Group #20275319 — subgroups

Subgroups — #20275319

Order 6784.

292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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×#13133146	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133129	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133129	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#11719921	C₂	1	1	yes	maximal, characteristic
6	#16810377	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂×#11719921	C₄	1	1	yes	characteristic
12	C₂×#11719921	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#13133146	—	2	2	no	—
15	#13133129	—	2	2	no	—
16	#13133129	—	2	2	no	—
17	C₄×#88322	—	2	2	no	—
18	#13133124	C₄	1	1	yes	characteristic
19	#13133124	C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	—	2	1	no	—
31	C₂×C₄×D₅₃	—	2	1	no	—
32	C₂×C₄×D₅₃	—	2	1	no	—
33	C₂×C₄×D₅₃	—	2	1	no	—
34	C₂×C₄×D₅₃	—	2	1	no	—
35	C₂×C₄×D₅₃	—	2	1	no	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₄²×C₅₃	—	2	1	no	—
43	C₄²×C₅₃	—	2	1	no	—
44	#11719921	—	2	2	no	—
45	#11719921	C₈	1	4	yes	complemented
46	C₄×Dic₅₃	—	2	1	no	—
47	C₄×Dic₅₃	—	2	1	no	—
48	C₂×#88322	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂×#88322	—	2	1	no	—
51	C₂×#88322	—	2	1	no	—
52	C₂×#88322	D₄	1	2	yes	—
53	C₂×#88322	D₄	1	2	yes	—
54	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×D₅₃	—	2	1	no	—
56	C₂²×D₅₃	—	2	1	no	—
57	C₂²×D₅₃	—	2	1	no	—
58	C₂²×D₅₃	—	2	1	no	—
59	C₂²×D₅₃	—	2	1	no	—
60	C₂²×D₅₃	—	2	1	no	—
61	C₂²×D₅₃	C₂×C₈	1	2	yes	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
67	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
74	C₂×Dic₅₃	—	2	1	no	—
75	C₂×Dic₅₃	—	2	1	no	—
76	C₂×Dic₅₃	—	2	1	no	—
77	C₂×Dic₅₃	—	2	1	no	—
78	C₂×Dic₅₃	—	2	1	no	—
79	C₂×Dic₅₃	—	2	1	no	—
80	C₂×Dic₅₃	—	2	1	no	—
81	C₂×Dic₅₃	—	2	1	no	—
82	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
83	C₄×D₅₃	—	4	1	no	—
84	C₄×D₅₃	—	4	1	no	—
85	C₄×D₅₃	—	4	1	no	—
86	C₄×D₅₃	—	4	1	no	—
87	C₈×C₅₃	—	4	1	no	—
88	C₈×C₅₃	—	4	1	no	—
89	#88322	—	2	2	no	—
90	#88322	—	2	2	no	—
91	C₂²×C₅₃	#117	1	1	yes	characteristic
92	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
95	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂²×C₅₃	#131	1	1	yes	characteristic
97	C₂²×C₅₃	#131	1	1	yes	characteristic
98	C₂×D₅₃	—	2	2	no	—
99	C₂×D₅₃	—	2	2	no	—
100	C₂×D₅₃	—	2	2	no	—
101	C₂×D₅₃	—	4	1	no	—
102	C₂×D₅₃	—	4	1	no	—
103	C₂×D₅₃	—	4	1	no	—
104	C₂×D₅₃	—	4	1	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	#121	1	1	yes	characteristic
108	C₄×C₅₃	#123	1	1	yes	characteristic
109	C₄×C₅₃	#124	1	1	yes	characteristic
110	C₄×C₅₃	#126	1	1	yes	characteristic
111	Dic₅₃	—	2	2	no	—
112	Dic₅₃	—	4	1	no	—
113	Dic₅₃	—	4	1	no	—
114	#1923	—	53	1	no	maximal, Sylow
115	C₂×C₅₃	#388	1	1	yes	characteristic
116	C₂×C₅₃	#431	1	1	yes	characteristic
117	C₂×C₅₃	#433	1	1	yes	characteristic
118	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
119	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
120	C₂×C₅₃	#435	1	1	yes	characteristic
121	C₂×C₅₃	#436	1	1	yes	characteristic
122	D₅₃	—	4	2	no	—
123	C₂×C₄⋊C₈	—	53	1	no	—
124	C₂×C₂²⋊C₈	—	53	1	no	—
125	C₂×C₂²⋊C₈	—	53	1	no	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₄×C₂²⋊C₄	—	53	1	no	—
128	#336	—	53	1	no	—
129	#336	D₅₃	1	1	yes	characteristic, complemented
130	C₅₃	#1923	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₄	—	53	1	no	—
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
136	C₂×C₂²⋊C₄	—	53	1	no	—
137	C₂×C₂²⋊C₄	—	53	1	no	—
138	C₂×C₄²	—	53	1	no	—
139	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
140	C₄⋊C₈	—	106	2	no	—
141	C₂²⋊C₈	—	106	2	no	—
142	C₂²⋊C₈	—	106	2	no	—
143	C₄×C₈	—	106	2	no	—
144	#95	—	53	1	no	—
145	#95	—	53	1	no	—
146	C₂⁴	—	53	1	no	—
147	C₂²×C₄	—	53	1	no	—
148	C₂²×C₄	—	53	1	no	—
149	C₂²×C₄	—	53	1	no	—
150	C₂²×C₄	—	106	1	no	—
151	C₂²×C₄	—	106	1	no	—
152	C₂²×C₄	—	106	1	no	—
153	C₂²×C₄	—	106	1	no	—
154	C₂²×C₄	—	106	1	no	—
155	C₂²×C₄	—	106	1	no	—
156	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	53	2	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	106	1	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	106	1	no	—
170	C₂×C₈	—	106	1	no	—
171	C₂²⋊C₄	—	53	4	no	—
172	C₂²⋊C₄	—	106	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	106	1	no	—
176	C₄²	—	106	1	no	—
177	C₂³	—	53	2	no	—
178	C₂³	—	106	1	no	—
179	C₂³	—	106	1	no	—
180	C₂³	—	106	1	no	—
181	C₂³	—	106	1	no	—
182	C₂³	—	106	1	no	—
183	C₂³	—	106	1	no	—
184	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	106	1	no	—
191	C₂×C₄	—	106	1	no	—
192	C₂×C₄	—	106	1	no	—
193	C₂×C₄	—	106	1	no	—
194	C₂×C₄	—	106	1	no	—
195	C₂×C₄	—	106	1	no	—
196	C₂×C₄	—	106	1	no	—
197	C₂×C₄	—	106	1	no	—
198	C₂×C₄	—	212	1	no	—
199	C₂×C₄	—	212	1	no	—
200	C₂×C₄	—	212	1	no	—
201	C₂×C₄	—	212	1	no	—
202	C₂×C₄	C₈×D₅₃	1	2	yes	—
203	C₂×C₄	#11719912	1	2	yes	—
204	C₂×C₄	#11719945	1	1	yes	characteristic
205	C₂×C₄	#11719945	1	1	yes	characteristic
206	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
207	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
208	C₂×C₄	#11719947	1	1	yes	characteristic
209	C₂×C₄	#11719949	1	1	yes	characteristic
210	C₈	—	4	1	no	—
211	C₈	—	4	1	no	—
212	C₈	—	106	2	no	—
213	C₈	—	106	2	no	—
214	C₂²	—	106	2	no	—
215	C₂²	—	106	2	no	—
216	C₂²	—	106	2	no	—
217	C₂²	—	212	1	no	—
218	C₂²	—	212	1	no	—
219	C₂²	—	212	1	no	—
220	C₂²	—	212	1	no	—
221	C₂²	#13133174	1	1	yes	characteristic
222	C₂²	#13133177	1	1	yes	characteristic
223	C₂²	#13133179	1	1	yes	characteristic
224	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
225	C₂²	C₂×#11719912	1	1	yes	characteristic
226	C₂²	#13133225	1	1	yes	characteristic
227	C₂²	#13133225	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	106	2	no	—
231	C₄	—	212	1	no	—
232	C₄	—	212	1	no	—
233	C₄	#13133190	1	1	yes	characteristic
234	C₄	#13133202	1	1	yes	characteristic
235	C₄	#13133203	1	1	yes	characteristic
236	C₄	#13133212	1	1	yes	characteristic
237	C₂	—	212	2	no	—
238	C₂	#16810512	1	1	yes	characteristic
239	C₂	#16810674	1	1	yes	characteristic
240	C₂	#16810685	1	1	yes	characteristic
241	C₂	#16810686	1	1	yes	characteristic
242	C₂	#16810688	1	1	yes	characteristic
243	C₂	#16810691	1	1	yes	characteristic
244	C₂	#16810700	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133146

C₂

yes

maximal, characteristic, complemented

C₂×#13133129

C₂

yes

maximal, characteristic

C₂×#13133129

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂

yes

maximal, characteristic, complemented

C₄×#11719921

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal, characteristic, complemented

C₅₃×#336

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×#11719921

C₄

yes

characteristic

C₂×#11719921

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

—

C₄

yes

characteristic

#13133124

C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

#11719921

—

#11719921

C₈

yes

complemented

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

D₄

yes

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

#117

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

100

C₂×D₅₃

—

101

C₂×D₅₃

—

102

C₂×D₅₃

—

103

C₂×D₅₃

—

104

C₂×D₅₃

—

105

C₄×C₅₃

—

106

C₄×C₅₃

—

107

C₄×C₅₃

#121

yes

characteristic

108

C₄×C₅₃

#123

yes

characteristic

109

C₄×C₅₃

#124

yes

characteristic

110

C₄×C₅₃

#126

yes

characteristic

111

Dic₅₃

—

112

Dic₅₃

—

113

Dic₅₃

—

114

#1923

—

maximal, Sylow

115

C₂×C₅₃

#388

yes

characteristic

116

C₂×C₅₃

#431

yes

characteristic

117

C₂×C₅₃

#433

yes

characteristic

118

C₂×C₅₃

C₈×D₄

yes

characteristic

119

C₂×C₅₃

C₈×D₄

yes

characteristic

120

C₂×C₅₃

#435

yes

characteristic

121

C₂×C₅₃

#436

yes

characteristic

122

D₅₃

—

123

C₂×C₄⋊C₈

—

124

C₂×C₂²⋊C₈

—

125

C₂×C₂²⋊C₈

—

126

C₂×C₄×C₈

—

127

C₄×C₂²⋊C₄

—

128

#336

—

129

#336

D₅₃

yes

characteristic, complemented

130

C₅₃

#1923

yes

characteristic, complemented, Sylow

131

C₂³×C₄

—

132

C₂²×C₈

—

133

C₂²×C₈

—

134

C₂²×C₈

C₂×D₅₃

yes

characteristic

135

C₂²×C₈

C₂×D₅₃

yes

characteristic

136

C₂×C₂²⋊C₄

—

137

C₂×C₂²⋊C₄

—

138

C₂×C₄²

—

139

C₂×C₄²

C₂×D₅₃

yes

characteristic

140

C₄⋊C₈

—

106

—

141

C₂²⋊C₈

—

106

—

142

C₂²⋊C₈

—

106

—

143

C₄×C₈

—

106

—

144

#95

—

145

#95

—

146

C₂⁴

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

106

—

151

C₂²×C₄

—

106

—

152

C₂²×C₄

—

106

—

153

C₂²×C₄

—

106

—

154

C₂²×C₄

—

106

—

155

C₂²×C₄

—

106

—

156

C₂²×C₄

C₄×D₅₃

yes

characteristic

157

C₂²×C₄

C₄×D₅₃

yes

characteristic

158

C₂²×C₄

C₂²×D₅₃

yes

characteristic

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

106

—

168

C₂×C₈

—

106

—

169

C₂×C₈

—

106

—

170

C₂×C₈

—

106

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

106

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

106

—

176

C₄²

—

106

—

177

C₂³

—

178

C₂³

—

106

—

179

C₂³

—

106

—

180

C₂³

—

106

—

181

C₂³

—

106

—

182

C₂³

—

106

—

183

C₂³

—

106

—

184

C₂³

C₂×C₄×D₅₃

yes

characteristic

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

106

—

191

C₂×C₄

—

106

—

192

C₂×C₄

—

106

—

193

C₂×C₄

—

106

—

194

C₂×C₄

—

106

—

195

C₂×C₄

—

106

—

196

C₂×C₄

—

106

—

197

C₂×C₄

—

106

—

198

C₂×C₄

—

212

—

199

C₂×C₄

—

212

—

200

C₂×C₄

—

212

—

201

C₂×C₄

—

212

—

202

C₂×C₄

C₈×D₅₃

yes

—

203

C₂×C₄

#11719912

yes

—

204

C₂×C₄

#11719945

yes

characteristic

205

C₂×C₄

#11719945

yes

characteristic

206

C₂×C₄

D₄×D₅₃

yes

characteristic

207

C₂×C₄

D₄×D₅₃

yes

characteristic

208

C₂×C₄

#11719947

yes

characteristic

209

C₂×C₄

#11719949

yes

characteristic

210

C₈

—

211

C₈

—

212

C₈

—

106

—

213

C₈

—

106

—

214

C₂²

—

106

—

215

C₂²

—

106

—

216

C₂²

—

106

—

217

C₂²

—

212

—

218

C₂²

—

212

—

219

C₂²

—

212

—

220

C₂²

—

212

—

221

C₂²

#13133174

yes

characteristic

222

C₂²

#13133177

yes

characteristic

223

C₂²

#13133179

yes

characteristic

224

C₂²

C₂×C₈×D₅₃

yes

characteristic

225

C₂²

C₂×#11719912

yes

characteristic

226

C₂²

#13133225

yes

characteristic

227

C₂²

#13133225

yes

characteristic

228

C₄

—

229

C₄

—

230

C₄

—

106

—

231

C₄

—

212

—

232

C₄

—

212

—

233

C₄

#13133190

yes

characteristic

234

C₄

#13133202

yes

characteristic

235

C₄

#13133203

yes

characteristic

236

C₄

#13133212

yes

characteristic

237

C₂

—

212

—

238

C₂

#16810512

yes

characteristic

239

C₂

#16810674

yes

characteristic

240

C₂

#16810685

yes

characteristic

241

C₂

#16810686

yes

characteristic

242

C₂

#16810688

yes

characteristic

243

C₂

#16810691

yes

characteristic

244

C₂

#16810700

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	#336	D₅₃
hypocenter	C₅₃	#1923
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂²×C₈

Fitting

C₅₃×#336

C₂

socle

C₂³×C₅₃

C₂²×C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
1. C₂×#13133146	C₂	semidirect
4. C₂×C₄×#88322	C₂	semidirect
6. #16810377	C₂	semidirect
7. C₅₃×#336	C₂	semidirect
45. #11719921	C₈	semidirect
129. #336	D₅₃	semidirect
130. C₅₃	#1923	semidirect

H ≅ G/N

type

1. C₂×#13133146

C₂

semidirect

4. C₂×C₄×#88322

C₂

semidirect

6. #16810377

C₂

semidirect

7. C₅₃×#336

C₂

semidirect

45. #11719921

C₈

semidirect

129. #336

D₅₃

semidirect

130. C₅₃

#1923

semidirect

Subgroups — #20275319

Conjugacy classes of subgroups

Complements