Group #27272366 — subgroups

Subgroups — #27272366

Order 13568.

468 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20274464	C₂	1	2	yes	maximal
2	#20272541	C₂	1	4	yes	maximal
3	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133128	—	2	2	no	—
5	#16810378	—	2	2	no	—
6	#16810378	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
9	C₅₃×#391	C₂²	1	2	yes	—
10	C₅₃×#389	—	2	2	no	—
11	C₅₃×#384	C₄	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#88322	—	2	4	no	—
15	#13133128	—	2	8	no	—
16	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₄⋊Q₈×C₅₃	—	2	4	no	—
18	C₄×Q₈×C₅₃	—	4	4	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	#95×C₅₃	—	2	4	no	—
30	C₂×Q₈×C₅₃	—	2	2	no	—
31	C₂×Q₈×C₅₃	—	4	2	no	—
32	C₂×Q₈×C₅₃	—	4	2	no	—
33	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₅₃	—	2	8	no	—
42	C₄⋊C₄×C₅₃	—	4	4	no	—
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₄⋊C₄×C₅₃	—	4	4	no	—
45	C₄²×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	—	2	4	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	4	no	—
49	C₄²×C₅₃	—	4	1	no	—
50	C₄²×C₅₃	—	4	4	no	—
51	C₂×#88322	—	2	8	no	—
52	C₂×#88322	—	4	4	no	—
53	C₂×#88322	—	4	4	no	—
54	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	Q₈×C₅₃	—	4	4	no	—
56	Q₈×C₅₃	—	4	4	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	4	no	—
71	C₂×C₄×C₅₃	#100	1	2	yes	—
72	C₂×C₄×C₅₃	#104	1	4	yes	—
73	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₅₃	#137	1	4	yes	—
75	#88322	—	4	8	no	—
76	#7639	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	#407	1	1	yes	characteristic
78	C₂²×C₅₃	#410	1	1	yes	characteristic
79	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
80	C₂²×C₅₃	#419	1	2	yes	—
81	C₂²×C₅₃	C₂×#104	1	2	yes	—
82	C₄×C₅₃	—	2	2	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	—	2	4	no	—
86	C₄×C₅₃	—	4	4	no	—
87	C₄×C₅₃	—	4	1	no	—
88	C₄×C₅₃	—	8	2	no	—
89	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
90	#1745	—	53	2	no	—
91	#1272	—	53	4	no	—
92	C₂×C₅₃	#1466	1	1	yes	characteristic
93	C₂×C₅₃	#1491	1	2	yes	—
94	C₂×C₅₃	#1506	1	2	yes	—
95	C₂×C₅₃	#1525	1	1	yes	characteristic
96	C₂×C₅₃	#1558	1	1	yes	characteristic
97	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
98	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
99	C₂×#97	—	106	2	no	—
100	#391	C₂×D₅₃	1	2	yes	—
101	#389	—	2	2	no	—
102	#384	Dic₅₃	1	1	yes	characteristic
103	C₄×C₄⋊C₄	—	2	2	no	—
104	C₄³	C₂×D₅₃	1	1	yes	characteristic
105	#336	—	53	4	no	—
106	#336	—	106	2	no	—
107	C₅₃	#7639	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
109	C₂²×C₈	—	106	4	no	—
110	C₄⋊Q₈	—	2	4	no	—
111	C₄×Q₈	—	4	4	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
116	C₂×C₄²	—	2	2	no	—
117	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
119	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
120	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
121	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	#97	—	106	8	no	—
123	#95	—	2	4	no	—
124	C₂×Q₈	—	2	2	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	#11719917	1	4	yes	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
131	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
133	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
135	C₂×C₈	—	106	8	no	—
136	C₂×C₈	—	212	4	no	—
137	C₂×C₈	—	212	4	no	—
138	C₄⋊C₄	—	2	8	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	4	1	no	—
147	C₄²	—	4	4	no	—
148	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
149	Q₈	—	4	4	no	—
150	Q₈	—	4	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	#13133134	1	2	yes	—
166	C₂×C₄	#13133144	1	4	yes	—
167	C₂×C₄	C₂×#11719917	1	2	yes	—
168	C₂×C₄	#13133248	1	4	yes	—
169	C₈	—	212	8	no	—
170	C₂²	#16810587	1	1	yes	characteristic
171	C₂²	#16810597	1	1	yes	characteristic
172	C₂²	C₂×#13133134	1	1	yes	characteristic
173	C₂²	#16810632	1	2	yes	—
174	C₂²	C₂×#13133144	1	2	yes	—
175	C₄	—	2	4	no	—
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	2	2	no	—
179	C₄	—	4	1	no	—
180	C₄	—	4	4	no	—
181	C₄	—	8	2	no	—
182	C₂	#20273056	1	1	yes	characteristic
183	C₂	#20273162	1	2	yes	—
184	C₂	#20273227	1	2	yes	—
185	C₂	#20273307	1	1	yes	characteristic
186	C₂	#20273441	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274464

C₂

yes

maximal

#20272541

C₂

yes

maximal

C₄×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133128

—

#16810378

—

#16810378

C₂²

yes

—

C₂×C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×C₄×Q₈×C₅₃

C₄

yes

—

C₅₃×#391

C₂²

yes

—

C₅₃×#389

—

C₅₃×#384

C₄

yes

characteristic

C₄×C₄⋊C₄×C₅₃

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

#13133128

—

C₂²×Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#100

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#137

yes

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#104

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₄⋊Q₈

D₅₃

yes

characteristic

#1745

—

#1272

—

C₂×C₅₃

#1466

yes

characteristic

C₂×C₅₃

#1491

yes

—

C₂×C₅₃

#1506

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

Dic₅₃

yes

characteristic

C₂×C₄×Q₈

Dic₅₃

yes

—

C₂×#97

—

106

—

100

#391

C₂×D₅₃

yes

—

101

#389

—

102

#384

Dic₅₃

yes

characteristic

103

C₄×C₄⋊C₄

—

104

C₄³

C₂×D₅₃

yes

characteristic

105

#336

—

106

#336

—

106

—

107

C₅₃

#7639

yes

characteristic, complemented, Sylow

108

C₂²×Q₈

C₂×Dic₅₃

yes

—

109

C₂²×C₈

—

106

—

110

C₄⋊Q₈

—

111

C₄×Q₈

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

118

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

119

C₂×C₄²

C₂²⋊D₅₃

yes

—

120

C₂×C₄²

C₂²⋊D₅₃

yes

—

121

C₂×C₄²

C₂²×D₅₃

yes

characteristic

122

#97

—

106

—

123

#95

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

#11719917

yes

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

131

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

132

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

133

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

134

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

135

C₂×C₈

—

106

—

136

C₂×C₈

—

212

—

137

C₂×C₈

—

212

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

149

Q₈

—

150

Q₈

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#13133134

yes

—

166

C₂×C₄

#13133144

yes

—

167

C₂×C₄

C₂×#11719917

yes

—

168

C₂×C₄

#13133248

yes

—

169

C₈

—

212

—

170

C₂²

#16810587

yes

characteristic

171

C₂²

#16810597

yes

characteristic

172

C₂²

C₂×#13133134

yes

characteristic

173

C₂²

#16810632

yes

—

174

C₂²

C₂×#13133144

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₂

#20273056

yes

characteristic

183

C₂

#20273162

yes

—

184

C₂

#20273227

yes

—

185

C₂

#20273307

yes

characteristic

186

C₂

#20273441

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#7639
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₄×C₄⋊Q₈×C₅₃

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

hypercenter

C₄×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#7639

LCS

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

N	H ≅ G/N	type
107. C₅₃	#7639	semidirect

H ≅ G/N

type

107. C₅₃

#7639

semidirect

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

subgroup	index
#20274464	2
#20272541	2
C₄×C₄⋊Q₈×C₅₃	2
#7639	53

subgroup

index

#20274464

#20272541

C₄×C₄⋊Q₈×C₅₃

#7639

Subgroups — #27272366

Conjugacy classes of subgroups

Complements

Maximal subgroups