Group #27302251 — subgroups

Subgroups — #27302251

Order 13568.

522 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 159 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×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810402	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
6	C₂×C₄×D₂₁₂	C₄	1	1	yes	characteristic
7	C₂²×#11719908	C₂²	1	2	yes	—
8	C₈×D₂₁₂	—	2	2	no	—
9	C₈×D₂₁₂	—	2	2	no	—
10	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
11	C₂×#13133129	C₄	1	2	yes	—
12	C₄⋊#11719908	—	2	2	no	—
13	#16810402	—	2	8	no	—
14	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#346	—	2	2	no	—
17	C₂²×D₂₁₂	C₈	1	1	yes	characteristic
18	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
19	C₂²×#88322	C₂×C₄	1	2	yes	—
20	C₂×C₈×D₅₃	—	2	4	no	—
21	C₂×C₈×D₅₃	—	2	2	no	—
22	C₂×C₈×D₅₃	—	2	2	no	—
23	C₂×C₈×D₅₃	—	2	4	no	—
24	C₄×D₂₁₂	—	2	2	no	—
25	C₄×D₂₁₂	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
27	C₂×#11719921	C₈	1	2	yes	—
28	C₂×#11719908	—	2	4	no	—
29	C₂×#11719908	—	2	2	no	—
30	C₄⋊#88322	—	2	2	no	—
31	#13133129	—	2	4	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	D₄	1	4	yes	complemented
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₂³×D₅₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
45	C₂×D₂₁₂	—	2	4	no	—
46	C₂×D₂₁₂	—	2	2	no	—
47	C₂×C₄×D₅₃	—	2	2	no	—
48	C₂×C₄×D₅₃	—	2	4	no	—
49	C₂×C₄×D₅₃	—	2	2	no	—
50	C₂×C₄×D₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	#41	1	2	yes	—
57	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₅₃	M₁₆	1	2	yes	—
59	#11719921	—	2	4	no	—
60	C₄⋊Dic₅₃	—	2	2	no	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	2	4	no	—
63	C₈×D₅₃	—	4	4	no	—
64	C₈×D₅₃	—	4	2	no	—
65	C₈×D₅₃	—	4	2	no	—
66	C₁₆×C₅₃	—	2	4	no	—
67	#11719908	—	4	2	no	—
68	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂²×D₅₃	—	2	2	no	—
70	C₂²×D₅₃	—	2	4	no	—
71	C₂²×D₅₃	—	2	2	no	—
72	C₂²×D₅₃	—	2	4	no	—
73	C₂²×D₅₃	M₃₂	1	4	yes	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#131	1	2	yes	—
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	—	2	4	no	—
84	D₂₁₂	—	2	8	no	—
85	C₄×D₅₃	—	4	2	no	—
86	C₄×D₅₃	—	4	2	no	—
87	C₄×D₅₃	—	4	4	no	—
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	#121	1	2	yes	—
90	C₈×C₅₃	#124	1	1	yes	characteristic
91	C₈×C₅₃	#127	1	1	yes	characteristic
92	#88322	—	4	2	no	—
93	#13190	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
95	C₂²×C₅₃	C₈×D₄	1	2	yes	—
96	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
97	C₂²×C₅₃	#504	1	2	yes	—
98	C₂×D₅₃	—	2	4	no	—
99	C₂×D₅₃	—	2	8	no	—
100	C₂×D₅₃	—	4	2	no	—
101	C₂×D₅₃	—	4	4	no	—
102	C₂×D₅₃	—	4	2	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	#390	1	1	yes	characteristic
106	C₄×C₅₃	#408	1	1	yes	characteristic
107	C₄×C₅₃	#436	1	2	yes	—
108	Dic₅₃	—	4	2	no	—
109	C₂×C₈×D₄	—	53	1	no	—
110	C₂×C₄⋊C₁₆	—	53	1	no	—
111	C₂×C₂²⋊C₁₆	—	53	4	no	—
112	C₂×#346	D₅₃	1	1	yes	characteristic, complemented
113	C₂×C₅₃	#1954	1	1	yes	characteristic
114	C₂×C₅₃	#2112	1	1	yes	characteristic
115	C₂×C₅₃	#2113	1	1	yes	characteristic
116	C₂×C₅₃	#2168	1	4	yes	—
117	D₅₃	—	4	4	no	—
118	C₂³×C₈	—	53	2	no	—
119	C₂×C₄×D₄	—	53	1	no	—
120	C₂²×C₁₆	—	53	2	no	—
121	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
122	C₈×D₄	—	106	2	no	—
123	C₈×D₄	—	106	2	no	—
124	C₂×C₄⋊C₈	—	53	1	no	—
125	C₂×C₂²⋊C₈	—	53	2	no	—
126	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
127	C₄⋊C₁₆	—	106	2	no	—
128	C₂²⋊C₁₆	—	106	8	no	—
129	#346	—	2	2	no	—
130	C₅₃	#13190	1	1	yes	characteristic, complemented, Sylow
131	C₂²×D₄	—	53	1	no	—
132	C₂³×C₄	—	53	2	no	—
133	C₂²×C₈	—	53	2	no	—
134	C₂²×C₈	—	106	4	no	—
135	C₂²×C₈	—	106	2	no	—
136	C₂²×C₈	—	106	4	no	—
137	C₂²×C₈	—	106	2	no	—
138	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
140	C₄×D₄	—	106	2	no	—
141	C₄×D₄	—	106	2	no	—
142	C₂×C₄⋊C₄	—	53	1	no	—
143	C₂×C₂²⋊C₄	—	53	2	no	—
144	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	106	4	no	—
148	C₂×C₁₆	—	106	2	no	—
149	C₂×C₁₆	D₂₁₂	1	4	yes	complemented
150	C₄⋊C₈	—	106	2	no	—
151	C₂²⋊C₈	—	106	4	no	—
152	C₄×C₈	D₂₁₂	1	2	yes	—
153	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
154	C₂⁴	—	53	2	no	—
155	C₂×D₄	—	106	4	no	—
156	C₂×D₄	—	106	2	no	—
157	C₂²×C₄	—	53	2	no	—
158	C₂²×C₄	—	106	2	no	—
159	C₂²×C₄	—	106	4	no	—
160	C₂²×C₄	—	106	2	no	—
161	C₂²×C₄	—	106	4	no	—
162	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
163	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	106	2	no	—
167	C₂×C₈	—	106	4	no	—
168	C₂×C₈	—	212	4	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	—	212	2	no	—
171	C₂×C₈	#11719921	1	2	yes	—
172	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
173	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
174	C₂×C₈	#11719945	1	2	yes	—
175	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
176	C₄⋊C₄	—	106	2	no	—
177	C₂²⋊C₄	—	106	4	no	—
178	C₄²	#11719912	1	2	yes	—
179	C₄²	#11719921	1	2	yes	—
180	C₁₆	—	2	4	no	—
181	C₁₆	—	212	2	no	—
182	C₂³	—	53	4	no	—
183	C₂³	—	106	2	no	—
184	C₂³	—	106	4	no	—
185	C₂³	—	106	2	no	—
186	C₂³	—	106	4	no	—
187	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
188	D₄	—	106	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	106	2	no	—
192	C₂×C₄	—	106	4	no	—
193	C₂×C₄	—	212	2	no	—
194	C₂×C₄	—	212	2	no	—
195	C₂×C₄	—	212	4	no	—
196	C₂×C₄	#13133129	1	2	yes	—
197	C₂×C₄	#13133129	1	2	yes	—
198	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
199	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
200	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
201	C₂×C₄	#13133223	1	2	yes	—
202	C₈	—	2	2	no	—
203	C₈	—	212	2	no	—
204	C₈	#13133191	1	2	yes	—
205	C₈	#13133205	1	1	yes	characteristic
206	C₈	#13133214	1	1	yes	characteristic
207	C₂²	—	106	8	no	—
208	C₂²	—	106	4	no	—
209	C₂²	—	212	2	no	—
210	C₂²	—	212	4	no	—
211	C₂²	—	212	2	no	—
212	C₂²	C₂×#13133129	1	1	yes	characteristic
213	C₂²	C₈×D₂₁₂	1	2	yes	—
214	C₂²	M₃₂×D₅₃	1	2	yes	—
215	C₂²	#16811041	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	212	2	no	—
219	C₄	#16810525	1	1	yes	characteristic
220	C₄	#16810589	1	1	yes	characteristic
221	C₄	#16810702	1	2	yes	—
222	C₂	—	212	4	no	—
223	C₂	#20275509	1	1	yes	characteristic
224	C₂	#20276325	1	1	yes	characteristic
225	C₂	#20276330	1	1	yes	characteristic
226	C₂	#20276542	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×D₂₁₂

C₂

yes

maximal, characteristic

C₂×C₄⋊#11719908

C₂

yes

maximal, characteristic, complemented

C₂×#16810402

C₂

yes

maximal, complemented

C₂×C₅₃×#346

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₅₃

C₂²

yes

—

C₂×C₄×D₂₁₂

C₄

yes

characteristic

C₂²×#11719908

C₂²

yes

—

C₈×D₂₁₂

—

C₈×D₂₁₂

—

C₂×C₄⋊#88322

C₄

yes

characteristic

C₂×#13133129

C₄

yes

—

C₄⋊#11719908

—

#16810402

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

—

C₂²×D₂₁₂

C₈

yes

characteristic

C₂²×C₄×D₅₃

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₄×D₂₁₂

—

C₄×D₂₁₂

—

C₂×C₄⋊Dic₅₃

C₈

yes

characteristic

C₂×#11719921

C₈

yes

—

—

—

—

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

complemented

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂³×D₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₈

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

M₁₆

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

M₃₂

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#121

yes

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

C₈×D₄

yes

—

C₂²×C₅₃

C₂×M₃₂

yes

—

C₂²×C₅₃

#504

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

100

C₂×D₅₃

—

101

C₂×D₅₃

—

102

C₂×D₅₃

—

103

C₄×C₅₃

—

104

C₄×C₅₃

—

105

C₄×C₅₃

#390

yes

characteristic

106

C₄×C₅₃

#408

yes

characteristic

107

C₄×C₅₃

#436

yes

—

108

Dic₅₃

—

109

C₂×C₈×D₄

—

110

C₂×C₄⋊C₁₆

—

111

C₂×C₂²⋊C₁₆

—

112

C₂×#346

D₅₃

yes

characteristic, complemented

113

C₂×C₅₃

#1954

yes

characteristic

114

C₂×C₅₃

#2112

yes

characteristic

115

C₂×C₅₃

#2113

yes

characteristic

116

C₂×C₅₃

#2168

yes

—

117

D₅₃

—

118

C₂³×C₈

—

119

C₂×C₄×D₄

—

120

C₂²×C₁₆

—

121

C₂²×C₁₆

C₂×D₅₃

yes

—

122

C₈×D₄

—

106

—

123

C₈×D₄

—

106

—

124

C₂×C₄⋊C₈

—

125

C₂×C₂²⋊C₈

—

126

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

127

C₄⋊C₁₆

—

106

—

128

C₂²⋊C₁₆

—

106

—

129

#346

—

130

C₅₃

#13190

yes

characteristic, complemented, Sylow

131

C₂²×D₄

—

132

C₂³×C₄

—

133

C₂²×C₈

—

134

C₂²×C₈

—

106

—

135

C₂²×C₈

—

106

—

136

C₂²×C₈

—

106

—

137

C₂²×C₈

—

106

—

138

C₂²×C₈

C₄×D₅₃

yes

characteristic

139

C₂²×C₈

C₂²×D₅₃

yes

characteristic

140

C₄×D₄

—

106

—

141

C₄×D₄

—

106

—

142

C₂×C₄⋊C₄

—

143

C₂×C₂²⋊C₄

—

144

C₂×C₄²

C₄×D₅₃

yes

characteristic

145

C₂×C₁₆

—

146

C₂×C₁₆

—

147

C₂×C₁₆

—

106

—

148

C₂×C₁₆

—

106

—

149

C₂×C₁₆

D₂₁₂

yes

complemented

150

C₄⋊C₈

—

106

—

151

C₂²⋊C₈

—

106

—

152

C₄×C₈

D₂₁₂

yes

—

153

C₄×C₈

C₂²⋊D₅₃

yes

—

154

C₂⁴

—

155

C₂×D₄

—

106

—

156

C₂×D₄

—

106

—

157

C₂²×C₄

—

158

C₂²×C₄

—

106

—

159

C₂²×C₄

—

106

—

160

C₂²×C₄

—

106

—

161

C₂²×C₄

—

106

—

162

C₂²×C₄

C₈×D₅₃

yes

characteristic

163

C₂²×C₄

C₈×D₅₃

yes

characteristic

164

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

165

C₂×C₈

—

166

C₂×C₈

—

106

—

167

C₂×C₈

—

106

—

168

C₂×C₈

—

212

—

169

C₂×C₈

—

212

—

170

C₂×C₈

—

212

—

171

C₂×C₈

#11719921

yes

—

172

C₂×C₈

C₂×D₂₁₂

yes

—

173

C₂×C₈

C₂×D₂₁₂

yes

characteristic

174

C₂×C₈

#11719945

yes

—

175

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

176

C₄⋊C₄

—

106

—

177

C₂²⋊C₄

—

106

—

178

C₄²

#11719912

yes

—

179

C₄²

#11719921

yes

—

180

C₁₆

—

181

C₁₆

—

212

—

182

C₂³

—

183

C₂³

—

106

—

184

C₂³

—

106

—

185

C₂³

—

106

—

186

C₂³

—

106

—

187

C₂³

C₂×C₈×D₅₃

yes

characteristic

188

D₄

—

106

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

106

—

192

C₂×C₄

—

106

—

193

C₂×C₄

—

212

—

194

C₂×C₄

—

212

—

195

C₂×C₄

—

212

—

196

C₂×C₄

#13133129

yes

—

197

C₂×C₄

#13133129

yes

—

198

C₂×C₄

C₂×#11719921

yes

characteristic

199

C₂×C₄

C₄×D₂₁₂

yes

—

200

C₂×C₄

C₂×#11719912

yes

characteristic

201

C₂×C₄

#13133223

yes

—

202

C₈

—

203

C₈

—

212

—

204

C₈

#13133191

yes

—

205

C₈

#13133205

yes

characteristic

206

C₈

#13133214

yes

characteristic

207

C₂²

—

106

—

208

C₂²

—

106

—

209

C₂²

—

212

—

210

C₂²

—

212

—

211

C₂²

—

212

—

212

C₂²

C₂×#13133129

yes

characteristic

213

C₂²

C₈×D₂₁₂

yes

—

214

C₂²

M₃₂×D₅₃

yes

—

215

C₂²

#16811041

yes

—

216

C₄

—

217

C₄

—

218

C₄

—

212

—

219

C₄

#16810525

yes

characteristic

220

C₄

#16810589

yes

characteristic

221

C₄

#16810702

yes

—

222

C₂

—

212

—

223

C₂

#20275509

yes

characteristic

224

C₂

#20276325

yes

characteristic

225

C₂

#20276330

yes

characteristic

226

C₂

#20276542

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₈
Fitting	C₂×C₅₃×#346	C₂
prime radical	C₂²×D₂₁₂	C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#13190
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂²×C₈

Fitting

C₂×C₅₃×#346

C₂

prime radical

C₂²×D₂₁₂

C₈

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. C₂×C₄⋊#11719908	C₂	semidirect
3. C₂×#16810402	C₂	semidirect
4. C₂×C₅₃×#346	C₂	semidirect
37. C₂×C₁₆×C₅₃	D₄	semidirect
112. C₂×#346	D₅₃	semidirect
130. C₅₃	#13190	semidirect
149. C₂×C₁₆	D₂₁₂	semidirect

H ≅ G/N

type

2. C₂×C₄⋊#11719908

C₂

semidirect

3. C₂×#16810402

C₂

semidirect

4. C₂×C₅₃×#346

C₂

semidirect

37. C₂×C₁₆×C₅₃

D₄

semidirect

112. C₂×#346

D₅₃

semidirect

130. C₅₃

#13190

semidirect

149. C₂×C₁₆

D₂₁₂

semidirect

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

subgroup	index
C₂×C₈×D₂₁₂	2
C₂×C₄⋊#11719908	2
C₂×#16810402	2
C₂×C₅₃×#346	2
#13190	53

subgroup

index

C₂×C₈×D₂₁₂

C₂×C₄⋊#11719908

C₂×#16810402

C₂×C₅₃×#346

#13190

Subgroups — #27302251

Conjugacy classes of subgroups

Complements

Maximal subgroups