Group #27301964 — subgroups

Subgroups — #27301964

Order 13568.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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	1	yes	maximal, characteristic
2	C₂×#16810440	C₂	1	1	yes	maximal, characteristic
3	#20276434	C₂	1	1	yes	maximal, characteristic
4	#20275584	C₂	1	1	yes	maximal, characteristic
5	#20274601	C₂	1	1	yes	maximal, characteristic
6	#20272794	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1386	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	1	yes	characteristic
10	C₂×#13133153	C₂²	1	1	yes	characteristic
11	C₂×#13133140	—	2	1	no	—
12	#16810579	—	2	1	no	—
13	C₂×#13133127	C₂²	1	1	yes	characteristic
14	#16810528	—	2	1	no	—
15	#16810445	—	2	2	no	—
16	#16810440	—	2	2	no	—
17	#16810426	C₄	1	4	yes	complemented
18	#16810382	—	2	1	no	—
19	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#427	C₂²	1	1	yes	characteristic
21	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₀₆	—	2	1	no	—
23	C₂×Dic₂₁₂	—	2	1	no	—
24	C₂×Dic₂₁₂	—	2	1	no	—
25	C₂×Dic₂₁₂	—	2	1	no	—
26	C₂×Dic₂₁₂	—	2	1	no	—
27	C₂×Dic₂₁₂	—	2	1	no	—
28	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
29	C₂×#11719917	—	2	1	no	—
30	C₂²×#88322	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
32	#13133157	—	2	1	no	—
33	#13133157	—	2	1	no	—
34	C₂×#11719908	C₂×C₄	1	2	yes	—
35	#13133153	—	2	1	no	—
36	#13133153	D₄	1	2	yes	—
37	#13133140	—	4	2	no	—
38	#13133128	—	4	1	no	—
39	#13133127	—	2	1	no	—
40	#13133127	—	2	1	no	—
41	C₄×#88322	—	4	1	no	—
42	#13133124	—	4	2	no	—
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	#117×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
46	M₃₂×C₅₃	—	2	1	no	—
47	M₃₂×C₅₃	—	2	1	no	—
48	C₂×C₁₆×C₅₃	—	2	1	no	—
49	#107×C₅₃	C₂×C₄	1	2	yes	—
50	#106×C₅₃	—	2	1	no	—
51	#106×C₅₃	—	2	1	no	—
52	#106×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	4	1	no	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	1	no	—
56	C₂²×Dic₅₃	—	4	1	no	—
57	C₂×Dic₁₀₆	—	2	2	no	—
58	C₂×Dic₁₀₆	—	4	1	no	—
59	C₂×Dic₁₀₆	—	4	1	no	—
60	C₂×Dic₁₀₆	—	4	1	no	—
61	C₂×Dic₁₀₆	—	4	1	no	—
62	C₂×Dic₁₀₆	—	4	1	no	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
66	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	—	4	1	no	—
71	C₄⋊C₄×C₅₃	—	4	1	no	—
72	C₂²⋊C₄×C₅₃	—	4	1	no	—
73	C₄²×C₅₃	—	4	1	no	—
74	C₄×Dic₅₃	—	2	1	no	—
75	C₄×Dic₅₃	—	2	1	no	—
76	#11719917	—	2	2	no	—
77	#11719917	—	4	1	no	—
78	C₂×#88322	—	2	1	no	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	4	1	no	—
81	C₂×#88322	SD₁₆	1	2	yes	—
82	Dic₂₁₂	—	2	2	no	—
83	Dic₂₁₂	—	2	2	no	—
84	Dic₂₁₂	—	4	1	no	—
85	Dic₂₁₂	—	4	1	no	—
86	C₁₆×C₅₃	—	4	1	no	—
87	#11719908	—	2	2	no	—
88	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	8	2	no	—
91	C₂×C₄×C₅₃	—	8	1	no	—
92	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
97	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×Dic₅₃	—	4	2	no	—
99	C₂×Dic₅₃	—	4	2	no	—
100	C₂×Dic₅₃	—	4	1	no	—
101	C₂×Dic₅₃	—	8	1	no	—
102	C₂×Dic₅₃	—	8	1	no	—
103	Dic₁₀₆	—	4	2	no	—
104	Dic₁₀₆	—	4	2	no	—
105	Dic₁₀₆	—	8	1	no	—
106	Dic₁₀₆	—	8	1	no	—
107	C₈×C₅₃	#117	1	1	yes	characteristic
108	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₈×C₅₃	#124	1	1	yes	characteristic
110	C₈×C₅₃	#126	1	1	yes	characteristic
111	#88322	—	2	2	no	—
112	#88322	—	4	2	no	—
113	#13141	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	#442	1	1	yes	characteristic
116	C₂²×C₅₃	#460	1	1	yes	characteristic
117	C₂²×C₅₃	#482	1	1	yes	characteristic
118	C₄×C₅₃	—	8	2	no	—
119	C₄×C₅₃	—	16	1	no	—
120	C₄×C₅₃	#388	1	1	yes	characteristic
121	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
122	C₄×C₅₃	#463	1	1	yes	characteristic
123	C₄×C₅₃	#481	1	1	yes	characteristic
124	Dic₅₃	—	8	1	no	—
125	Dic₅₃	—	8	2	no	—
126	C₂×#365	—	53	1	no	—
127	C₂×#362	—	53	1	no	—
128	#2141	—	53	1	no	—
129	#1971	—	53	1	no	—
130	#1773	—	53	1	no	—
131	#1386	D₅₃	1	1	yes	characteristic
132	#1375	—	53	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	#1926	1	1	yes	characteristic
136	C₂×C₅₃	#2180	1	2	yes	—
137	C₂²×Q₁₆	—	53	1	no	—
138	C₂×M₃₂	—	53	1	no	—
139	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
140	C₂×#108	—	53	1	no	—
141	#427	C₂×D₅₃	1	1	yes	characteristic
142	C₂×#106	C₂×D₅₃	1	1	yes	characteristic
143	C₂×#103	—	106	1	no	—
144	#405	—	106	1	no	—
145	C₂×#97	—	53	1	no	—
146	#391	—	106	1	no	—
147	#365	—	106	2	no	—
148	#362	—	106	2	no	—
149	#358	—	53	4	no	—
150	#339	—	106	1	no	—
151	C₅₃	#13141	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₈	—	106	1	no	—
153	C₂×Q₁₆	—	53	2	no	—
154	C₂×Q₁₆	—	106	1	no	—
155	C₂×Q₁₆	—	106	1	no	—
156	C₂×Q₁₆	—	106	1	no	—
157	C₂×Q₁₆	—	106	1	no	—
158	C₂×Q₁₆	—	106	1	no	—
159	C₂×M₁₆	—	106	1	no	—
160	C₂²×C₈	—	53	1	no	—
161	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
162	#117	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄²	—	53	1	no	—
165	M₃₂	—	2	1	no	—
166	M₃₂	—	2	1	no	—
167	M₃₂	—	106	1	no	—
168	M₃₂	—	106	1	no	—
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	—	53	2	no	—
171	#108	—	53	2	no	—
172	#108	—	106	1	no	—
173	#107	C₄×D₅₃	1	2	yes	—
174	#106	—	2	1	no	—
175	#106	—	2	1	no	—
176	#106	—	2	1	no	—
177	#103	—	212	2	no	—
178	#97	—	106	1	no	—
179	#97	—	106	1	no	—
180	#97	—	212	1	no	—
181	C₄×C₈	—	212	1	no	—
182	#95	—	212	2	no	—
183	C₂×Q₈	—	106	2	no	—
184	C₂×Q₈	—	212	1	no	—
185	C₂×Q₈	—	212	1	no	—
186	C₂×Q₈	—	212	1	no	—
187	C₂×Q₈	—	212	1	no	—
188	C₂×Q₈	—	212	1	no	—
189	C₂²×C₄	—	4	1	no	—
190	C₂²×C₄	—	106	1	no	—
191	C₂²×C₄	—	212	1	no	—
192	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
193	Q₁₆	—	106	2	no	—
194	Q₁₆	—	106	2	no	—
195	Q₁₆	—	212	1	no	—
196	Q₁₆	—	212	1	no	—
197	M₁₆	—	106	2	no	—
198	M₁₆	—	212	1	no	—
199	C₂×C₈	—	53	2	no	—
200	C₂×C₈	—	106	1	no	—
201	C₂×C₈	—	106	1	no	—
202	C₂×C₈	—	212	1	no	—
203	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂×C₈	#11719945	1	1	yes	characteristic
205	C₂×C₈	#11719945	1	1	yes	characteristic
206	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
207	C₂×C₈	#11719947	1	1	yes	characteristic
208	C₂×C₈	#11719949	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₂²⋊C₄	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	106	1	no	—
215	C₄²	—	106	1	no	—
216	C₁₆	—	4	1	no	—
217	C₁₆	—	106	2	no	—
218	C₂³	#13133241	1	1	yes	characteristic
219	Q₈	—	424	1	no	—
220	Q₈	—	424	1	no	—
221	Q₈	—	212	2	no	—
222	Q₈	—	212	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	424	1	no	—
227	C₂×C₄	—	424	1	no	—
228	C₂×C₄	—	212	2	no	—
229	C₂×C₄	—	212	1	no	—
230	C₂×C₄	—	212	2	no	—
231	C₂×C₄	#13133177	1	1	yes	characteristic
232	C₂×C₄	#13133190	1	1	yes	characteristic
233	C₂×C₄	#13133202	1	1	yes	characteristic
234	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
235	C₂×C₄	#13133240	1	1	yes	characteristic
236	C₂×C₄	#13133244	1	1	yes	characteristic
237	C₈	—	106	2	no	—
238	C₈	—	212	2	no	—
239	C₈	#13133174	1	1	yes	characteristic
240	C₈	#13133179	1	1	yes	characteristic
241	C₈	#13133203	1	1	yes	characteristic
242	C₈	#13133212	1	1	yes	characteristic
243	C₂²	—	2	2	no	—
244	C₂²	#16810732	1	1	yes	characteristic
245	C₂²	#16810822	1	1	yes	characteristic
246	C₂²	#16810950	1	1	yes	characteristic
247	C₄	—	8	2	no	—
248	C₄	—	16	1	no	—
249	C₄	—	424	1	no	—
250	C₄	—	424	2	no	—
251	C₄	#16810512	1	1	yes	characteristic
252	C₄	#16810708	1	1	yes	characteristic
253	C₄	#16810841	1	1	yes	characteristic
254	C₄	#16810943	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	#20275340	1	1	yes	characteristic
258	C₂	#20276603	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810445

C₂

yes

maximal, characteristic

C₂×#16810440

C₂

yes

maximal, characteristic

#20276434

C₂

yes

maximal, characteristic

#20275584

C₂

yes

maximal, characteristic

#20274601

C₂

yes

maximal, characteristic

#20272794

C₂

yes

maximal, characteristic

C₅₃×#1386

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

C₂²

yes

characteristic

C₂×#13133157

C₂²

yes

characteristic

C₂×#13133153

C₂²

yes

—

—

C₂²

yes

—

—

—

C₄

yes

complemented

#16810382

—

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₅₃×#427

C₂²

yes

characteristic

C₂×#106×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂×C₄

yes

—

C₂×#11719917

—

C₂²×#88322

D₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

—

—

C₂×C₄

yes

—

#13133153

—

#13133153

D₄

yes

—

—

—

—

—

—

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

#107×C₅₃

C₂×C₄

yes

—

#106×C₅₃

—

#106×C₅₃

—

#106×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

SD₁₆

yes

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×SD₁₆

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

100

C₂×Dic₅₃

—

101

C₂×Dic₅₃

—

102

C₂×Dic₅₃

—

103

Dic₁₀₆

—

104

Dic₁₀₆

—

105

Dic₁₀₆

—

106

Dic₁₀₆

—

107

C₈×C₅₃

#117

yes

characteristic

108

C₈×C₅₃

C₄×D₄

yes

characteristic

109

C₈×C₅₃

#124

yes

characteristic

110

C₈×C₅₃

#126

yes

characteristic

111

#88322

—

112

#88322

—

113

#13141

—

maximal, Sylow

114

C₂²×C₅₃

—

115

C₂²×C₅₃

#442

yes

characteristic

116

C₂²×C₅₃

#460

yes

characteristic

117

C₂²×C₅₃

#482

yes

characteristic

118

C₄×C₅₃

—

119

C₄×C₅₃

—

120

C₄×C₅₃

#388

yes

characteristic

121

C₄×C₅₃

C₄×SD₁₆

yes

characteristic

122

C₄×C₅₃

#463

yes

characteristic

123

C₄×C₅₃

#481

yes

characteristic

124

Dic₅₃

—

125

Dic₅₃

—

126

C₂×#365

—

127

C₂×#362

—

128

#2141

—

129

#1971

—

130

#1773

—

131

#1386

D₅₃

yes

characteristic

132

#1375

—

133

C₂×C₅₃

—

134

C₂×C₅₃

—

135

C₂×C₅₃

#1926

yes

characteristic

136

C₂×C₅₃

#2180

yes

—

137

C₂²×Q₁₆

—

138

C₂×M₃₂

—

139

C₂×M₃₂

C₂×D₅₃

yes

characteristic

140

C₂×#108

—

141

#427

C₂×D₅₃

yes

characteristic

142

C₂×#106

C₂×D₅₃

yes

characteristic

143

C₂×#103

—

106

—

144

#405

—

106

—

145

C₂×#97

—

146

#391

—

106

—

147

#365

—

106

—

148

#362

—

106

—

149

#358

—

150

#339

—

106

—

151

C₅₃

#13141

yes

characteristic, complemented, Sylow

152

C₂²×Q₈

—

106

—

153

C₂×Q₁₆

—

154

C₂×Q₁₆

—

106

—

155

C₂×Q₁₆

—

106

—

156

C₂×Q₁₆

—

106

—

157

C₂×Q₁₆

—

106

—

158

C₂×Q₁₆

—

106

—

159

C₂×M₁₆

—

106

—

160

C₂²×C₈

—

161

C₂²×C₈

C₂²×D₅₃

yes

characteristic

162

#117

—

163

C₂×C₄⋊C₄

—

164

C₂×C₄²

—

165

M₃₂

—

166

M₃₂

—

167

M₃₂

—

106

—

168

M₃₂

—

106

—

169

C₂×C₁₆

—

170

C₂×C₁₆

—

171

#108

—

172

#108

—

106

—

173

#107

C₄×D₅₃

yes

—

174

#106

—

175

#106

—

176

#106

—

177

#103

—

212

—

178

#97

—

106

—

179

#97

—

106

—

180

#97

—

212

—

181

C₄×C₈

—

212

—

182

#95

—

212

—

183

C₂×Q₈

—

106

—

184

C₂×Q₈

—

212

—

185

C₂×Q₈

—

212

—

186

C₂×Q₈

—

212

—

187

C₂×Q₈

—

212

—

188

C₂×Q₈

—

212

—

189

C₂²×C₄

—

190

C₂²×C₄

—

106

—

191

C₂²×C₄

—

212

—

192

C₂²×C₄

D₄×D₅₃

yes

characteristic

193

Q₁₆

—

106

—

194

Q₁₆

—

106

—

195

Q₁₆

—

212

—

196

Q₁₆

—

212

—

197

M₁₆

—

106

—

198

M₁₆

—

212

—

199

C₂×C₈

—

200

C₂×C₈

—

106

—

201

C₂×C₈

—

106

—

202

C₂×C₈

—

212

—

203

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

204

C₂×C₈

#11719945

yes

characteristic

205

C₂×C₈

#11719945

yes

characteristic

206

C₂×C₈

D₄×D₅₃

yes

characteristic

207

C₂×C₈

#11719947

yes

characteristic

208

C₂×C₈

#11719949

yes

characteristic

209

C₄⋊C₄

—

210

C₄⋊C₄

—

211

C₄⋊C₄

—

212

C₂²⋊C₄

—

213

C₄²

—

214

C₄²

—

106

—

215

C₄²

—

106

—

216

C₁₆

—

217

C₁₆

—

106

—

218

C₂³

#13133241

yes

characteristic

219

Q₈

—

424

—

220

Q₈

—

424

—

221

Q₈

—

212

—

222

Q₈

—

212

—

223

C₂×C₄

—

224

C₂×C₄

—

225

C₂×C₄

—

226

C₂×C₄

—

424

—

227

C₂×C₄

—

424

—

228

C₂×C₄

—

212

—

229

C₂×C₄

—

212

—

230

C₂×C₄

—

212

—

231

C₂×C₄

#13133177

yes

characteristic

232

C₂×C₄

#13133190

yes

characteristic

233

C₂×C₄

#13133202

yes

characteristic

234

C₂×C₄

SD₁₆×D₅₃

yes

characteristic

235

C₂×C₄

#13133240

yes

characteristic

236

C₂×C₄

#13133244

yes

characteristic

237

C₈

—

106

—

238

C₈

—

212

—

239

C₈

#13133174

yes

characteristic

240

C₈

#13133179

yes

characteristic

241

C₈

#13133203

yes

characteristic

242

C₈

#13133212

yes

characteristic

243

C₂²

—

244

C₂²

#16810732

yes

characteristic

245

C₂²

#16810822

yes

characteristic

246

C₂²

#16810950

yes

characteristic

247

C₄

—

248

C₄

—

249

C₄

—

424

—

250

C₄

—

424

—

251

C₄

#16810512

yes

characteristic

252

C₄

#16810708

yes

characteristic

253

C₄

#16810841

yes

characteristic

254

C₄

#16810943

yes

characteristic

255

C₂

—

256

C₂

—

257

C₂

#20275340

yes

characteristic

258

C₂

#20276603

yes

—

	subgroup	G/H
center	C₂²	#16810732
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1386	C₂
socle	C₂²×C₅₃	#442
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1386	D₅₃
hypocenter	C₅₃	#13141
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1386

subgroup

G/H

center

C₂²

#16810732

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1386

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 ◁ C₂²×C₈ ◁ #1386

N	H ≅ G/N	type
17. #16810426	C₄	semidirect
151. C₅₃	#13141	semidirect

H ≅ G/N

type

17. #16810426

C₄

semidirect

151. C₅₃

#13141

semidirect

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

subgroup	index
C₂×#16810445	2
C₂×#16810440	2
#20276434	2
#20275584	2
#20274601	2
#20272794	2
C₅₃×#1386	2
#13141	53

subgroup

index

C₅₃×#1386

Subgroups — #27301964

Conjugacy classes of subgroups

Complements

Maximal subgroups