Group #27302231 — subgroups

Subgroups — #27302231

Order 13568.

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

normal subgroups only

This group has 310 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	C₂×#16810445	C₂	1	2	yes	maximal
2	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
3	#20274594	C₂	1	1	yes	maximal, characteristic
4	#20272822	C₂	1	2	yes	maximal
5	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133153	C₂²	1	1	yes	characteristic
8	#16810661	C₂²	1	1	yes	characteristic
9	#16810578	—	2	1	no	—
10	#16810445	—	2	4	no	—
11	#16810444	C₄	1	4	yes	complemented
12	#16810383	—	2	1	no	—
13	C₂×#106×C₅₃	—	2	1	no	—
14	C₂×#106×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#384	—	2	1	no	—
17	C₂×#11719917	—	2	1	no	—
18	#13133176	—	2	1	no	—
19	#13133157	—	2	2	no	—
20	#13133157	C₂×C₄	1	4	yes	—
21	C₂×#11719908	—	2	2	no	—
22	#13133153	—	2	1	no	—
23	#13133153	D₄	1	2	yes	—
24	#13133151	Q₈	1	2	yes	—
25	#13133149	—	2	1	no	—
26	#13133127	—	4	1	no	—
27	C₈×Dic₅₃	—	4	1	no	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	#106×C₅₃	—	2	2	no	—
34	#106×C₅₃	—	2	2	no	—
35	#106×C₅₃	—	4	2	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	—	4	2	no	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
43	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
50	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₄⋊C₄×C₅₃	—	8	4	no	—
54	C₄²×C₅₃	—	2	1	no	—
55	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂²⋊Dic₅₃	—	4	1	no	—
57	C₄⋊Dic₅₃	—	2	2	no	—
58	C₄×Dic₅₃	—	4	1	no	—
59	#11719917	—	2	2	no	—
60	#11719917	—	4	1	no	—
61	C₂×#88322	—	4	1	no	—
62	#11719908	—	2	4	no	—
63	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	8	4	no	—
67	C₂×C₄×C₅₃	—	8	2	no	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
74	C₂×Dic₅₃	—	4	2	no	—
75	C₈×C₅₃	#106	1	2	yes	—
76	C₈×C₅₃	#106	1	2	yes	—
77	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
78	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₅₃	#125	1	1	yes	characteristic
80	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
81	#88322	—	4	2	no	—
82	#13186	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	#428	1	1	yes	characteristic
85	C₂²×C₅₃	#460	1	1	yes	characteristic
86	C₂²×C₅₃	#476	1	1	yes	characteristic
87	C₄×C₅₃	—	2	4	no	—
88	C₄×C₅₃	—	16	2	no	—
89	C₄×C₅₃	#389	1	1	yes	characteristic
90	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
91	C₄×C₅₃	#461	1	1	yes	characteristic
92	C₄×C₅₃	#475	1	1	yes	characteristic
93	Dic₅₃	—	8	2	no	—
94	C₂×#365	—	53	2	no	—
95	C₂×#364	—	53	1	no	—
96	#1844	D₅₃	1	1	yes	characteristic
97	#1772	—	53	1	no	—
98	#1386	—	53	2	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	#1948	1	1	yes	characteristic
102	C₂×C₅₃	#2230	1	2	yes	—
103	C₂×M₃₂	—	53	2	no	—
104	C₂×#108	—	53	1	no	—
105	#427	—	53	1	no	—
106	C₂×#106	—	2	1	no	—
107	C₂×#106	C₂×D₅₃	1	2	yes	—
108	#405	—	106	1	no	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	#384	—	2	1	no	—
111	#365	—	106	4	no	—
112	#364	—	53	4	no	—
113	#339	—	106	1	no	—
114	C₅₃	#13186	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	—	106	1	no	—
116	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	#117	—	106	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	2	no	—
121	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
122	M₃₂	—	53	4	no	—
123	M₃₂	—	106	2	no	—
124	C₂×C₁₆	—	106	2	no	—
125	#108	—	53	2	no	—
126	#108	—	106	1	no	—
127	#107	—	53	2	no	—
128	#106	—	2	2	no	—
129	#106	—	2	2	no	—
130	#106	—	4	2	no	—
131	#106	—	106	1	no	—
132	#97	—	212	1	no	—
133	C₄×C₈	—	212	1	no	—
134	C₄×C₈	Dic₁₀₆	1	2	yes	—
135	C₄×C₈	C₄×D₅₃	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	M₁₆	—	106	2	no	—
140	M₁₆	—	212	1	no	—
141	C₂×C₈	—	212	1	no	—
142	C₂×C₈	#11719919	1	2	yes	—
143	C₂×C₈	#11719923	1	2	yes	—
144	C₂×C₈	#11719924	1	1	yes	characteristic
145	C₂×C₈	#11719924	1	1	yes	characteristic
146	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
147	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
149	C₂×C₈	#11719947	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
151	C₂×C₈	#11719949	1	1	yes	characteristic
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	8	4	no	—
155	C₄⋊C₄	—	106	2	no	—
156	C₂²⋊C₄	—	212	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	212	1	no	—
159	C₄²	#11719919	1	2	yes	—
160	C₁₆	—	106	4	no	—
161	C₂³	#13133243	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	212	2	no	—
167	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
168	C₂×C₄	#13133188	1	1	yes	characteristic
169	C₂×C₄	#13133193	1	1	yes	characteristic
170	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
172	C₂×C₄	#13133248	1	1	yes	characteristic
173	C₈	—	212	2	no	—
174	C₈	#13133148	1	2	yes	—
175	C₈	#13133148	1	2	yes	—
176	C₈	#13133179	1	1	yes	characteristic
177	C₈	#13133182	1	1	yes	characteristic
178	C₈	#13133208	1	1	yes	characteristic
179	C₈	#13133216	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	#16810663	1	1	yes	characteristic
182	C₂²	#16810820	1	1	yes	characteristic
183	C₂²	#16810910	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	16	2	no	—
186	C₄	—	424	2	no	—
187	C₄	#16810519	1	1	yes	characteristic
188	C₄	C₂×#13133148	1	1	yes	characteristic
189	C₄	#16810825	1	1	yes	characteristic
190	C₄	#16810905	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#20275476	1	1	yes	characteristic
194	C₂	#20276883	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810445

C₂

yes

maximal

C₂×#16810444

C₂

yes

maximal, characteristic

#20274594

C₂

yes

maximal, characteristic

#20272822

C₂

yes

maximal

C₅₃×#1844

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×#13133153

C₂²

yes

characteristic

#16810661

C₂²

yes

—

—

C₄

yes

complemented

#16810383

—

C₂×#106×C₅₃

—

C₂×#106×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

—

—

—

—

C₂×C₄

yes

—

C₂×#11719908

—

#13133153

—

#13133153

D₄

yes

—

#13133151

Q₈

yes

—

#13133149

—

#13133127

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

#106×C₅₃

—

#106×C₅₃

—

#106×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

characteristic

C₂×C₈×C₅₃

SD₁₆

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

C₂³×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

#137

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

C₄×D₄

yes

characteristic

C₈×C₅₃

C₄×Q₈

yes

characteristic

C₈×C₅₃

#125

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

C₂×#365

—

C₂×#364

—

#1844

D₅₃

yes

characteristic

#1772

—

#1386

—

C₂×C₅₃

—

100

C₂×C₅₃

—

101

C₂×C₅₃

#1948

yes

characteristic

102

C₂×C₅₃

#2230

yes

—

103

C₂×M₃₂

—

104

C₂×#108

—

105

#427

—

106

C₂×#106

—

107

C₂×#106

C₂×D₅₃

yes

—

108

#405

—

106

—

109

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

110

#384

—

111

#365

—

106

—

112

#364

—

113

#339

—

106

—

114

C₅₃

#13186

yes

characteristic, complemented, Sylow

115

C₂×M₁₆

—

106

—

116

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

117

C₂²×C₈

C₂²×D₅₃

yes

characteristic

118

#117

—

106

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

122

M₃₂

—

123

M₃₂

—

106

—

124

C₂×C₁₆

—

106

—

125

#108

—

126

#108

—

106

—

127

#107

—

128

#106

—

129

#106

—

130

#106

—

131

#106

—

106

—

132

#97

—

212

—

133

C₄×C₈

—

212

—

134

C₄×C₈

Dic₁₀₆

yes

—

135

C₄×C₈

C₄×D₅₃

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

139

M₁₆

—

106

—

140

M₁₆

—

212

—

141

C₂×C₈

—

212

—

142

C₂×C₈

#11719919

yes

—

143

C₂×C₈

#11719923

yes

—

144

C₂×C₈

#11719924

yes

characteristic

145

C₂×C₈

#11719924

yes

characteristic

146

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

147

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

148

C₂×C₈

D₄×D₅₃

yes

characteristic

149

C₂×C₈

#11719947

yes

characteristic

150

C₂×C₈

Q₈×D₅₃

yes

characteristic

151

C₂×C₈

#11719949

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

106

—

156

C₂²⋊C₄

—

212

—

157

C₄²

—

158

C₄²

—

212

—

159

C₄²

#11719919

yes

—

160

C₁₆

—

106

—

161

C₂³

#13133243

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

212

—

167

C₂×C₄

C₂×#11719919

yes

characteristic

168

C₂×C₄

#13133188

yes

characteristic

169

C₂×C₄

#13133193

yes

characteristic

170

C₂×C₄

C₂×#11719924

yes

characteristic

171

C₂×C₄

C₂×#11719923

yes

characteristic

172

C₂×C₄

#13133248

yes

characteristic

173

C₈

—

212

—

174

C₈

#13133148

yes

—

175

C₈

#13133148

yes

—

176

C₈

#13133179

yes

characteristic

177

C₈

#13133182

yes

characteristic

178

C₈

#13133208

yes

characteristic

179

C₈

#13133216

yes

characteristic

180

C₂²

—

181

C₂²

#16810663

yes

characteristic

182

C₂²

#16810820

yes

characteristic

183

C₂²

#16810910

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

—

424

—

187

C₄

#16810519

yes

characteristic

188

C₄

C₂×#13133148

yes

characteristic

189

C₄

#16810825

yes

characteristic

190

C₄

#16810905

yes

characteristic

191

C₂

—

192

C₂

—

193

C₂

#20275476

yes

characteristic

194

C₂

#20276883

yes

—

	subgroup	G/H
center	C₂²	#16810910
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1844	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1844	D₅₃
hypocenter	C₅₃	#13186
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

subgroup

G/H

center

C₂²

#16810910

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1844

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

N	H ≅ G/N	type
11. #16810444	C₄	semidirect
114. C₅₃	#13186	semidirect

H ≅ G/N

type

11. #16810444

C₄

semidirect

114. C₅₃

#13186

semidirect

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

subgroup	index
C₂×#16810445	2
C₂×#16810444	2
#20274594	2
#20272822	2
C₅₃×#1844	2
#13186	53

subgroup

index

C₅₃×#1844

Subgroups — #27302231

Conjugacy classes of subgroups

Complements

Maximal subgroups