Group #27283206 — subgroups

Subgroups — #27283206

Order 13568.

466 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 151 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	#20274832	C₂	1	1	yes	maximal, characteristic
2	#20274568	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810372	C₂	1	2	yes	maximal
4	C₂×#16810369	C₂	1	2	yes	maximal
5	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133128	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	#16810378	—	2	2	no	—
11	#16810372	—	2	4	no	—
12	#16810369	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#391	—	2	1	no	—
16	C₅₃×#391	—	2	1	no	—
17	C₅₃×#391	C₄	1	4	yes	—
18	C₅₃×#382	—	2	2	no	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	D₄	1	4	yes	—
26	#13133128	—	4	2	no	—
27	C₂²×Q₈×C₅₃	—	2	1	no	—
28	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#95×C₅₃	—	2	4	no	—
43	#95×C₅₃	—	2	4	no	—
44	C₂×Q₈×C₅₃	—	2	4	no	—
45	C₂×Q₈×C₅₃	—	4	4	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	4	1	no	—
49	C₂×Q₈×C₅₃	—	4	1	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	1	no	—
60	C₄⋊C₄×C₅₃	—	4	1	no	—
61	C₄⋊C₄×C₅₃	—	4	1	no	—
62	C₄⋊C₄×C₅₃	—	4	1	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	1	no	—
65	C₄²×C₅₃	—	2	1	no	—
66	C₄²×C₅₃	—	2	4	no	—
67	C₄²×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	C₂×D₄	1	2	yes	—
69	C₄²×C₅₃	#41	1	2	yes	—
70	C₂×#88322	—	2	4	no	—
71	C₂×#88322	—	4	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₂×#88322	—	4	2	no	—
74	C₂×#88322	—	4	4	no	—
75	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Q₈×C₅₃	—	8	4	no	—
77	Q₈×C₅₃	—	8	2	no	—
78	Q₈×C₅₃	—	8	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	4	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₅₃	#121	1	2	yes	—
93	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#135	1	2	yes	—
96	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#137	1	4	yes	—
99	#88322	—	4	4	no	—
100	#88322	—	8	2	no	—
101	#9709	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	#390	1	1	yes	characteristic
103	C₂²×C₅₃	#413	1	2	yes	—
104	C₂²×C₅₃	#416	1	1	yes	characteristic
105	C₂²×C₅₃	#417	1	1	yes	characteristic
106	C₂²×C₅₃	#419	1	2	yes	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	4	2	no	—
109	C₄×C₅₃	—	4	2	no	—
110	C₄×C₅₃	—	8	2	no	—
111	C₄×C₅₃	—	8	2	no	—
112	C₄×C₅₃	#463	1	2	yes	—
113	C₄×C₅₃	#464	1	2	yes	—
114	C₄×C₅₃	#491	1	2	yes	—
115	C₄×C₅₃	#497	1	2	yes	—
116	#2862	D₅₃	1	1	yes	characteristic
117	#1831	—	53	1	no	—
118	#1767	—	53	1	no	—
119	C₂×#333	—	53	2	no	—
120	C₂×#332	—	53	2	no	—
121	C₂×C₅₃	#1554	1	2	yes	—
122	C₂×C₅₃	#1559	1	2	yes	—
123	C₂×C₅₃	#1960	1	1	yes	characteristic
124	C₂×C₅₃	#1967	1	1	yes	characteristic
125	C₂×C₅₃	#1983	1	1	yes	characteristic
126	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	53	2	no	—
129	C₂×C₄⋊C₈	—	53	2	no	—
130	C₂×#97	—	106	1	no	—
131	#391	—	2	1	no	—
132	#391	—	2	1	no	—
133	#391	Dic₅₃	1	4	yes	—
134	#382	—	2	2	no	—
135	C₄³	C₂×D₅₃	1	1	yes	characteristic
136	#336	—	106	2	no	—
137	#336	—	106	1	no	—
138	#333	—	106	4	no	—
139	#332	—	106	4	no	—
140	C₅₃	#9709	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	2	1	no	—
142	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
143	C₂²×C₈	—	106	2	no	—
144	C₂²×C₈	—	106	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
153	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
154	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
156	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
158	C₄⋊C₈	—	53	4	no	—
159	C₄⋊C₈	—	106	2	no	—
160	C₄⋊C₈	—	106	2	no	—
161	C₄⋊C₈	—	106	2	no	—
162	#97	—	212	2	no	—
163	#95	—	2	4	no	—
164	#95	—	2	4	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	1	no	—
170	C₂×Q₈	—	4	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
174	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂×C₈	—	106	4	no	—
179	C₂×C₈	—	212	2	no	—
180	C₂×C₈	—	212	4	no	—
181	C₂×C₈	—	212	2	no	—
182	C₂×C₈	—	212	2	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	4	2	no	—
194	C₄²	D₄×D₅₃	1	2	yes	—
195	C₄²	#11719947	1	2	yes	—
196	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
197	Q₈	—	8	4	no	—
198	Q₈	—	8	2	no	—
199	Q₈	—	8	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
213	C₂×C₄	#13133188	1	2	yes	—
214	C₂×C₄	#13133204	1	1	yes	characteristic
215	C₂×C₄	#13133213	1	1	yes	characteristic
216	C₂×C₄	#13133237	1	2	yes	—
217	C₂×C₄	#13133242	1	1	yes	characteristic
218	C₂×C₄	#13133247	1	1	yes	characteristic
219	C₂×C₄	#13133248	1	4	yes	—
220	C₈	—	424	2	no	—
221	C₈	—	212	4	no	—
222	C₂²	#16810526	1	1	yes	characteristic
223	C₂²	#16810607	1	2	yes	—
224	C₂²	#16810619	1	1	yes	characteristic
225	C₂²	#16810624	1	1	yes	characteristic
226	C₂²	#16810632	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	2	no	—
232	C₄	#16810835	1	2	yes	—
233	C₄	#16810842	1	2	yes	—
234	C₄	#16810991	1	2	yes	—
235	C₄	#16811015	1	2	yes	—
236	C₂	#20273425	1	2	yes	—
237	C₂	#20273445	1	2	yes	—
238	C₂	#20275536	1	1	yes	characteristic
239	C₂	#20275565	1	1	yes	characteristic
240	C₂	#20275648	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274832

C₂

yes

maximal, characteristic

#20274568

C₂

yes

maximal, characteristic

C₂×#16810372

C₂

yes

maximal

C₂×#16810369

C₂

yes

maximal

C₅₃×#2862

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

—

—

—

—

—

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×#391

—

C₅₃×#391

—

C₅₃×#391

C₄

yes

—

C₅₃×#382

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₂²×#88322

—

—

—

—

D₄

yes

—

#13133128

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

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×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

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×D₄

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

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

C₄×D₄

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

#88322

—

100

#88322

—

101

#9709

—

maximal, Sylow

102

C₂²×C₅₃

#390

yes

characteristic

103

C₂²×C₅₃

#413

yes

—

104

C₂²×C₅₃

#416

yes

characteristic

105

C₂²×C₅₃

#417

yes

characteristic

106

C₂²×C₅₃

#419

yes

—

107

C₄×C₅₃

—

108

C₄×C₅₃

—

109

C₄×C₅₃

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

#463

yes

—

113

C₄×C₅₃

#464

yes

—

114

C₄×C₅₃

#491

yes

—

115

C₄×C₅₃

#497

yes

—

116

#2862

D₅₃

yes

characteristic

117

#1831

—

118

#1767

—

119

C₂×#333

—

120

C₂×#332

—

121

C₂×C₅₃

#1554

yes

—

122

C₂×C₅₃

#1559

yes

—

123

C₂×C₅₃

#1960

yes

characteristic

124

C₂×C₅₃

#1967

yes

characteristic

125

C₂×C₅₃

#1983

yes

characteristic

126

C₂×C₄⋊Q₈

C₂×D₅₃

yes

characteristic

127

C₂×C₄⋊Q₈

C₂×D₅₃

yes

characteristic

128

C₂×C₄⋊C₈

—

129

C₂×C₄⋊C₈

—

130

C₂×#97

—

106

—

131

#391

—

132

#391

—

133

#391

Dic₅₃

yes

—

134

#382

—

135

C₄³

C₂×D₅₃

yes

characteristic

136

#336

—

106

—

137

#336

—

106

—

138

#333

—

106

—

139

#332

—

106

—

140

C₅₃

#9709

yes

characteristic, complemented, Sylow

141

C₂²×Q₈

—

142

C₂²×Q₈

C₂×Dic₅₃

yes

—

143

C₂²×C₈

—

106

—

144

C₂²×C₈

—

106

—

145

C₄⋊Q₈

—

146

C₄⋊Q₈

—

147

C₄⋊Q₈

—

148

C₄⋊Q₈

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄⋊C₄

—

152

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

153

C₂×C₄²

C₂×Dic₅₃

yes

—

154

C₂×C₄²

C₂²⋊D₅₃

yes

—

155

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

156

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

157

C₂×C₄²

C₂²×D₅₃

yes

characteristic

158

C₄⋊C₈

—

159

C₄⋊C₈

—

106

—

160

C₄⋊C₈

—

106

—

161

C₄⋊C₈

—

106

—

162

#97

—

212

—

163

#95

—

164

#95

—

165

C₂×Q₈

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂²×C₄

—

172

C₂²×C₄

—

173

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

174

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

175

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

176

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

177

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

178

C₂×C₈

—

106

—

179

C₂×C₈

—

212

—

180

C₂×C₈

—

212

—

181

C₂×C₈

—

212

—

182

C₂×C₈

—

212

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄²

—

190

C₄²

—

191

C₄²

—

192

C₄²

—

193

C₄²

—

194

C₄²

D₄×D₅₃

yes

—

195

C₄²

#11719947

yes

—

196

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

197

Q₈

—

198

Q₈

—

199

Q₈

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

D₄×Dic₅₃

yes

—

213

C₂×C₄

#13133188

yes

—

214

C₂×C₄

#13133204

yes

characteristic

215

C₂×C₄

#13133213

yes

characteristic

216

C₂×C₄

#13133237

yes

—

217

C₂×C₄

#13133242

yes

characteristic

218

C₂×C₄

#13133247

yes

characteristic

219

C₂×C₄

#13133248

yes

—

220

C₈

—

424

—

221

C₈

—

212

—

222

C₂²

#16810526

yes

characteristic

223

C₂²

#16810607

yes

—

224

C₂²

#16810619

yes

characteristic

225

C₂²

#16810624

yes

characteristic

226

C₂²

#16810632

yes

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

#16810835

yes

—

233

C₄

#16810842

yes

—

234

C₄

#16810991

yes

—

235

C₄

#16811015

yes

—

236

C₂

#20273425

yes

—

237

C₂

#20273445

yes

—

238

C₂

#20275536

yes

characteristic

239

C₂

#20275565

yes

characteristic

240

C₂

#20275648

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2862	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2862	D₅₃
hypocenter	C₅₃	#9709
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2862

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

N	H ≅ G/N	type
140. C₅₃	#9709	semidirect

H ≅ G/N

type

140. C₅₃

#9709

semidirect

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

subgroup	index
#20274832	2
#20274568	2
C₂×#16810372	2
C₂×#16810369	2
C₅₃×#2862	2
#9709	53

subgroup

index

C₅₃×#2862

Subgroups — #27283206

Conjugacy classes of subgroups

Complements

Maximal subgroups