Group #27538592 — subgroups

Subgroups — #27538592

Order 13568.

650 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 171 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×#16811039	C₂	1	1	yes	maximal, characteristic
2	#20284366	C₂	1	2	yes	maximal
3	#20276820	C₂	1	4	yes	maximal, complemented
4	#20276803	C₂	1	4	yes	maximal, complemented
5	#20276499	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276435	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	1	yes	characteristic
9	#16811197	—	2	2	no	—
10	#16811039	—	2	4	no	—
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133156	C₂²	1	2	yes	—
13	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
14	#16810878	—	2	4	no	—
15	#16810861	C₂²	1	8	yes	—
16	#16810661	C₂²	1	2	yes	—
17	#16810632	—	2	2	no	—
18	#16810451	C₂²	1	2	yes	—
19	#16810449	C₂²	1	2	yes	—
20	#16810427	C₂²	1	8	yes	—
21	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂×Q₃₂×C₅₃	—	2	4	no	—
23	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
24	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#11719945	D₄	1	1	yes	characteristic
27	#13133223	—	2	2	no	—
28	#13133223	—	2	2	no	—
29	C₂×#11719917	D₄	1	1	yes	characteristic
30	C₂×#11719912	D₄	1	2	yes	—
31	C₂×C₈×D₅₃	C₂³	1	2	yes	—
32	#13133215	—	4	2	no	—
33	#13133206	—	4	2	no	—
34	#13133200	—	4	4	no	—
35	#13133193	—	2	8	no	—
36	#13133176	—	2	2	no	—
37	#13133157	—	2	2	no	—
38	#13133156	—	2	4	no	—
39	C₂×#11719908	C₂³	1	2	yes	—
40	C₁₆×D₅₃	—	2	4	no	—
41	#13133151	C₂³	1	4	yes	—
42	#13133149	—	2	2	no	—
43	#13133141	—	2	8	no	—
44	C₂²×Q₈×C₅₃	—	2	2	no	—
45	C₂×Q₁₆×C₅₃	—	2	2	no	—
46	C₂×Q₁₆×C₅₃	—	2	4	no	—
47	C₂×Q₁₆×C₅₃	—	2	4	no	—
48	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	Q₃₂×C₅₃	—	2	4	no	—
51	Q₃₂×C₅₃	—	2	4	no	—
52	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
53	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×Q₈×C₅₃	—	2	4	no	—
56	C₂×Q₈×C₅₃	—	4	2	no	—
57	C₂×Q₈×C₅₃	—	4	4	no	—
58	C₂×Q₈×C₅₃	—	4	4	no	—
59	C₂²×C₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₃	D₈	1	2	yes	—
62	#11719945	—	2	2	no	—
63	#11719945	—	2	2	no	—
64	C₂×D₂₁₂	D₈	1	1	yes	characteristic
65	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
66	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
67	Q₁₆×C₅₃	—	2	4	no	—
68	Q₁₆×C₅₃	—	2	4	no	—
69	Q₁₆×C₅₃	—	4	4	no	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	#41	1	2	yes	—
73	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂²⋊Dic₅₃	—	4	2	no	—
75	#11719921	—	4	8	no	—
76	C₄⋊Dic₅₃	—	2	4	no	—
77	#11719919	—	4	8	no	—
78	C₄×Dic₅₃	—	4	2	no	—
79	#11719917	—	2	2	no	—
80	C₂×#88322	C₂×D₄	1	2	yes	—
81	#11719912	—	2	4	no	—
82	C₈×D₅₃	—	2	2	no	—
83	C₈×D₅₃	—	2	2	no	—
84	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
85	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
86	#11719908	—	2	2	no	—
87	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
88	C₂²×D₅₃	C₂×D₈	1	2	yes	—
89	Q₈×C₅₃	—	4	4	no	—
90	Q₈×C₅₃	—	4	4	no	—
91	Q₈×C₅₃	—	8	4	no	—
92	C₂×C₄×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	8	4	no	—
94	C₂×C₄×C₅₃	#121	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#135	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₂²⋊D₅₃	—	2	4	no	—
99	C₂×Dic₅₃	—	4	4	no	—
100	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
101	D₂₁₂	—	2	2	no	—
102	C₄×D₅₃	—	2	2	no	—
103	C₄×D₅₃	—	2	2	no	—
104	Dic₁₀₆	—	2	2	no	—
105	C₈×C₅₃	#121	1	2	yes	—
106	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
108	#88322	—	2	2	no	—
109	#33608	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	#466	1	2	yes	—
112	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
113	C₂×D₅₃	—	2	4	no	—
114	C₂×D₅₃	—	2	2	no	—
115	C₄×C₅₃	—	8	4	no	—
116	C₄×C₅₃	#466	1	2	yes	—
117	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
118	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
119	Dic₅₃	—	2	2	no	—
120	Dic₅₃	—	8	4	no	—
121	C₂²×Q₃₂	D₅₃	1	1	yes	characteristic, complemented
122	C₂×#504	—	53	1	no	—
123	#3059	—	53	2	no	—
124	#2219	—	53	4	no	—
125	#2216	—	53	4	no	—
126	#2158	—	53	1	no	—
127	#2141	—	53	2	no	—
128	C₂×C₅₃	—	2	2	no	—
129	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
130	C₂×C₅₃	#3417	1	2	yes	—
131	D₅₃	—	4	2	no	—
132	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
133	C₂×#131	—	53	1	no	—
134	#536	—	106	2	no	—
135	C₂×Q₃₂	—	2	4	no	—
136	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
137	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
138	#504	—	106	4	no	—
139	C₂×M₃₂	—	53	1	no	—
140	C₂×M₃₂	—	53	2	no	—
141	C₂²×C₁₆	—	53	2	no	—
142	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
143	#470	—	106	4	no	—
144	#467	—	53	8	no	—
145	#427	—	53	2	no	—
146	#419	—	106	2	no	—
147	#367	—	53	2	no	—
148	#366	—	53	2	no	—
149	#358	—	53	8	no	—
150	C₅₃	#33608	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	53	1	no	—
152	C₂²×Q₈	—	2	2	no	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×Q₁₆	—	2	4	no	—
156	C₂×Q₁₆	C₂²×D₅₃	1	4	yes	—
157	#131	—	106	2	no	—
158	#131	—	106	2	no	—
159	C₂×M₁₆	—	53	1	no	—
160	C₂×M₁₆	—	53	2	no	—
161	C₂²×C₈	—	53	2	no	—
162	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	C₄⋊Q₈	—	212	2	no	—
164	#124	—	212	2	no	—
165	#123	—	212	4	no	—
166	C₂²⋊Q₈	—	106	8	no	—
167	#117	—	106	2	no	—
168	Q₃₂	—	2	4	no	—
169	Q₃₂	—	2	4	no	—
170	M₃₂	—	106	2	no	—
171	M₃₂	—	106	4	no	—
172	C₂×C₁₆	—	53	2	no	—
173	C₂×C₁₆	—	106	4	no	—
174	C₂×C₁₆	C₂²⋊D₅₃	1	4	yes	—
175	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
176	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
177	#107	—	53	4	no	—
178	#106	—	106	2	no	—
179	#103	—	106	8	no	—
180	#41	—	106	2	no	—
181	#41	—	106	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	53	1	no	—
187	C₂×D₄	—	53	2	no	—
188	C₂×D₄	—	53	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	53	2	no	—
191	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	4	4	no	—
195	M₁₆	—	106	2	no	—
196	M₁₆	—	106	4	no	—
197	C₂×C₈	—	53	2	no	—
198	C₂×C₈	—	106	2	no	—
199	C₂×C₈	—	106	2	no	—
200	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
201	C₂×C₈	#11719947	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
203	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
204	C₄⋊C₄	—	106	4	no	—
205	C₄⋊C₄	—	212	8	no	—
206	C₂²⋊C₄	—	212	8	no	—
207	C₂²⋊C₄	—	212	2	no	—
208	C₄²	—	212	2	no	—
209	C₁₆	—	106	2	no	—
210	C₁₆	C₂×C₂²⋊D₅₃	1	2	yes	—
211	C₁₆	C₂×C₂²⋊D₅₃	1	2	yes	—
212	C₂³	—	53	2	no	—
213	C₂³	D₈×D₅₃	1	1	yes	characteristic
214	Q₈	—	4	4	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	8	4	no	—
217	Q₈	—	106	2	no	—
218	D₄	—	106	4	no	—
219	D₄	—	106	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	53	2	no	—
223	C₂×C₄	—	106	2	no	—
224	C₂×C₄	—	106	2	no	—
225	C₂×C₄	—	212	4	no	—
226	C₂×C₄	#13133188	1	2	yes	—
227	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
228	C₂×C₄	#13133236	1	2	yes	—
229	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
230	C₈	—	106	2	no	—
231	C₈	#13133188	1	2	yes	—
232	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
233	C₈	C₂×#11719947	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	106	2	no	—
236	C₂²	—	106	4	no	—
237	C₂²	#16810856	1	2	yes	—
238	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
239	C₄	—	8	4	no	—
240	C₄	—	106	2	no	—
241	C₄	—	424	4	no	—
242	C₄	#16810856	1	2	yes	—
243	C₄	C₂×#13133188	1	1	yes	characteristic
244	C₄	C₂×#13133236	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	212	2	no	—
247	C₂	C₂×#16810856	1	1	yes	characteristic
248	C₂	#20287864	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811039

C₂

yes

maximal, characteristic

#20284366

C₂

yes

maximal

#20276820

C₂

yes

maximal, complemented

#20276803

C₂

yes

maximal, complemented

#20276499

C₂

yes

maximal, characteristic, complemented

#20276435

C₂

yes

maximal, complemented

C₂²×Q₃₂×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#13133223

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#13133156

C₂²

yes

—

C₂×C₁₆×D₅₃

C₂²

yes

—

#16810878

—

#16810861

C₂²

yes

—

#16810661

C₂²

yes

—

#16810632

—

#16810451

C₂²

yes

—

#16810449

C₂²

yes

—

#16810427

C₂²

yes

—

C₂²×Q₁₆×C₅₃

C₂²

yes

—

C₂×Q₃₂×C₅₃

—

C₂×Q₃₂×C₅₃

C₂²

yes

—

C₂×Q₃₂×C₅₃

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#11719945

D₄

yes

—

—

D₄

yes

characteristic

C₂×#11719912

D₄

yes

—

C₂×C₈×D₅₃

C₂³

yes

—

—

—

—

—

—

—

—

C₂³

yes

—

C₁₆×D₅₃

—

#13133151

C₂³

yes

—

#13133149

—

#13133141

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂³

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

Q₃₂×C₅₃

—

Q₃₂×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₅₃

D₈

yes

—

#11719945

—

#11719945

—

C₂×D₂₁₂

D₈

yes

characteristic

C₂×C₄×D₅₃

C₂×D₄

yes

—

C₂×Dic₁₀₆

D₈

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

C₂⁴

yes

characteristic

C₂²⋊Dic₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

C₂×D₄

yes

—

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

C₂×D₄

yes

—

C₁₆×C₅₃

C₂×D₄

yes

—

#11719908

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂²×D₅₃

C₂×D₈

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

100

C₂×Dic₅₃

C₂×D₈

yes

—

101

D₂₁₂

—

102

C₄×D₅₃

—

103

C₄×D₅₃

—

104

Dic₁₀₆

—

105

C₈×C₅₃

#121

yes

—

106

C₈×C₅₃

C₂²×D₄

yes

characteristic

107

C₈×C₅₃

C₂×#41

yes

characteristic

108

#88322

—

109

#33608

—

maximal, Sylow

110

C₂²×C₅₃

—

111

C₂²×C₅₃

#466

yes

—

112

C₂²×C₅₃

C₂²×D₈

yes

characteristic

113

C₂×D₅₃

—

114

C₂×D₅₃

—

115

C₄×C₅₃

—

116

C₄×C₅₃

#466

yes

—

117

C₄×C₅₃

C₂×#121

yes

characteristic

118

C₄×C₅₃

C₂×#135

yes

characteristic

119

Dic₅₃

—

120

Dic₅₃

—

121

C₂²×Q₃₂

D₅₃

yes

characteristic, complemented

122

C₂×#504

—

123

#3059

—

124

#2219

—

125

#2216

—

126

#2158

—

127

#2141

—

128

C₂×C₅₃

—

129

C₂×C₅₃

C₂×#466

yes

characteristic

130

C₂×C₅₃

#3417

yes

—

131

D₅₃

—

132

C₂²×Q₁₆

C₂×D₅₃

yes

—

133

C₂×#131

—

134

#536

—

106

—

135

C₂×Q₃₂

—

136

C₂×Q₃₂

C₂×D₅₃

yes

—

137

C₂×Q₃₂

C₂×D₅₃

yes

—

138

#504

—

106

—

139

C₂×M₃₂

—

140

C₂×M₃₂

—

141

C₂²×C₁₆

—

142

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

143

#470

—

106

—

144

#467

—

145

#427

—

146

#419

—

106

—

147

#367

—

148

#366

—

149

#358

—

150

C₅₃

#33608

yes

characteristic, complemented, Sylow

151

C₂×#41

—

152

C₂²×Q₈

—

153

C₂×Q₁₆

—

154

C₂×Q₁₆

—

155

C₂×Q₁₆

—

156

C₂×Q₁₆

C₂²×D₅₃

yes

—

157

#131

—

106

—

158

#131

—

106

—

159

C₂×M₁₆

—

160

C₂×M₁₆

—

161

C₂²×C₈

—

162

C₂²×C₈

C₂²×D₅₃

yes

characteristic

163

C₄⋊Q₈

—

212

—

164

#124

—

212

—

165

#123

—

212

—

166

C₂²⋊Q₈

—

106

—

167

#117

—

106

—

168

Q₃₂

—

169

Q₃₂

—

170

M₃₂

—

106

—

171

M₃₂

—

106

—

172

C₂×C₁₆

—

173

C₂×C₁₆

—

106

—

174

C₂×C₁₆

C₂²⋊D₅₃

yes

—

175

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

176

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

177

#107

—

178

#106

—

106

—

179

#103

—

106

—

180

#41

—

106

—

181

#41

—

106

—

182

C₂×Q₈

—

183

C₂×Q₈

—

184

C₂×Q₈

—

185

C₂×Q₈

—

186

C₂×Q₈

—

187

C₂×D₄

—

188

C₂×D₄

—

189

C₂²×C₄

—

190

C₂²×C₄

—

191

C₂²×C₄

D₄×D₅₃

yes

characteristic

192

Q₁₆

—

193

Q₁₆

—

194

Q₁₆

—

195

M₁₆

—

106

—

196

M₁₆

—

106

—

197

C₂×C₈

—

198

C₂×C₈

—

106

—

199

C₂×C₈

—

106

—

200

C₂×C₈

D₄×D₅₃

yes

characteristic

201

C₂×C₈

#11719947

yes

—

202

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

203

C₂×C₈

C₂³×D₅₃

yes

characteristic

204

C₄⋊C₄

—

106

—

205

C₄⋊C₄

—

212

—

206

C₂²⋊C₄

—

212

—

207

C₂²⋊C₄

—

212

—

208

C₄²

—

212

—

209

C₁₆

—

106

—

210

C₁₆

C₂×C₂²⋊D₅₃

yes

—

211

C₁₆

C₂×C₂²⋊D₅₃

yes

—

212

C₂³

—

213

C₂³

D₈×D₅₃

yes

characteristic

214

Q₈

—

215

Q₈

—

216

Q₈

—

217

Q₈

—

106

—

218

D₄

—

106

—

219

D₄

—

106

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

—

106

—

224

C₂×C₄

—

106

—

225

C₂×C₄

—

212

—

226

C₂×C₄

#13133188

yes

—

227

C₂×C₄

D₈×D₅₃

yes

characteristic

228

C₂×C₄

#13133236

yes

—

229

C₂×C₄

C₂×D₄×D₅₃

yes

characteristic

230

C₈

—

106

—

231

C₈

#13133188

yes

—

232

C₈

C₂²×C₂²⋊D₅₃

yes

characteristic

233

C₈

C₂×#11719947

yes

characteristic

234

C₂²

—

235

C₂²

—

106

—

236

C₂²

—

106

—

237

C₂²

#16810856

yes

—

238

C₂²

C₂×D₈×D₅₃

yes

characteristic

239

C₄

—

240

C₄

—

106

—

241

C₄

—

424

—

242

C₄

#16810856

yes

—

243

C₄

C₂×#13133188

yes

characteristic

244

C₄

C₂×#13133236

yes

characteristic

245

C₂

—

246

C₂

—

212

—

247

C₂

C₂×#16810856

yes

characteristic

248

C₂

#20287864

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#11719945	D₄
hypercenter	C₂²×Q₃₂	D₅₃
hypocenter	C₅₃	#33608
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂²

C₂×D₈×D₅₃

commutator, cosocle

C₂×C₈×C₅₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₃

Fitting

C₂²×Q₃₂×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₈

prime radical

C₂×#11719945

D₄

hypercenter

C₂²×Q₃₂

D₅₃

hypocenter

C₅₃

#33608

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
3. #20276820	C₂	semidirect
4. #20276803	C₂	semidirect
5. #20276499	C₂	semidirect
6. #20276435	C₂	semidirect
7. C₂²×Q₃₂×C₅₃	C₂	semidirect
121. C₂²×Q₃₂	D₅₃	semidirect
150. C₅₃	#33608	semidirect

H ≅ G/N

type

3. #20276820

C₂

semidirect

4. #20276803

C₂

semidirect

5. #20276499

C₂

semidirect

6. #20276435

C₂

semidirect

7. C₂²×Q₃₂×C₅₃

C₂

semidirect

121. C₂²×Q₃₂

D₅₃

semidirect

150. C₅₃

#33608

semidirect

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

subgroup	index
C₂×#16811039	2
#20284366	2
#20276820	2
#20276803	2
#20276499	2
#20276435	2
C₂²×Q₃₂×C₅₃	2
#33608	53

subgroup

index

C₂²×Q₃₂×C₅₃

Subgroups — #27538592

Conjugacy classes of subgroups

Complements

Maximal subgroups