Group #27536692 — subgroups

Subgroups — #27536692

Order 13568.

822 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 267 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×#16811029	C₂	1	4	yes	maximal
2	#20283078	C₂	1	1	yes	maximal, characteristic
3	#20274884	C₂	1	4	yes	maximal
4	#20274877	C₂	1	4	yes	maximal
5	C₄×#13133128	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133209	C₂²	1	2	yes	—
8	#16811029	—	2	16	no	—
9	C₂×#13133150	C₂²	1	8	yes	—
10	C₂×#13133148	C₂²	1	8	yes	—
11	C₂×#13133128	C₂²	1	2	yes	—
12	C₂×#13133128	C₂²	1	2	yes	—
13	C₂×C₄×#88322	C₂²	1	2	yes	—
14	#16810521	—	2	4	no	—
15	#16810493	C₂²	1	2	yes	—
16	#16810489	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#391	—	2	4	no	—
19	C₅₃×#382	C₂²	1	2	yes	—
20	C₅₃×#382	C₂²	1	2	yes	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×#88322	C₂³	1	4	yes	—
23	#13133209	—	2	8	no	—
24	C₂×C₄⋊Dic₅₃	—	2	4	no	—
25	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
26	C₂×#11719919	—	2	4	no	—
27	#13133150	—	2	16	no	—
28	#13133148	—	2	16	no	—
29	#13133128	D₄	1	8	yes	—
30	#13133128	Q₈	1	8	yes	complemented
31	C₄×#88322	Q₈	1	8	yes	complemented
32	#13133124	—	2	4	no	—
33	C₂²×Q₈×C₅₃	—	2	2	no	—
34	C₄⋊Q₈×C₅₃	—	2	8	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₅₃	D₄	1	4	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#95×C₅₃	—	2	4	no	—
42	#95×C₅₃	—	2	4	no	—
43	C₂×Q₈×C₅₃	—	4	8	no	—
44	C₂×Q₈×C₅₃	—	4	4	no	—
45	C₂²×C₄×C₅₃	—	2	4	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂²×Dic₅₃	—	2	4	no	—
51	C₄⋊C₄×C₅₃	—	2	8	no	—
52	C₄⋊C₄×C₅₃	—	4	4	no	—
53	C₄²×C₅₃	—	4	4	no	—
54	C₄²×C₅₃	C₂×D₄	1	4	yes	—
55	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
56	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
57	C₄⋊Dic₅₃	—	2	4	no	—
58	C₄⋊Dic₅₃	—	2	4	no	—
59	C₄⋊Dic₅₃	—	4	8	no	—
60	#11719919	—	4	8	no	—
61	C₂×#88322	C₂×D₄	1	8	yes	—
62	C₂×#88322	C₂×Q₈	1	8	yes	—
63	C₂×#88322	C₂×Q₈	1	8	yes	—
64	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
65	Q₈×C₅₃	—	8	8	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	—	4	8	no	—
71	C₂×C₄×C₅₃	#127	1	4	yes	—
72	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
73	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
74	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
77	C₂×Dic₅₃	—	4	4	no	—
78	C₂×Dic₅₃	—	4	8	no	—
79	#88322	C₄⋊Q₈	1	16	yes	—
80	#33317	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
84	C₂²×C₅₃	#575	1	4	yes	—
85	C₄×C₅₃	—	2	4	no	—
86	C₄×C₅₃	—	2	4	no	—
87	C₄×C₅₃	—	2	4	no	—
88	C₄×C₅₃	—	8	4	no	—
89	C₄×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
90	Dic₅₃	—	8	4	no	—
91	C₂×#501	—	53	4	no	—
92	#2862	D₅₃	1	1	yes	characteristic
93	#2859	—	53	1	no	—
94	#1847	—	53	4	no	—
95	#1846	—	53	4	no	—
96	C₄×#97	—	53	1	no	—
97	C₂×C₅₃	#2868	1	1	yes	characteristic
98	C₂×C₅₃	#3153	1	2	yes	—
99	C₂×C₅₃	#3162	1	2	yes	—
100	C₂×C₅₃	#3166	1	2	yes	—
101	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
102	C₂×#125	—	53	2	no	—
103	#501	—	106	16	no	—
104	C₂×#107	—	53	8	no	—
105	C₂×#106	—	53	8	no	—
106	C₂×#97	—	53	2	no	—
107	C₂×#97	—	53	2	no	—
108	C₂×C₄×C₈	—	53	2	no	—
109	#391	—	2	4	no	—
110	#389	—	106	4	no	—
111	#384	—	53	2	no	—
112	#382	—	53	2	no	—
113	#382	C₂×D₅₃	1	2	yes	—
114	#382	C₂×D₅₃	1	2	yes	—
115	C₄³	C₂×D₅₃	1	1	yes	characteristic
116	C₅₃	#33317	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	—	2	2	no	—
118	C₂²×C₈	—	53	4	no	—
119	C₄⋊Q₈	—	2	8	no	—
120	#125	—	106	8	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	53	4	no	—
123	C₂×C₄⋊C₄	—	106	4	no	—
124	C₂×C₄⋊C₄	—	106	4	no	—
125	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
126	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
127	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
129	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	#107	—	106	16	no	—
131	#106	—	106	16	no	—
132	#97	—	53	8	no	—
133	#97	—	53	8	no	—
134	C₄×C₈	—	53	8	no	—
135	#95	—	2	4	no	—
136	#95	—	2	4	no	—
137	#95	—	106	4	no	—
138	C₂×Q₈	—	4	8	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	106	4	no	—
142	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
144	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
145	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
146	C₂×C₈	—	53	8	no	—
147	C₂×C₈	—	53	8	no	—
148	C₂×C₈	—	53	8	no	—
149	C₄⋊C₄	—	2	8	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	106	4	no	—
152	C₄⋊C₄	—	106	4	no	—
153	C₄⋊C₄	—	212	8	no	—
154	C₄⋊C₄	—	212	8	no	—
155	C₄²	—	4	4	no	—
156	C₄²	D₄×D₅₃	1	4	yes	—
157	C₄²	Q₈×D₅₃	1	4	yes	—
158	C₄²	Q₈×D₅₃	1	4	yes	—
159	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
160	Q₈	—	8	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	212	4	no	—
167	C₂×C₄	—	212	8	no	—
168	C₂×C₄	#13133213	1	4	yes	—
169	C₂×C₄	#13133215	1	4	yes	—
170	C₂×C₄	#13133215	1	4	yes	—
171	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
172	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
173	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
174	C₈	—	53	16	no	—
175	C₂²	C₂×#13133213	1	1	yes	characteristic
176	C₂²	C₂×#13133215	1	1	yes	characteristic
177	C₂²	C₂×#13133215	1	1	yes	characteristic
178	C₂²	#16811421	1	4	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	2	4	no	—
182	C₄	—	8	4	no	—
183	C₄	—	424	4	no	—
184	C₄	C₄⋊Q₈×D₅₃	1	4	yes	—
185	C₂	#20283111	1	1	yes	characteristic
186	C₂	#20285179	1	2	yes	—
187	C₂	#20285237	1	2	yes	—
188	C₂	#20285268	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811029

C₂

yes

maximal

#20283078

C₂

yes

maximal, characteristic

#20274884

C₂

yes

maximal

#20274877

C₂

yes

maximal

C₄×#13133128

C₂

yes

maximal, characteristic

C₅₃×#2862

C₂

yes

maximal, characteristic

C₂×#13133209

C₂²

yes

—

#16811029

—

C₂×#13133150

C₂²

yes

—

C₂×#13133148

C₂²

yes

—

C₂×#13133128

C₂²

yes

—

C₂×#13133128

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810521

—

#16810493

C₂²

yes

—

#16810489

C₂²

yes

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#391

—

C₅₃×#382

C₂²

yes

—

C₅₃×#382

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂³

yes

—

#13133209

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂³

yes

—

—

—

—

D₄

yes

—

#13133128

Q₈

yes

complemented

C₄×#88322

Q₈

yes

complemented

#13133124

—

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂³

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₂×#88322

C₂×D₄

yes

—

C₂×#88322

C₂×Q₈

yes

—

C₂×#88322

C₂×Q₈

yes

—

C₂³×C₅₃

C₂²×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×C₄×C₅₃

C₂²×Q₈

yes

—

C₂×C₄×C₅₃

C₂²×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

C₄⋊Q₈

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

#575

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊Q₈

yes

—

Dic₅₃

—

C₂×#501

—

#2862

D₅₃

yes

—

—

—

C₄×#97

—

C₂×C₅₃

#2868

yes

characteristic

C₂×C₅₃

#3153

yes

—

C₂×C₅₃

#3162

yes

—

100

C₂×C₅₃

#3166

yes

—

101

C₂×C₄⋊Q₈

C₂×D₅₃

yes

—

102

C₂×#125

—

103

#501

—

106

—

104

C₂×#107

—

105

C₂×#106

—

106

C₂×#97

—

107

C₂×#97

—

108

C₂×C₄×C₈

—

109

#391

—

110

#389

—

106

—

111

#384

—

112

#382

—

113

#382

C₂×D₅₃

yes

—

114

#382

C₂×D₅₃

yes

—

115

C₄³

C₂×D₅₃

yes

characteristic

116

C₅₃

#33317

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

—

118

C₂²×C₈

—

119

C₄⋊Q₈

—

120

#125

—

106

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

106

—

124

C₂×C₄⋊C₄

—

106

—

125

C₂×C₄⋊C₄

C₂²×D₅₃

yes

—

126

C₂×C₄²

C₂²⋊D₅₃

yes

—

127

C₂×C₄²

C₂²×D₅₃

yes

characteristic

128

C₂×C₄²

C₂²×D₅₃

yes

characteristic

129

C₂×C₄²

C₂²×D₅₃

yes

characteristic

130

#107

—

106

—

131

#106

—

106

—

132

#97

—

133

#97

—

134

C₄×C₈

—

135

#95

—

136

#95

—

137

#95

—

106

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂²×C₄

—

141

C₂²×C₄

—

106

—

142

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

144

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

145

C₂²×C₄

C₂³×D₅₃

yes

characteristic

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

106

—

152

C₄⋊C₄

—

106

—

153

C₄⋊C₄

—

212

—

154

C₄⋊C₄

—

212

—

155

C₄²

—

156

C₄²

D₄×D₅₃

yes

—

157

C₄²

Q₈×D₅₃

yes

—

158

C₄²

Q₈×D₅₃

yes

—

159

C₂³

C₂²×C₂²⋊D₅₃

yes

characteristic

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

212

—

167

C₂×C₄

—

212

—

168

C₂×C₄

#13133213

yes

—

169

C₂×C₄

#13133215

yes

—

170

C₂×C₄

#13133215

yes

—

171

C₂×C₄

C₂×D₄×D₅₃

yes

—

172

C₂×C₄

C₂×Q₈×D₅₃

yes

—

173

C₂×C₄

C₂×Q₈×D₅₃

yes

—

174

C₈

—

175

C₂²

C₂×#13133213

yes

characteristic

176

C₂²

C₂×#13133215

yes

characteristic

177

C₂²

C₂×#13133215

yes

characteristic

178

C₂²

#16811421

yes

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

424

—

184

C₄

C₄⋊Q₈×D₅₃

yes

—

185

C₂

#20283111

yes

characteristic

186

C₂

#20285179

yes

—

187

C₂

#20285237

yes

—

188

C₂

#20285268

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2862	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2862	D₅₃
hypocenter	C₅₃	#33317
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

subgroup

G/H

center

C₂³

C₂²×C₂²⋊D₅₃

commutator, cosocle

C₂²×C₄×C₅₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting

C₅₃×#2862

C₂

socle, prime radical

C₂³×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

N	H ≅ G/N	type
30. #13133128	Q₈	semidirect
31. C₄×#88322	Q₈	semidirect
116. C₅₃	#33317	semidirect

H ≅ G/N

type

30. #13133128

Q₈

semidirect

31. C₄×#88322

Q₈

semidirect

116. C₅₃

#33317

semidirect

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

subgroup	index
C₂×#16811029	2
#20283078	2
#20274884	2
#20274877	2
C₄×#13133128	2
C₅₃×#2862	2
#33317	53

subgroup

index

C₅₃×#2862

Subgroups — #27536692

Conjugacy classes of subgroups

Complements

Maximal subgroups