133148	—	2	8	no	—
39	#13133142	—	2	8	no	—
40	C₄×#88322	C₂³	1	1	yes	characteristic
41	C₄×#88322	C₂³	1	2	yes	—
42	C₈×Dic₅₃	D₄	1	2	yes	—
43	C₈×Dic₅₃	D₄	1	4	yes	—
44	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
45	#107×C₅₃	—	2	4	no	—
46	#103×C₅₃	—	2	8	no	—
47	C₄×C₈×C₅₃	C₂³	1	2	yes	—
48	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₅₃	—	2	4	no	—
50	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₀₆	—	2	4	no	—
53	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
54	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
58	#11719925	—	4	16	no	—
59	#11719921	—	4	4	no	—
60	C₄⋊Dic₅₃	—	2	4	no	—
61	C₄⋊Dic₅₃	—	2	4	no	—
62	C₄⋊Dic₅₃	—	4	4	no	—
63	#11719919	—	4	4	no	—
64	C₄×Dic₅₃	—	4	4	no	—
65	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
66	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×#88322	C₂×D₄	1	2	yes	—
68	C₂×#88322	C₂×D₄	1	4	yes	—
69	C₈×D₅₃	—	2	8	no	—
70	C₈×D₅₃	Q₁₆	1	8	yes	complemented
71	C₂²×D₅₃	#127	1	1	yes	characteristic
72	Q₈×C₅₃	—	4	8	no	—
73	C₂×C₄×C₅₃	—	4	4	no	—
74	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
76	C₂×Dic₅₃	—	4	4	no	—
77	C₂×Dic₅₃	#127	1	2	yes	—
78	C₂×Dic₅₃	#127	1	1	yes	characteristic
79	C₄×D₅₃	—	2	4	no	—
80	C₄×D₅₃	C₂×Q₁₆	1	4	yes	—
81	Dic₁₀₆	—	4	8	no	—
82	C₈×C₅₃	—	2	4	no	—
83	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
84	#88322	C₂×Q₁₆	1	4	yes	—
85	#88322	#135	1	8	yes	—
86	#19604	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
88	C₂×D₅₃	—	2	2	no	—
89	C₂×D₅₃	#494	1	2	yes	—
90	C₄×C₅₃	—	8	4	no	—
91	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
92	C₄×C₅₃	C₂×#135	1	4	yes	—
93	Dic₅₃	—	2	2	no	—
94	Dic₅₃	—	8	4	no	—
95	Dic₅₃	#494	1	2	yes	—
96	#3170	—	53	4	no	—
97	#3160	—	53	2	no	—
98	#1713	—	53	2	no	—
99	#1710	—	53	1	no	—
100	#1710	D₅₃	1	1	yes	characteristic, complemented
101	#1707	—	53	4	no	—
102	C₂×C₈²	—	53	1	no	—
103	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
104	C₂×C₅₃	#3147	1	2	yes	—
105	D₅₃	—	2	2	no	—
106	#533	—	53	4	no	—
107	#500	—	53	4	no	—
108	#500	—	53	2	no	—
109	#500	C₂×D₅₃	1	2	yes	—
110	#499	—	106	4	no	—
111	#498	—	53	4	no	—
112	#494	—	53	4	no	—
113	#487	—	53	4	no	—
114	#487	C₂×D₅₃	1	4	yes	—
115	#467	—	106	8	no	—
116	C₄×Q₁₆	—	106	8	no	—
117	#427	—	106	4	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	—	53	2	no	—
120	C₈²	—	53	2	no	—
121	C₈²	—	53	1	no	—
122	C₈²	C₂×D₅₃	1	1	yes	characteristic
123	C₅₃	#19604	1	1	yes	characteristic, complemented, Sylow
124	C₂×Q₁₆	—	106	8	no	—
125	C₂²×C₈	—	53	2	no	—
126	C₂²×C₈	—	53	4	no	—
127	C₄⋊Q₈	—	53	4	no	—
128	C₄⋊Q₈	C₂²×D₅₃	1	4	yes	—
129	#125	—	106	4	no	—
130	C₂²⋊Q₈	—	106	4	no	—
131	C₂²⋊Q₈	—	106	4	no	—
132	C₄×Q₈	—	106	4	no	—
133	C₄×Q₈	—	106	4	no	—
134	#117	—	106	4	no	—
135	C₂×C₄²	—	53	1	no	—
136	#107	—	2	4	no	—
137	#107	—	106	8	no	—
138	#107	—	106	4	no	—
139	#106	—	106	8	no	—
140	#103	—	2	8	no	—
141	#103	—	106	8	no	—
142	C₄×C₈	—	53	4	no	—
143	C₄×C₈	—	53	2	no	—
144	C₄×C₈	—	53	2	no	—
145	C₄×C₈	—	53	1	no	—
146	C₄×C₈	C₂²×D₅₃	1	2	yes	—
147	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	106	4	no	—
150	C₂²×C₄	—	53	1	no	—
151	C₂²×C₄	—	53	2	no	—
152	Q₁₆	—	212	16	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	4	no	—
155	C₂×C₈	—	53	8	no	—
156	C₂×C₈	—	106	8	no	—
157	C₂×C₈	D₄×D₅₃	1	4	yes	—
158	C₂×C₈	D₄×D₅₃	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	106	4	no	—
162	C₄⋊C₄	—	106	4	no	—
163	C₄⋊C₄	—	212	4	no	—
164	C₄⋊C₄	—	212	4	no	—
165	C₂²⋊C₄	—	212	4	no	—
166	C₄²	—	53	2	no	—
167	C₄²	—	53	1	no	—
168	C₄²	—	212	4	no	—
169	C₄²	C₂³×D₅₃	1	1	yes	characteristic
170	C₂³	—	53	1	no	—
171	Q₈	—	4	8	no	—
172	Q₈	—	212	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	53	1	no	—
175	C₂×C₄	—	53	4	no	—
176	C₂×C₄	—	53	2	no	—
177	C₂×C₄	—	106	4	no	—
178	C₂×C₄	—	212	4	no	—
179	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	53	4	no	—
183	C₈	—	53	8	no	—
184	C₈	Q₁₆×D₅₃	1	4	yes	—
185	C₂²	—	53	2	no	—
186	C₂²	—	106	2	no	—
187	C₂²	#127×D₅₃	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	53	2	no	—
190	C₄	—	106	2	no	—
191	C₄	—	424	4	no	—
192	C₄	C₂×Q₁₆×D₅₃	1	2	yes	—
193	C₄	#135×D₅₃	1	4	yes	—
194	C₂	—	106	2	no	—
195	C₂	#494×D₅₃	1	1	yes	characteristic
196	C₂	#20285140	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20285300

C₂

yes

maximal

#20285226

C₂

yes

maximal

#20274332

C₂

yes

maximal, complemented

#20274325

C₂

yes

maximal, characteristic, complemented

#20274311

C₂

yes

maximal, complemented

C₈²×D₅₃

C₂

yes

maximal, characteristic

C₅₃×#1710

C₂

yes

maximal, characteristic, complemented

#16811178

C₂²

yes

—

#16811028

C₂²

yes

—

#16811026

C₂²

yes

—

#16811024

—

#16811019

C₂²

yes

—

#16811002

C₂²

yes

—

#16810974

C₂²

yes

—

#16810863

—

#16810715

—

#16810660

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₄×C₈×D₅₃

C₂²

yes

—

C₈×#88322

C₂²

yes

—

C₈×#88322

C₂²

yes

characteristic

C₅₃×#500

C₂²

yes

—

C₅₃×#487

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×#11719925

—

C₂×C₈×D₅₃

D₄

yes

—

C₂×C₈×D₅₃

D₄

yes

—

#13133217

C₂³

yes

—

#13133208

—

#13133197

—

#13133193

—

Q₈×Dic₅₃

—

#13133182

—

#13133175

—

C₄²×D₅₃

C₂³

yes

—

—

—

—

C₂³

yes

characteristic

C₄×#88322

C₂³

yes

—

C₈×Dic₅₃

D₄

yes

—

C₈×Dic₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

C₂³

yes

—

#107×C₅₃

—

#103×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×C₄×D₅₃

C₂×D₄

yes

—

C₂×C₄×D₅₃

C₂×D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂⁴

yes

characteristic

#11719925

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×D₄

yes

—

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₂×#88322

C₂×D₄

yes

—

C₂×#88322

C₂×D₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

Q₁₆

yes

complemented

C₂²×D₅₃

#127

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#127

yes

—

C₂×Dic₅₃

#127

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂×Q₁₆

yes

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₁₆

yes

—

#88322

C₂×Q₁₆

yes

—

#88322

#135

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

#494

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×Q₁₆

yes

—

C₄×C₅₃

C₂×#135

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#494

yes

—

—

—

—

—

100

#1710

D₅₃

yes

characteristic, complemented

101

#1707

—

102

C₂×C₈²

—

103

C₂×C₅₃

C₂×#494

yes

characteristic

104

C₂×C₅₃

#3147

yes

—

105

D₅₃

—

106

#533

—

107

#500

—

108

#500

—

109

#500

C₂×D₅₃

yes

—

110

#499

—

106

—

111

#498

—

112

#494

—

113

#487

—

114

#487

C₂×D₅₃

yes

—

115

#467

—

106

—

116

C₄×Q₁₆

—

106

—

117

#427

—

106

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

C₈²

—

121

C₈²

—

122

C₈²

C₂×D₅₃

yes

characteristic

123

C₅₃

#19604

yes

characteristic, complemented, Sylow

124

C₂×Q₁₆

—

106

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂²×D₅₃

yes

—

129

#125

—

106

—

130

C₂²⋊Q₈

—

106

—

131

C₂²⋊Q₈

—

106

—

132

C₄×Q₈

—

106

—

133

C₄×Q₈

—

106

—

134

#117

—

106

—

135

C₂×C₄²

—

136

#107

—

137

#107

—

106

—

138

#107

—

106

—

139

#106

—

106

—

140

#103

—

141

#103

—

106

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂²×D₅₃

yes

—

147

C₄×C₈

C₂²×D₅₃

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

106

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

Q₁₆

—

212

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

106

—

157

C₂×C₈

D₄×D₅₃

yes

—

158

C₂×C₈

D₄×D₅₃

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

106

—

162

C₄⋊C₄

—

106

—

163

C₄⋊C₄

—

212

—

164

C₄⋊C₄

—

212

—

165

C₂²⋊C₄

—

212

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

212

—

169

C₄²

C₂³×D₅₃

yes

characteristic

170

C₂³

—

171

Q₈

—

172

Q₈

—

212

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

106

—

178

C₂×C₄

—

212

—

179

C₂×C₄

C₂×D₄×D₅₃

yes

characteristic

180

C₂×C₄

C₂×D₄×D₅₃

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

Q₁₆×D₅₃

yes

—

185

C₂²

—

186

C₂²

—

106

—

187

C₂²

#127×D₅₃

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

106

—

191

C₄

—

424

—

192

C₄

C₂×Q₁₆×D₅₃

yes

—

193

C₄

#135×D₅₃

yes

—

194

C₂

—

106

—

195

C₂

#494×D₅₃

yes

characteristic

196

C₂

#20285140

yes

—

	subgroup	G/H
center	C₂²	#127×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1710	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	C₂²×D₅₃	#127
hypercenter	#1710	D₅₃
hypocenter	C₅₃	#19604
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1710

subgroup

G/H

center

C₂²

#127×D₅₃

commutator, cosocle

C₄²×C₅₃

C₂⁴

Frattini

C₄²

C₂³×D₅₃

Fitting

C₅₃×#1710

C₂

C₂²×C₅₃

C₂²×D₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #1710

N	H ≅ G/N	type
3. #20274332	C₂	semidirect
4. #20274325	C₂	semidirect
5. #20274311	C₂	semidirect
7. C₅₃×#1710	C₂	semidirect
70. C₈×D₅₃	Q₁₆	semidirect
100. #1710	D₅₃	semidirect
123. C₅₃	#19604	semidirect

H ≅ G/N

type

3. #20274332

C₂

semidirect

4. #20274325

C₂

semidirect

5. #20274311

C₂

semidirect

7. C₅₃×#1710

C₂

semidirect

70. C₈×D₅₃

Q₁₆

semidirect

100. #1710

D₅₃

semidirect

123. C₅₃

#19604

semidirect

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

subgroup	index
#20285300	2
#20285226	2
#20274332	2
#20274325	2
#20274311	2
C₈²×D₅₃	2
C₅₃×#1710	2
#19604	53

subgroup

index

C₈²×D₅₃

C₅₃×#1710

#19604

Subgroups — #27371799

Conjugacy classes of subgroups

Complements

Maximal subgroups