Group #27536666 — subgroups

Subgroups — #27536666

Order 13568.

1006 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 395 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	4	yes	maximal
2	#20283113	C₂	1	1	yes	maximal, characteristic
3	#20274874	C₂	1	8	yes	maximal
4	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2868	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	—	2	2	no	—
7	C₂×#13133217	C₂²	1	2	yes	—
8	#16811026	—	2	16	no	—
9	C₂×#13133150	C₂²	1	16	yes	—
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	C₂×C₄×#88322	C₂²	1	4	yes	—
12	#16810493	—	2	2	no	—
13	#16810493	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#384	—	2	2	no	—
17	C₅₃×#384	C₂²	1	4	yes	—
18	C₄³×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₀₆	—	2	2	no	—
20	C₂²×#88322	C₂³	1	4	yes	—
21	#13133217	—	2	8	no	—
22	#13133217	—	4	8	no	—
23	C₂×C₄⋊Dic₅₃	—	2	8	no	—
24	C₂×C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
26	#13133150	—	2	32	no	—
27	C₄×#88322	D₄	1	8	yes	—
28	C₄×#88322	Q₈	1	16	yes	complemented
29	C₂²×Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	—	2	8	no	—
31	C₄⋊Q₈×C₅₃	—	4	8	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
39	C₂×Q₈×C₅₃	—	4	4	no	—
40	C₂×Q₈×C₅₃	—	4	8	no	—
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂²×Dic₅₃	—	2	4	no	—
47	C₂×Dic₁₀₆	—	4	4	no	—
48	C₂×Dic₁₀₆	—	4	8	no	—
49	C₄⋊C₄×C₅₃	—	2	8	no	—
50	C₄⋊C₄×C₅₃	—	4	4	no	—
51	C₄⋊C₄×C₅₃	—	4	16	no	—
52	C₄²×C₅₃	D₈	1	8	yes	—
53	C₄²×C₅₃	Q₁₆	1	8	yes	—
54	C₄²×C₅₃	C₂×D₄	1	4	yes	—
55	C₄²×C₅₃	C₂×Q₈	1	8	yes	—
56	C₄⋊Dic₅₃	—	2	8	no	—
57	C₄⋊Dic₅₃	—	4	4	no	—
58	C₄⋊Dic₅₃	—	4	16	no	—
59	C₂×#88322	C₂×D₄	1	8	yes	—
60	C₂×#88322	C₂×Q₈	1	16	yes	—
61	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₅₃	—	8	8	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	4	8	no	—
65	C₂×C₄×C₅₃	#127	1	2	yes	—
66	C₂×C₄×C₅₃	#127	1	2	yes	—
67	C₂×C₄×C₅₃	C₄⋊Q₈	1	8	yes	—
68	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×D₈	1	8	yes	—
70	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
71	C₂×C₄×C₅₃	C₂×Q₁₆	1	8	yes	—
72	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
74	C₂×Dic₅₃	—	4	4	no	—
75	C₂×Dic₅₃	—	4	8	no	—
76	Dic₁₀₆	—	8	8	no	—
77	#88322	C₄⋊Q₈	1	16	yes	complemented
78	#33312	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
80	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
81	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
82	C₂²×C₅₃	C₂²×Q₁₆	1	2	yes	—
83	C₄×C₅₃	—	8	4	no	—
84	C₄×C₅₃	#493	1	4	yes	—
85	C₄×C₅₃	#494	1	4	yes	—
86	C₄×C₅₃	#500	1	16	yes	complemented
87	C₄×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
88	Dic₅₃	—	8	4	no	—
89	C₂×#500	—	53	4	no	—
90	#2868	—	53	1	no	—
91	#2868	D₅₃	1	1	yes	characteristic
92	#1845	—	53	8	no	—
93	C₄²×C₈	—	53	1	no	—
94	C₂×C₅₃	#2868	1	1	yes	characteristic
95	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
96	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
97	C₂×C₅₃	C₂×#500	1	4	yes	—
98	C₂×C₄⋊Q₈	—	2	2	no	—
99	C₂×C₄⋊Q₈	—	53	2	no	—
100	C₂×C₄⋊Q₈	—	106	2	no	—
101	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
102	#500	—	106	16	no	—
103	C₂×#107	—	53	16	no	—
104	C₂×C₄×C₈	—	53	4	no	—
105	C₂×C₄×C₈	—	53	2	no	—
106	#384	—	2	2	no	—
107	#384	—	53	4	no	—
108	#384	—	106	2	no	—
109	#384	C₂×D₅₃	1	4	yes	—
110	C₄³	C₂×D₅₃	1	1	yes	characteristic
111	C₅₃	#33312	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	2	2	no	—
113	C₂²×Q₈	—	106	2	no	—
114	C₂²×C₈	—	53	4	no	—
115	C₄⋊Q₈	—	2	8	no	—
116	C₄⋊Q₈	—	4	8	no	—
117	C₄⋊Q₈	—	106	8	no	—
118	C₄⋊Q₈	—	212	8	no	—
119	C₂×C₄⋊C₄	—	2	8	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	53	4	no	—
122	C₂×C₄⋊C₄	—	106	2	no	—
123	C₂×C₄⋊C₄	—	106	8	no	—
124	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
125	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
126	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
129	#107	—	106	32	no	—
130	C₄×C₈	—	53	8	no	—
131	C₄×C₈	—	53	16	no	—
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	4	8	no	—
134	C₂×Q₈	—	212	4	no	—
135	C₂×Q₈	—	212	8	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	106	4	no	—
138	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
141	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
142	C₂×C₈	—	53	8	no	—
143	C₂×C₈	—	53	16	no	—
144	C₄⋊C₄	—	2	8	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	16	no	—
147	C₄⋊C₄	—	106	8	no	—
148	C₄⋊C₄	—	212	4	no	—
149	C₄⋊C₄	—	212	16	no	—
150	C₄²	#11719922	1	8	yes	—
151	C₄²	#11719925	1	8	yes	—
152	C₄²	D₄×D₅₃	1	4	yes	—
153	C₄²	Q₈×D₅₃	1	8	yes	—
154	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
155	Q₈	—	8	8	no	—
156	Q₈	—	424	8	no	—
157	C₂×C₄	—	4	8	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	212	8	no	—
160	C₂×C₄	—	212	4	no	—
161	C₂×C₄	#13133213	1	2	yes	—
162	C₂×C₄	#13133213	1	2	yes	—
163	C₂×C₄	#13133215	1	8	yes	—
164	C₂×C₄	C₂×#11719922	1	8	yes	—
165	C₂×C₄	C₂×#11719922	1	4	yes	—
166	C₂×C₄	C₂×#11719925	1	8	yes	—
167	C₂×C₄	C₂×#11719925	1	4	yes	—
168	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
169	C₂×C₄	C₂×Q₈×D₅₃	1	4	yes	—
170	C₈	—	53	16	no	—
171	C₂²	C₂×#13133213	1	1	yes	characteristic
172	C₂²	C₂×#13133215	1	2	yes	—
173	C₂²	C₂²×#11719922	1	2	yes	—
174	C₂²	C₂²×#11719925	1	2	yes	—
175	C₄	—	8	4	no	—
176	C₄	—	424	4	no	—
177	C₄	#16810999	1	4	yes	—
178	C₄	#16811002	1	4	yes	—
179	C₄	#16811026	1	16	yes	complemented
180	C₄	C₄⋊Q₈×D₅₃	1	4	yes	—
181	C₂	#20283111	1	1	yes	characteristic
182	C₂	C₂×#16810999	1	1	yes	characteristic
183	C₂	C₂×#16811002	1	1	yes	characteristic
184	C₂	C₂×#16811026	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811026

C₂

yes

maximal

#20283113

C₂

yes

maximal, characteristic

#20274874

C₂

yes

maximal

C₄²×#88322

C₂

yes

maximal, characteristic

C₅₃×#2868

C₂

yes

maximal, characteristic

C₂×#13133217

—

C₂×#13133217

C₂²

yes

—

#16811026

—

C₂×#13133150

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

—

#16810493

C₂²

yes

—

C₂×C₄⋊Q₈×C₅₃

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#384

—

C₅₃×#384

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

C₂²×#88322

C₂³

yes

—

#13133217

—

#13133217

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂³

yes

—

#13133150

—

C₄×#88322

D₄

yes

—

C₄×#88322

Q₈

yes

complemented

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂³

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

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

D₈

yes

—

C₄²×C₅₃

Q₁₆

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×Q₈

yes

—

C₂³×C₅₃

C₂²×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

—

C₂×C₄×C₅₃

#127

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×C₄×C₅₃

C₂²×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

C₄⋊Q₈

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₄⋊Q₈

yes

—

C₂²×C₅₃

C₂²×D₈

yes

—

C₂²×C₅₃

C₂²×Q₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

#493

yes

—

C₄×C₅₃

#494

yes

—

C₄×C₅₃

#500

yes

complemented

C₄×C₅₃

C₂×C₄⋊Q₈

yes

—

Dic₅₃

—

C₂×#500

—

#2868

—

#2868

D₅₃

yes

characteristic

#1845

—

C₄²×C₈

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

100

C₂×C₄⋊Q₈

—

106

—

101

C₂×C₄⋊Q₈

C₂×D₅₃

yes

—

102

#500

—

106

—

103

C₂×#107

—

104

C₂×C₄×C₈

—

105

C₂×C₄×C₈

—

106

#384

—

107

#384

—

108

#384

—

106

—

109

#384

C₂×D₅₃

yes

—

110

C₄³

C₂×D₅₃

yes

characteristic

111

C₅₃

#33312

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂²×Q₈

—

106

—

114

C₂²×C₈

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

106

—

118

C₄⋊Q₈

—

212

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

106

—

123

C₂×C₄⋊C₄

—

106

—

124

C₂×C₄⋊C₄

C₂²×D₅₃

yes

—

125

C₂×C₄²

C₂²⋊D₅₃

yes

—

126

C₂×C₄²

C₂²⋊D₅₃

yes

—

127

C₂×C₄²

C₂²×D₅₃

yes

characteristic

128

C₂×C₄²

C₂²×D₅₃

yes

—

129

#107

—

106

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

212

—

135

C₂×Q₈

—

212

—

136

C₂²×C₄

—

137

C₂²×C₄

—

106

—

138

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

141

C₂²×C₄

C₂³×D₅₃

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

106

—

148

C₄⋊C₄

—

212

—

149

C₄⋊C₄

—

212

—

150

C₄²

#11719922

yes

—

151

C₄²

#11719925

yes

—

152

C₄²

D₄×D₅₃

yes

—

153

C₄²

Q₈×D₅₃

yes

—

154

C₂³

C₂²×C₂²⋊D₅₃

yes

characteristic

155

Q₈

—

156

Q₈

—

424

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

212

—

160

C₂×C₄

—

212

—

161

C₂×C₄

#13133213

yes

—

162

C₂×C₄

#13133213

yes

—

163

C₂×C₄

#13133215

yes

—

164

C₂×C₄

C₂×#11719922

yes

—

165

C₂×C₄

C₂×#11719922

yes

—

166

C₂×C₄

C₂×#11719925

yes

—

167

C₂×C₄

C₂×#11719925

yes

—

168

C₂×C₄

C₂×D₄×D₅₃

yes

—

169

C₂×C₄

C₂×Q₈×D₅₃

yes

—

170

C₈

—

171

C₂²

C₂×#13133213

yes

characteristic

172

C₂²

C₂×#13133215

yes

—

173

C₂²

C₂²×#11719922

yes

—

174

C₂²

C₂²×#11719925

yes

—

175

C₄

—

176

C₄

—

424

—

177

C₄

#16810999

yes

—

178

C₄

#16811002

yes

—

179

C₄

#16811026

yes

complemented

180

C₄

C₄⋊Q₈×D₅₃

yes

—

181

C₂

#20283111

yes

characteristic

182

C₂

C₂×#16810999

yes

characteristic

183

C₂

C₂×#16811002

yes

characteristic

184

C₂

C₂×#16811026

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2868	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2868	D₅₃
hypocenter	C₅₃	#33312
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂²×C₂²⋊D₅₃

commutator, cosocle

C₂²×C₄×C₅₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting

C₅₃×#2868

C₂

socle, prime radical

C₂³×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

N	H ≅ G/N	type
28. C₄×#88322	Q₈	semidirect
77. #88322	C₄⋊Q₈	semidirect
86. C₄×C₅₃	#500	semidirect
111. C₅₃	#33312	semidirect
179. C₄	#16811026	semidirect

H ≅ G/N

type

28. C₄×#88322

Q₈

semidirect

77. #88322

C₄⋊Q₈

semidirect

86. C₄×C₅₃

#500

semidirect

111. C₅₃

#33312

semidirect

179. C₄

#16811026

semidirect

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

subgroup	index
C₂×#16811026	2
#20283113	2
#20274874	2
C₄²×#88322	2
C₅₃×#2868	2
#33312	53

subgroup

index

C₂×#16811026

#20283113

#20274874

C₄²×#88322

C₅₃×#2868

#33312

Subgroups — #27536666

Conjugacy classes of subgroups

Complements

Maximal subgroups