Group #27284065 — subgroups

Subgroups — #27284065

Order 13568.

514 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 159 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	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273166	C₂	1	4	yes	maximal, complemented
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133153	—	2	2	no	—
6	#16810647	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	D₄⋊#88322	—	2	8	no	—
9	#16810344	—	2	2	no	—
10	C₂×#135×C₅₃	—	2	2	no	—
11	C₅₃×#532	C₂²	1	2	yes	—
12	C₅₃×#495	—	2	2	no	—
13	C₅₃×#494	C₄	1	1	yes	characteristic
14	C₅₃×#493	C₄	1	1	yes	characteristic
15	C₅₃×#492	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719917	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	#13133153	—	4	4	no	—
20	C₄⋊#88322	—	2	4	no	—
21	C₄×#88322	—	2	2	no	—
22	C₄×#88322	D₄	1	4	yes	—
23	C₂×#41×C₅₃	—	2	2	no	—
24	#135×C₅₃	—	4	4	no	—
25	#135×C₅₃	—	4	4	no	—
26	C₂×Q₁₆×C₅₃	—	2	2	no	—
27	C₂×SD₁₆×C₅₃	—	2	4	no	—
28	C₂×D₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	D₄	1	2	yes	—
30	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
31	#127×C₅₃	C₂×C₄	1	2	yes	—
32	#124×C₅₃	—	2	2	no	—
33	#121×C₅₃	—	2	4	no	—
34	C₄×D₄×C₅₃	—	2	4	no	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	#41×C₅₃	—	4	4	no	—
40	#41×C₅₃	—	4	4	no	—
41	C₂×Q₈×C₅₃	—	2	2	no	—
42	C₂×D₄×C₅₃	—	2	2	no	—
43	C₂×D₄×C₅₃	—	2	4	no	—
44	C₂×D₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	#41	1	2	yes	—
48	Q₁₆×C₅₃	—	4	4	no	—
49	SD₁₆×C₅₃	—	4	8	no	—
50	D₈×C₅₃	—	4	4	no	—
51	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₃	D₈	1	4	yes	—
54	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₂²⋊C₄×C₅₃	—	4	4	no	—
57	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×D₄	1	2	yes	—
60	#11719917	—	4	4	no	—
61	C₂×#88322	—	2	4	no	—
62	C₂×#88322	—	4	4	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂³×C₅₃	—	4	2	no	—
65	C₂³×C₅₃	#124	1	1	yes	characteristic
66	Q₈×C₅₃	—	4	4	no	—
67	D₄×C₅₃	—	4	4	no	—
68	D₄×C₅₃	—	4	8	no	—
69	D₄×C₅₃	—	8	4	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₅₃	#121	1	2	yes	—
77	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
80	C₈×C₅₃	#102	1	4	yes	—
81	C₈×C₅₃	#102	1	4	yes	—
82	#88322	—	4	4	no	—
83	#88322	—	8	2	no	—
84	#9862	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	8	2	no	—
87	C₂²×C₅₃	—	8	4	no	—
88	C₂²×C₅₃	#390	1	1	yes	characteristic
89	C₂²×C₅₃	#486	1	2	yes	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	8	2	no	—
92	C₄×C₅₃	C₂×#102	1	2	yes	—
93	C₄×C₅₃	#444	1	4	yes	—
94	C₄×C₅₃	#492	1	1	yes	characteristic
95	C₄×C₅₃	#493	1	1	yes	characteristic
96	#3147	D₅₃	1	1	yes	characteristic
97	#1848	—	53	1	no	—
98	#1492	—	53	4	no	—
99	C₂×#322	—	53	1	no	—
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	—	16	2	no	—
102	C₂×C₅₃	#1957	1	1	yes	characteristic
103	C₂×C₅₃	#1975	1	2	yes	—
104	C₂×#135	—	2	2	no	—
105	#532	C₂×D₅₃	1	2	yes	—
106	#495	—	2	2	no	—
107	#494	Dic₅₃	1	1	yes	characteristic
108	#493	Dic₅₃	1	1	yes	characteristic
109	#492	Dic₅₃	1	2	yes	—
110	C₂×#108	—	106	2	no	—
111	#423	—	53	2	no	—
112	C₂×C₄×C₈	—	53	2	no	—
113	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	D₄⋊C₈	—	106	8	no	—
115	#322	—	106	2	no	—
116	C₅₃	#9862	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	—	2	2	no	—
118	#135	—	4	4	no	—
119	#135	—	4	4	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×SD₁₆	—	2	4	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×M₁₆	—	106	2	no	—
124	C₂²×C₈	—	106	2	no	—
125	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
126	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
127	#127	C₂×Dic₅₃	1	2	yes	—
128	#124	—	2	2	no	—
129	#121	—	2	4	no	—
130	C₄×D₄	—	2	4	no	—
131	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
132	#108	—	212	4	no	—
133	C₄⋊C₈	—	106	4	no	—
134	C₄×C₈	—	53	4	no	—
135	C₄×C₈	—	106	2	no	—
136	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
139	#41	—	4	4	no	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	#11719947	1	2	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	Q₁₆	—	4	4	no	—
149	SD₁₆	—	4	8	no	—
150	D₈	—	4	4	no	—
151	M₁₆	—	212	4	no	—
152	C₂×C₈	—	106	4	no	—
153	C₂×C₈	—	212	2	no	—
154	C₂×C₈	—	212	4	no	—
155	C₂×C₈	#11719922	1	4	yes	—
156	C₂×C₈	#11719923	1	4	yes	—
157	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
158	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	D₄×D₅₃	1	2	yes	—
162	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
163	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₂³	—	4	2	no	—
165	C₂³	#13133204	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	D₄	—	4	4	no	—
168	D₄	—	4	8	no	—
169	D₄	—	8	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
176	C₂×C₄	#13133188	1	2	yes	—
177	C₂×C₄	#13133213	1	1	yes	characteristic
178	C₂×C₄	C₂×#11719922	1	2	yes	—
179	C₂×C₄	C₂×#11719923	1	2	yes	—
180	C₈	—	424	2	no	—
181	C₈	—	212	4	no	—
182	C₈	#13133138	1	4	yes	—
183	C₈	#13133138	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	2	no	—
186	C₂²	—	8	4	no	—
187	C₂²	#16810526	1	1	yes	characteristic
188	C₂²	#16810969	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	8	2	no	—
191	C₄	C₂×#13133138	1	2	yes	—
192	C₄	#16810745	1	4	yes	—
193	C₄	#16810995	1	1	yes	characteristic
194	C₄	#16810999	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	2	no	—
197	C₂	#20275522	1	1	yes	characteristic
198	C₂	#20275609	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274887

C₂

yes

maximal, characteristic, complemented

#20273166

C₂

yes

maximal, complemented

C₂×#16810344

C₂

yes

maximal, characteristic, complemented

C₅₃×#3147

C₂

yes

maximal, characteristic

C₂×#13133153

—

#16810647

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

D₄⋊#88322

—

#16810344

—

C₂×#135×C₅₃

—

C₅₃×#532

C₂²

yes

—

C₅₃×#495

—

C₅₃×#494

C₄

yes

characteristic

C₅₃×#493

C₄

yes

characteristic

C₅₃×#492

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×#11719917

—

C₂²×#88322

—

—

—

—

D₄

yes

—

C₂×#41×C₅₃

—

#135×C₅₃

—

#135×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

#127×C₅₃

C₂×C₄

yes

—

#124×C₅₃

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

Q₁₆×C₅₃

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

#124

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

#102

yes

—

C₈×C₅₃

#102

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#486

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#102

yes

—

C₄×C₅₃

#444

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

—

—

—

100

C₂×C₅₃

—

101

C₂×C₅₃

—

102

C₂×C₅₃

#1957

yes

characteristic

103

C₂×C₅₃

#1975

yes

—

104

C₂×#135

—

105

#532

C₂×D₅₃

yes

—

106

#495

—

107

#494

Dic₅₃

yes

characteristic

108

#493

Dic₅₃

yes

characteristic

109

#492

Dic₅₃

yes

—

110

C₂×#108

—

106

—

111

#423

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

114

D₄⋊C₈

—

106

—

115

#322

—

106

—

116

C₅₃

#9862

yes

characteristic, complemented, Sylow

117

C₂×#41

—

118

#135

—

119

#135

—

120

C₂×Q₁₆

—

121

C₂×SD₁₆

—

122

C₂×D₈

—

123

C₂×M₁₆

—

106

—

124

C₂²×C₈

—

106

—

125

C₂²×C₈

C₂²⋊D₅₃

yes

—

126

C₄⋊Q₈

C₂×Dic₅₃

yes

—

127

#127

C₂×Dic₅₃

yes

—

128

#124

—

129

#121

—

130

C₄×D₄

—

131

C₂×C₄²

C₂²×D₅₃

yes

characteristic

132

#108

—

212

—

133

C₄⋊C₈

—

106

—

134

C₄×C₈

—

135

C₄×C₈

—

106

—

136

C₄×C₈

C₂×Dic₅₃

yes

characteristic

137

C₄×C₈

C₂×Dic₅₃

yes

characteristic

138

C₄×C₈

C₂²⋊D₅₃

yes

—

139

#41

—

140

#41

—

141

C₂×Q₈

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

#11719947

yes

—

147

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

148

Q₁₆

—

149

SD₁₆

—

150

D₈

—

151

M₁₆

—

212

—

152

C₂×C₈

—

106

—

153

C₂×C₈

—

212

—

154

C₂×C₈

—

212

—

155

C₂×C₈

#11719922

yes

—

156

C₂×C₈

#11719923

yes

—

157

C₂×C₈

C₂²⋊Dic₅₃

yes

—

158

C₂×C₈

C₂²⋊Dic₅₃

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

D₄×D₅₃

yes

—

162

C₄²

C₂²×Dic₅₃

yes

characteristic

163

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

164

C₂³

—

165

C₂³

#13133204

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

175

C₂×C₄

D₄×Dic₅₃

yes

—

176

C₂×C₄

#13133188

yes

—

177

C₂×C₄

#13133213

yes

characteristic

178

C₂×C₄

C₂×#11719922

yes

—

179

C₂×C₄

C₂×#11719923

yes

—

180

C₈

—

424

—

181

C₈

—

212

—

182

C₈

#13133138

yes

—

183

C₈

#13133138

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#16810526

yes

characteristic

188

C₂²

#16810969

yes

—

189

C₄

—

190

C₄

—

191

C₄

C₂×#13133138

yes

—

192

C₄

#16810745

yes

—

193

C₄

#16810995

yes

characteristic

194

C₄

#16810999

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#20275522

yes

characteristic

198

C₂

#20275609

yes

—

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3147	C₂
socle	C₂²×C₅₃	#390
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3147	D₅₃
hypocenter	C₅₃	#9862
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting, prime radical

C₅₃×#3147

C₂

socle

C₂²×C₅₃

#390

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

N	H ≅ G/N	type
1. #20274887	C₂	semidirect
2. #20273166	C₂	semidirect
3. C₂×#16810344	C₂	semidirect
116. C₅₃	#9862	semidirect

H ≅ G/N

type

1. #20274887

C₂

semidirect

2. #20273166

C₂

semidirect

3. C₂×#16810344

C₂

semidirect

116. C₅₃

#9862

semidirect

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

subgroup	index
#20274887	2
#20273166	2
C₂×#16810344	2
C₅₃×#3147	2
#9862	53

subgroup

index

C₅₃×#3147

Subgroups — #27284065

Conjugacy classes of subgroups

Complements

Maximal subgroups