Group #27275232 — subgroups

Subgroups — #27275232

Order 13568.

470 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 199 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	C₄×#13133146	C₂	1	2	yes	maximal
2	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810345	C₂	1	2	yes	maximal
4	#20272542	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133146	C₄	1	4	yes	—
7	C₂×#13133127	C₄	1	2	yes	—
8	C₂×#13133127	C₄	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	2	yes	—
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810377	C₄	1	4	yes	—
13	#16810345	—	2	4	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	C₂×C₄	1	2	yes	—
17	C₂²×#88322	C₂×C₄	1	4	yes	—
18	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	#13133146	—	2	8	no	—
23	#13133127	—	2	4	no	—
24	#13133127	—	2	4	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	—	2	2	no	—
27	C₄×#88322	D₄	1	2	yes	—
28	C₄×#88322	Q₈	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
37	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
38	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₄²	1	2	yes	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
46	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅₃	#41	1	2	yes	—
49	C₄×Dic₅₃	—	2	2	no	—
50	C₄×Dic₅₃	—	2	4	no	—
51	C₄×Dic₅₃	—	2	4	no	—
52	C₄×Dic₅₃	M₁₆	1	4	yes	—
53	C₂×#88322	—	2	4	no	—
54	C₂×#88322	—	2	2	no	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	2	4	no	—
57	C₂×#88322	C₄⋊C₄	1	4	yes	—
58	C₂×#88322	M₁₆	1	8	yes	—
59	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
66	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₅₃	#131	1	4	yes	—
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	—	2	1	no	—
71	C₂×Dic₅₃	—	2	1	no	—
72	C₂×Dic₅₃	—	2	2	no	—
73	C₂×Dic₅₃	—	2	2	no	—
74	C₂×Dic₅₃	#97	1	4	yes	—
75	C₈×C₅₃	—	2	4	no	—
76	C₈×C₅₃	—	4	4	no	—
77	#88322	—	2	4	no	—
78	#88322	—	2	8	no	—
79	#8263	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
82	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
83	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
84	C₂²×C₅₃	#405	1	2	yes	—
85	C₄×C₅₃	#423	1	2	yes	—
86	C₄×C₅₃	#432	1	2	yes	—
87	C₄×C₅₃	#435	1	4	yes	—
88	C₄×C₅₃	#446	1	4	yes	—
89	Dic₅₃	—	2	4	no	—
90	Dic₅₃	—	4	1	no	—
91	Dic₅₃	—	4	1	no	—
92	C₄×C₄⋊C₈	—	53	2	no	—
93	C₄×#97	—	53	1	no	—
94	C₂×#322	—	53	2	no	—
95	#1273	—	53	1	no	—
96	#1273	D₅₃	1	1	yes	characteristic
97	C₂×C₅₃	#1451	1	2	yes	—
98	C₂×C₅₃	#1453	1	1	yes	characteristic
99	C₂×C₅₃	#1455	1	1	yes	characteristic
100	C₂×C₅₃	#1767	1	1	yes	characteristic
101	C₂×C₅₃	#1771	1	2	yes	—
102	C₂×C₄⋊C₈	—	53	4	no	—
103	C₂×#97	—	53	2	no	—
104	C₂×#97	—	53	2	no	—
105	C₂×C₄×C₈	—	53	1	no	—
106	C₂×C₄×C₈	—	53	2	no	—
107	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₄³	—	53	1	no	—
110	#336	—	53	4	no	—
111	#322	—	106	4	no	—
112	C₅₃	#8263	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	53	4	no	—
114	C₂²×C₈	—	53	2	no	—
115	C₂²×C₈	C₄×D₅₃	1	4	yes	—
116	C₂²×C₈	C₄×D₅₃	1	2	yes	—
117	C₂×C₄²	—	53	1	no	—
118	C₂×C₄²	—	53	2	no	—
119	C₂×C₄²	—	53	1	no	—
120	C₂×C₄²	—	53	2	no	—
121	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	C₄⋊C₈	—	106	8	no	—
123	#97	—	106	4	no	—
124	#97	—	106	4	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	53	2	no	—
129	C₄×C₈	—	53	2	no	—
130	C₄×C₈	—	106	2	no	—
131	C₄×C₈	—	106	2	no	—
132	C₂²×C₄	—	53	2	no	—
133	C₂²×C₄	—	53	1	no	—
134	C₂²×C₄	—	53	1	no	—
135	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	53	4	no	—
143	C₂×C₈	—	53	8	no	—
144	C₂×C₈	—	106	4	no	—
145	C₂×C₈	—	106	2	no	—
146	C₂×C₈	—	106	2	no	—
147	C₂×C₈	—	106	4	no	—
148	C₂×C₈	#11719912	1	4	yes	—
149	C₄²	—	53	4	no	—
150	C₄²	—	106	4	no	—
151	C₄²	—	106	2	no	—
152	C₄²	—	106	4	no	—
153	C₄²	D₄×D₅₃	1	1	yes	characteristic
154	C₄²	#11719947	1	2	yes	—
155	C₄²	Q₈×D₅₃	1	1	yes	characteristic
156	C₂³	C₄²×D₅₃	1	1	yes	characteristic
157	C₂×C₄	—	53	4	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	—	106	2	no	—
160	C₂×C₄	—	106	1	no	—
161	C₂×C₄	—	106	1	no	—
162	C₂×C₄	—	106	2	no	—
163	C₂×C₄	—	106	2	no	—
164	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
165	C₂×C₄	#13133173	1	1	yes	characteristic
166	C₂×C₄	#13133177	1	2	yes	—
167	C₂×C₄	#13133182	1	2	yes	—
168	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
169	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
170	C₂×C₄	C₂×#11719912	1	2	yes	—
171	C₂×C₄	#13133223	1	4	yes	—
172	C₈	—	2	4	no	—
173	C₈	—	4	4	no	—
174	C₈	—	106	4	no	—
175	C₈	—	106	8	no	—
176	C₂²	#16810474	1	1	yes	characteristic
177	C₂²	#97×D₅₃	1	1	yes	characteristic
178	C₂²	C₄×#11719912	1	1	yes	characteristic
179	C₂²	#16810575	1	2	yes	—
180	C₂²	#16810577	1	2	yes	—
181	C₄	—	106	4	no	—
182	C₄	—	212	1	no	—
183	C₄	—	212	1	no	—
184	C₄	#16810645	1	2	yes	—
185	C₄	#16810677	1	2	yes	—
186	C₄	#16810697	1	4	yes	—
187	C₄	#16810752	1	4	yes	—
188	C₂	#20273003	1	2	yes	—
189	C₂	#20273009	1	1	yes	characteristic
190	C₂	#20273019	1	1	yes	characteristic
191	C₂	#20274567	1	1	yes	characteristic
192	C₂	#20274588	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133146

C₂

yes

maximal

C₄×#13133127

C₂

yes

maximal, characteristic

C₂×#16810345

C₂

yes

maximal

#20272542

C₂

yes

maximal, characteristic

C₅₃×#1273

C₂

yes

maximal, characteristic

C₂×#13133146

C₄

yes

—

C₂×#13133127

C₄

yes

—

C₂×#13133127

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₄²×Dic₅₃

C₂²

yes

characteristic

#16810377

C₄

yes

—

#16810345

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#88322

Q₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×Dic₅₃

C₄²

yes

characteristic

C₂²×Dic₅₃

C₄²

yes

characteristic

C₂²×Dic₅₃

C₄²

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

M₁₆

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#88322

M₁₆

yes

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₄×Q₈

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#97

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂²×C₅₃

C₄×M₁₆

yes

characteristic

C₂²×C₅₃

C₄×M₁₆

yes

—

C₂²×C₅₃

#405

yes

—

C₄×C₅₃

#423

yes

—

C₄×C₅₃

#432

yes

—

C₄×C₅₃

#435

yes

—

C₄×C₅₃

#446

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₄×C₄⋊C₈

—

C₄×#97

—

C₂×#322

—

#1273

—

#1273

D₅₃

yes

characteristic

C₂×C₅₃

#1451

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

100

C₂×C₅₃

#1767

yes

characteristic

101

C₂×C₅₃

#1771

yes

—

102

C₂×C₄⋊C₈

—

103

C₂×#97

—

104

C₂×#97

—

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

C₂×D₅₃

yes

—

108

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

109

C₄³

—

110

#336

—

111

#322

—

106

—

112

C₅₃

#8263

yes

characteristic, complemented, Sylow

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₄×D₅₃

yes

—

116

C₂²×C₈

C₄×D₅₃

yes

—

117

C₂×C₄²

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

C₂²×D₅₃

yes

characteristic

122

C₄⋊C₈

—

106

—

123

#97

—

106

—

124

#97

—

106

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

106

—

131

C₄×C₈

—

106

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

136

C₂²×C₄

C₂×C₄×D₅₃

yes

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

106

—

145

C₂×C₈

—

106

—

146

C₂×C₈

—

106

—

147

C₂×C₈

—

106

—

148

C₂×C₈

#11719912

yes

—

149

C₄²

—

150

C₄²

—

106

—

151

C₄²

—

106

—

152

C₄²

—

106

—

153

C₄²

D₄×D₅₃

yes

characteristic

154

C₄²

#11719947

yes

—

155

C₄²

Q₈×D₅₃

yes

characteristic

156

C₂³

C₄²×D₅₃

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

106

—

159

C₂×C₄

—

106

—

160

C₂×C₄

—

106

—

161

C₂×C₄

—

106

—

162

C₂×C₄

—

106

—

163

C₂×C₄

—

106

—

164

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

165

C₂×C₄

#13133173

yes

characteristic

166

C₂×C₄

#13133177

yes

—

167

C₂×C₄

#13133182

yes

—

168

C₂×C₄

M₁₆×D₅₃

yes

—

169

C₂×C₄

M₁₆×D₅₃

yes

—

170

C₂×C₄

C₂×#11719912

yes

—

171

C₂×C₄

#13133223

yes

—

172

C₈

—

173

C₈

—

174

C₈

—

106

—

175

C₈

—

106

—

176

C₂²

#16810474

yes

characteristic

177

C₂²

#97×D₅₃

yes

characteristic

178

C₂²

C₄×#11719912

yes

characteristic

179

C₂²

#16810575

yes

—

180

C₂²

#16810577

yes

—

181

C₄

—

106

—

182

C₄

—

212

—

183

C₄

—

212

—

184

C₄

#16810645

yes

—

185

C₄

#16810677

yes

—

186

C₄

#16810697

yes

—

187

C₄

#16810752

yes

—

188

C₂

#20273003

yes

—

189

C₂

#20273009

yes

characteristic

190

C₂

#20273019

yes

characteristic

191

C₂

#20274567

yes

characteristic

192

C₂

#20274588

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂×C₄²
Fitting	C₅₃×#1273	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1273	D₅₃
hypocenter	C₅₃	#8263
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₅₃

commutator, socle, prime radical

C₂³×C₅₃

C₂×C₄²

Fitting

C₅₃×#1273

C₂

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
112. C₅₃	#8263	semidirect

H ≅ G/N

type

112. C₅₃

#8263

semidirect

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

subgroup	index
C₄×#13133146	2
C₄×#13133127	2
C₂×#16810345	2
#20272542	2
C₅₃×#1273	2
#8263	53

subgroup

index

C₅₃×#1273

Subgroups — #27275232

Conjugacy classes of subgroups

Complements

Maximal subgroups