Group #27282851 — subgroups

Subgroups — #27282851

Order 13568.

268 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20273504	C₂	1	2	yes	maximal
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	C₈×#11719917	C₂	1	1	yes	maximal, characteristic
4	#20272606	C₂	1	2	yes	maximal
5	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	1	yes	characteristic
7	#16810647	C₂²	1	2	yes	—
8	C₄×#11719917	C₂²	1	1	yes	characteristic
9	#16810376	—	2	2	no	—
10	#16810376	—	2	1	no	—
11	#16810376	—	2	1	no	—
12	#16810374	—	2	2	no	—
13	#16810345	—	2	1	no	—
14	C₈×#88322	—	2	1	no	—
15	C₅₃×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	C₄	1	2	yes	—
18	C₅₃×#334	C₄	1	2	yes	—
19	C₂×#11719917	—	2	1	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	D₄	1	1	yes	characteristic
24	C₄⋊#88322	D₄	1	1	yes	characteristic
25	C₂²⋊#88322	—	2	1	no	—
26	#13133128	Q₈	1	2	yes	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	—	2	1	no	—
29	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²⋊C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	—	4	1	no	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	#11719917	—	2	4	no	—
54	#11719917	—	8	2	no	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	2	1	no	—
57	C₂×#88322	—	2	1	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂³×C₅₃	#125	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	—	2	1	no	—
73	C₈×C₅₃	—	2	1	no	—
74	C₈×C₅₃	—	4	4	no	—
75	C₈×C₅₃	#108	1	2	yes	—
76	#88322	—	2	4	no	—
77	#88322	—	4	1	no	—
78	#88322	—	4	1	no	—
79	#88322	—	8	2	no	—
80	#9634	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	—	2	1	no	—
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	#389	1	1	yes	characteristic
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
89	C₄×C₅₃	#430	1	1	yes	characteristic
90	C₄×C₅₃	#443	1	2	yes	—
91	C₄×C₅₃	#472	1	1	yes	characteristic
92	C₄×C₅₃	#473	1	1	yes	characteristic
93	#1574	—	53	2	no	—
94	#1567	—	53	1	no	—
95	#1564	D₅₃	1	1	yes	characteristic
96	C₈×M₁₆	—	53	1	no	—
97	#1299	—	53	2	no	—
98	C₂×C₅₃	—	4	1	no	—
99	C₂×C₅₃	#1945	1	1	yes	characteristic
100	C₂×C₅₃	#1950	1	1	yes	characteristic
101	C₂×C₅₃	#1952	1	1	yes	characteristic
102	#431	—	53	1	no	—
103	#431	C₂×D₅₃	1	2	yes	—
104	#423	—	53	2	no	—
105	C₄×M₁₆	—	53	1	no	—
106	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
107	#335	—	106	2	no	—
108	#335	—	106	1	no	—
109	#335	—	106	1	no	—
110	#335	Dic₅₃	1	2	yes	—
111	#334	—	106	2	no	—
112	#334	Dic₅₃	1	2	yes	—
113	#322	—	106	1	no	—
114	C₈²	—	106	1	no	—
115	C₅₃	#9634	1	1	yes	characteristic, complemented, Sylow
116	C₂×M₁₆	—	106	1	no	—
117	C₂×M₁₆	—	106	2	no	—
118	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
119	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
121	C₄⋊C₈	—	53	1	no	—
122	C₄⋊C₈	—	53	1	no	—
123	C₄⋊C₈	—	106	2	no	—
124	C₄⋊C₈	—	106	2	no	—
125	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	106	1	no	—
128	#97	—	53	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	106	1	no	—
131	C₄×C₈	—	106	1	no	—
132	C₄×C₈	D₂₁₂	1	1	yes	characteristic
133	C₄×C₈	D₂₁₂	1	1	yes	characteristic
134	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
135	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
137	C₂²×C₄	#11719947	1	1	yes	characteristic
138	C₂²×C₄	#11719949	1	1	yes	characteristic
139	M₁₆	—	106	4	no	—
140	M₁₆	—	424	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	106	2	no	—
145	C₂×C₈	—	106	1	no	—
146	C₂×C₈	—	106	1	no	—
147	C₂×C₈	—	212	2	no	—
148	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
149	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
150	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
151	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
152	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
153	C₄²	D₄×D₅₃	1	1	yes	characteristic
154	C₄²	Q₈×D₅₃	1	1	yes	characteristic
155	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
156	C₂³	#13133208	1	1	yes	characteristic
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
163	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
165	C₂×C₄	#13133192	1	1	yes	characteristic
166	C₂×C₄	#13133197	1	1	yes	characteristic
167	C₂×C₄	#13133216	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	4	4	no	—
172	C₈	—	106	4	no	—
173	C₈	—	424	2	no	—
174	C₈	—	212	1	no	—
175	C₈	—	212	1	no	—
176	C₈	#13133153	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	#16810522	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₂×#13133153	1	1	yes	characteristic
185	C₄	#16810672	1	1	yes	characteristic
186	C₄	#16810740	1	2	yes	—
187	C₄	#16810894	1	1	yes	characteristic
188	C₄	#16810899	1	1	yes	characteristic
189	C₂	—	4	1	no	—
190	C₂	#20275464	1	1	yes	characteristic
191	C₂	#20275491	1	1	yes	characteristic
192	C₂	#20275501	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273504

C₂

yes

maximal

#20273474

C₂

yes

maximal, characteristic

C₈×#11719917

C₂

yes

maximal, characteristic

#20272606

C₂

yes

maximal

C₅₃×#1564

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

characteristic

#16810647

C₂²

yes

—

C₄×#11719917

C₂²

yes

—

—

—

—

—

—

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

C₄

yes

—

C₅₃×#334

C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄⋊#88322

D₄

yes

characteristic

C₂²⋊#88322

—

#13133128

Q₈

yes

—

C₄×#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

C₂²×C₄×C₅₃

yes

C₂²×C₄×C₅₃

yes

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

—

—

—

—

—

C₂³×C₅₃

#125

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#108

yes

—

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#389

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

D₅₃

yes

characteristic

C₈×M₁₆

—

#1299

—

C₂×C₅₃

—

C₂×C₅₃

#1945

yes

characteristic

100

C₂×C₅₃

#1950

yes

characteristic

101

C₂×C₅₃

#1952

yes

characteristic

102

#431

—

103

#431

C₂×D₅₃

yes

—

104

#423

—

105

C₄×M₁₆

—

106

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

107

#335

—

106

—

108

#335

—

106

—

109

#335

—

106

—

110

#335

Dic₅₃

yes

—

111

#334

—

106

—

112

#334

Dic₅₃

yes

—

113

#322

—

106

—

114

C₈²

—

106

—

115

C₅₃

#9634

yes

characteristic, complemented, Sylow

116

C₂×M₁₆

—

106

—

117

C₂×M₁₆

—

106

—

118

C₂²×C₈

Dic₁₀₆

yes

characteristic

119

C₂²×C₈

Dic₁₀₆

yes

characteristic

120

C₂×C₄²

C₂²×D₅₃

yes

characteristic

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

106

—

124

C₄⋊C₈

—

106

—

125

C₄⋊C₈

C₂×Dic₅₃

yes

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

106

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

—

106

—

131

C₄×C₈

—

106

—

132

C₄×C₈

D₂₁₂

yes

characteristic

133

C₄×C₈

D₂₁₂

yes

characteristic

134

C₄×C₈

C₂×Dic₅₃

yes

characteristic

135

C₄×C₈

C₂×Dic₅₃

yes

characteristic

136

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

137

C₂²×C₄

#11719947

yes

characteristic

138

C₂²×C₄

#11719949

yes

characteristic

139

M₁₆

—

106

—

140

M₁₆

—

424

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

106

—

145

C₂×C₈

—

106

—

146

C₂×C₈

—

106

—

147

C₂×C₈

—

212

—

148

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

149

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

150

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

151

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

152

C₄²

C₂×D₂₁₂

yes

characteristic

153

C₄²

D₄×D₅₃

yes

characteristic

154

C₄²

Q₈×D₅₃

yes

characteristic

155

C₄²

C₂²×Dic₅₃

yes

characteristic

156

C₂³

#13133208

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

163

C₂×C₄

D₄×Dic₅₃

yes

characteristic

164

C₂×C₄

Q₈×Dic₅₃

yes

characteristic

165

C₂×C₄

#13133192

yes

characteristic

166

C₂×C₄

#13133197

yes

characteristic

167

C₂×C₄

#13133216

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

106

—

173

C₈

—

424

—

174

C₈

—

212

—

175

C₈

—

212

—

176

C₈

#13133153

yes

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

#16810522

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

C₂×#13133153

yes

characteristic

185

C₄

#16810672

yes

characteristic

186

C₄

#16810740

yes

—

187

C₄

#16810894

yes

characteristic

188

C₄

#16810899

yes

characteristic

189

C₂

—

190

C₂

#20275464

yes

characteristic

191

C₂

#20275491

yes

characteristic

192

C₂

#20275501

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	#125
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#9634
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

subgroup

G/H

center

C₂×C₄

#13133192

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1564

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

N	H ≅ G/N	type
115. C₅₃	#9634	semidirect

H ≅ G/N

type

115. C₅₃

#9634

semidirect

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

subgroup	index
#20273504	2
#20273474	2
C₈×#11719917	2
#20272606	2
C₅₃×#1564	2
#9634	53

subgroup

index

C₅₃×#1564

Subgroups — #27282851

Conjugacy classes of subgroups

Complements

Maximal subgroups