Group #27536670 — subgroups

Subgroups — #27536670

Order 13568.

862 conjugacy classes of subgroups; only the 299 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 299 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×#16811026	C₂	1	2	yes	maximal
2	C₂×#16811019	C₂	1	2	yes	maximal
3	#20283093	C₂	1	1	yes	maximal, characteristic
4	#20274877	C₂	1	8	yes	maximal
5	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133217	C₂²	1	2	yes	—
8	#16811026	—	2	8	no	—
9	#16811019	—	2	8	no	—
10	C₂×#13133150	C₂²	1	8	yes	—
11	C₂×#13133148	C₂²	1	8	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄×#88322	C₂²	1	4	yes	—
15	#16810529	—	2	4	no	—
16	#16810489	C₂²	1	4	yes	—
17	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#391	—	2	4	no	—
19	C₅₃×#382	C₂²	1	4	yes	—
20	C₄³×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×Dic₁₀₆	—	2	2	no	—
22	C₂²×#88322	C₂³	1	2	yes	—
23	C₂²×#88322	C₂³	1	2	yes	—
24	#13133217	—	2	4	no	—
25	#13133217	—	2	4	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
28	#13133150	—	2	16	no	—
29	#13133148	—	2	16	no	—
30	C₄×#88322	D₄	1	4	yes	—
31	C₄×#88322	D₄	1	4	yes	—
32	C₄×#88322	Q₈	1	16	yes	complemented
33	#13133124	—	2	8	no	—
34	C₂²×Q₈×C₅₃	—	2	2	no	—
35	C₄⋊Q₈×C₅₃	—	2	4	no	—
36	C₄⋊Q₈×C₅₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
39	C₂×C₄²×C₅₃	D₄	1	4	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
42	#95×C₅₃	—	2	8	no	—
43	C₂×Q₈×C₅₃	—	4	4	no	—
44	C₂×Q₈×C₅₃	—	4	8	no	—
45	C₂²×C₄×C₅₃	—	2	4	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	2	4	no	—
50	C₂×Dic₁₀₆	—	4	4	no	—
51	C₂×Dic₁₀₆	—	4	8	no	—
52	C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₄⋊C₄×C₅₃	—	4	4	no	—
55	C₄²×C₅₃	—	2	8	no	—
56	C₄²×C₅₃	C₂×D₄	1	2	yes	—
57	C₄²×C₅₃	C₂×D₄	1	2	yes	—
58	C₄²×C₅₃	C₂×Q₈	1	8	yes	—
59	C₄⋊Dic₅₃	—	2	4	no	—
60	C₄⋊Dic₅₃	—	2	4	no	—
61	C₄⋊Dic₅₃	—	4	4	no	—
62	C₂×#88322	C₂×D₄	1	4	yes	—
63	C₂×#88322	C₂×D₄	1	4	yes	—
64	C₂×#88322	C₂×Q₈	1	8	yes	—
65	C₂×#88322	C₂×Q₈	1	8	yes	—
66	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
67	Q₈×C₅₃	—	8	8	no	—
68	C₂×C₄×C₅₃	—	2	8	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	8	yes	—
73	C₂×C₄×C₅₃	#135	1	8	yes	—
74	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
77	C₂×Dic₅₃	—	4	4	no	—
78	C₂×Dic₅₃	—	4	8	no	—
79	Dic₁₀₆	—	8	8	no	—
80	#88322	C₄⋊Q₈	1	8	yes	—
81	#88322	C₄⋊Q₈	1	8	yes	—
82	#33313	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
85	C₂²×C₅₃	C₂×#135	1	4	yes	—
86	C₄×C₅₃	—	2	8	no	—
87	C₄×C₅₃	—	8	4	no	—
88	C₄×C₅₃	#495	1	8	yes	—
89	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
90	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
91	Dic₅₃	—	8	4	no	—
92	C₂×#500	—	53	2	no	—
93	C₂×#498	—	53	2	no	—
94	#2862	—	53	1	no	—
95	#2862	D₅₃	1	1	yes	characteristic
96	#1846	—	53	8	no	—
97	C₄²×C₈	—	53	1	no	—
98	C₂×C₅₃	#2868	1	1	yes	characteristic
99	C₂×C₅₃	C₂×#495	1	2	yes	—
100	C₂×C₅₃	#3160	1	4	yes	—
101	C₂×C₄⋊Q₈	—	53	2	no	—
102	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
103	#500	—	106	8	no	—
104	#498	—	106	8	no	—
105	C₂×#107	—	53	8	no	—
106	C₂×#106	—	53	8	no	—
107	C₂×C₄×C₈	—	53	4	no	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	#391	—	2	4	no	—
111	#391	—	106	4	no	—
112	#382	—	53	4	no	—
113	#382	C₂×D₅₃	1	4	yes	—
114	C₄³	C₂×D₅₃	1	1	yes	characteristic
115	C₅₃	#33313	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	2	2	no	—
117	C₂²×Q₈	—	106	2	no	—
118	C₂²×C₈	—	53	2	no	—
119	C₂²×C₈	—	53	2	no	—
120	C₄⋊Q₈	—	2	4	no	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	106	4	no	—
123	C₄⋊Q₈	—	106	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	53	4	no	—
126	C₂×C₄⋊C₄	—	106	2	no	—
127	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
128	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
129	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
131	#107	—	106	16	no	—
132	#106	—	106	16	no	—
133	C₄×C₈	—	53	4	no	—
134	C₄×C₈	—	53	4	no	—
135	C₄×C₈	—	53	16	no	—
136	#95	—	2	8	no	—
137	#95	—	106	8	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂×Q₈	—	4	8	no	—
140	C₂×Q₈	—	212	4	no	—
141	C₂×Q₈	—	212	8	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	106	4	no	—
144	C₂²×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
146	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
147	C₂×C₈	—	53	4	no	—
148	C₂×C₈	—	53	4	no	—
149	C₂×C₈	—	53	8	no	—
150	C₂×C₈	—	53	8	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	—	106	4	no	—
155	C₄⋊C₄	—	106	4	no	—
156	C₄⋊C₄	—	212	4	no	—
157	C₄²	—	2	8	no	—
158	C₄²	D₄×D₅₃	1	2	yes	—
159	C₄²	D₄×D₅₃	1	2	yes	—
160	C₄²	Q₈×D₅₃	1	8	yes	—
161	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
162	Q₈	—	8	8	no	—
163	Q₈	—	424	8	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	212	8	no	—
168	C₂×C₄	—	212	4	no	—
169	C₂×C₄	#13133213	1	4	yes	—
170	C₂×C₄	#13133215	1	8	yes	—
171	C₂×C₄	#13133237	1	8	yes	—
172	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
173	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
174	C₂×C₄	C₂×Q₈×D₅₃	1	4	yes	—
175	C₈	—	53	8	no	—
176	C₈	—	53	8	no	—
177	C₂²	C₂×#13133213	1	1	yes	characteristic
178	C₂²	C₂×#13133215	1	2	yes	—
179	C₂²	C₂×#13133237	1	4	yes	—
180	C₄	—	2	8	no	—
181	C₄	—	8	4	no	—
182	C₄	—	424	4	no	—
183	C₄	#16811005	1	8	yes	—
184	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
185	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
186	C₂	#20283111	1	1	yes	characteristic
187	C₂	C₂×#16811005	1	2	yes	—
188	C₂	#20285225	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811026

C₂

yes

maximal

C₂×#16811019

C₂

yes

maximal

#20283093

C₂

yes

maximal, characteristic

#20274877

C₂

yes

maximal

C₄²×#88322

C₂

yes

maximal, characteristic

C₅₃×#2862

C₂

yes

maximal, characteristic

C₂×#13133217

C₂²

yes

—

#16811026

—

#16811019

—

C₂×#13133150

C₂²

yes

—

C₂×#13133148

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

—

#16810529

—

#16810489

C₂²

yes

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#391

—

C₅₃×#382

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

C₂²×#88322

C₂³

yes

—

C₂²×#88322

C₂³

yes

—

#13133217

—

#13133217

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂³

yes

—

#13133150

—

#13133148

—

C₄×#88322

D₄

yes

—

C₄×#88322

D₄

yes

—

C₄×#88322

Q₈

yes

complemented

#13133124

—

C₂²×Q₈×C₅₃

—

C₄⋊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

—

#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

C₂⁴

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

C₂×D₄

yes

—

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

#127

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

C₄⋊Q₈

yes

—

#88322

C₄⋊Q₈

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₄⋊Q₈

yes

—

C₂²×C₅₃

C₂×#135

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#495

yes

—

C₄×C₅₃

C₂×C₄⋊Q₈

yes

—

C₄×C₅₃

C₂×C₄⋊Q₈

yes

—

Dic₅₃

—

—

—

—

D₅₃

yes

characteristic

#1846

—

C₄²×C₈

—

C₂×C₅₃

#2868

yes

characteristic

C₂×C₅₃

C₂×#495

yes

—

100

C₂×C₅₃

#3160

yes

—

101

C₂×C₄⋊Q₈

—

102

C₂×C₄⋊Q₈

C₂×D₅₃

yes

—

103

#500

—

106

—

104

#498

—

106

—

105

C₂×#107

—

106

C₂×#106

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

#391

—

111

#391

—

106

—

112

#382

—

113

#382

C₂×D₅₃

yes

—

114

C₄³

C₂×D₅₃

yes

characteristic

115

C₅₃

#33313

yes

characteristic, complemented, Sylow

116

C₂²×Q₈

—

117

C₂²×Q₈

—

106

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

—

122

C₄⋊Q₈

—

106

—

123

C₄⋊Q₈

—

106

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

106

—

127

C₂×C₄⋊C₄

C₂²×D₅₃

yes

—

128

C₂×C₄²

C₂²⋊D₅₃

yes

—

129

C₂×C₄²

C₂²×D₅₃

yes

characteristic

130

C₂×C₄²

C₂²×D₅₃

yes

—

131

#107

—

106

—

132

#106

—

106

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

#95

—

137

#95

—

106

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

212

—

141

C₂×Q₈

—

212

—

142

C₂²×C₄

—

143

C₂²×C₄

—

106

—

144

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

145

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

146

C₂²×C₄

C₂³×D₅₃

yes

characteristic

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

106

—

155

C₄⋊C₄

—

106

—

156

C₄⋊C₄

—

212

—

157

C₄²

—

158

C₄²

D₄×D₅₃

yes

—

159

C₄²

D₄×D₅₃

yes

—

160

C₄²

Q₈×D₅₃

yes

—

161

C₂³

C₂²×C₂²⋊D₅₃

yes

characteristic

162

Q₈

—

163

Q₈

—

424

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

212

—

168

C₂×C₄

—

212

—

169

C₂×C₄

#13133213

yes

—

170

C₂×C₄

#13133215

yes

—

171

C₂×C₄

#13133237

yes

—

172

C₂×C₄

C₂×D₄×D₅₃

yes

characteristic

173

C₂×C₄

C₂×D₄×D₅₃

yes

characteristic

174

C₂×C₄

C₂×Q₈×D₅₃

yes

—

175

C₈

—

176

C₈

—

177

C₂²

C₂×#13133213

yes

characteristic

178

C₂²

C₂×#13133215

yes

—

179

C₂²

C₂×#13133237

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

424

—

183

C₄

#16811005

yes

—

184

C₄

C₄⋊Q₈×D₅₃

yes

—

185

C₄

C₄⋊Q₈×D₅₃

yes

—

186

C₂

#20283111

yes

characteristic

187

C₂

C₂×#16811005

yes

—

188

C₂

#20285225

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	#33313
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
32. C₄×#88322	Q₈	semidirect
115. C₅₃	#33313	semidirect

H ≅ G/N

type

32. C₄×#88322

Q₈

semidirect

115. C₅₃

#33313

semidirect

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

subgroup	index
C₂×#16811026	2
C₂×#16811019	2
#20283093	2
#20274877	2
C₄²×#88322	2
C₅₃×#2862	2
#33313	53

subgroup

index

C₄²×#88322

C₅₃×#2862

#33313

Subgroups — #27536670

Conjugacy classes of subgroups

Complements

Maximal subgroups