Group #27302465 — subgroups

Subgroups — #27302465

Order 13568.

438 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₂²×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810426	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20275574	C₂	1	2	yes	maximal
5	C₂×C₅₃×#366	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
7	C₂×Dic₄₂₄	—	2	2	no	—
8	C₂×Dic₄₂₄	—	2	2	no	—
9	C₂×Dic₄₂₄	C₄	1	4	yes	complemented
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	C₂×#13133140	—	2	2	no	—
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810528	—	2	2	no	—
14	#16810426	—	2	4	no	—
15	C₁₆×Dic₅₃	—	2	2	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#366	—	2	2	no	—
19	C₂²×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₂₁₂	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	2	no	—
24	C₂×Dic₂₁₂	—	2	2	no	—
25	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
26	C₂²×#88322	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
28	Dic₄₂₄	—	2	8	no	—
29	C₂×#11719908	—	2	1	no	—
30	C₂×#11719908	—	2	1	no	—
31	C₂×#11719908	D₄	1	2	yes	—
32	#13133140	—	4	4	no	—
33	C₈×Dic₅₃	—	2	1	no	—
34	C₈×Dic₅₃	—	2	1	no	—
35	#13133124	—	4	4	no	—
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
40	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
41	#107×C₅₃	—	2	2	no	—
42	#107×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂²×Dic₅₃	D₈	1	2	yes	—
47	C₂×Dic₁₀₆	—	2	4	no	—
48	C₂×Dic₁₀₆	—	4	2	no	—
49	C₂×Dic₁₀₆	—	4	2	no	—
50	C₂×Dic₁₀₆	—	4	2	no	—
51	C₂×Dic₁₀₆	—	4	2	no	—
52	C₂×Dic₁₀₆	—	4	2	no	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
58	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄×Dic₅₃	—	2	1	no	—
62	C₄×Dic₅₃	—	2	1	no	—
63	C₂×#88322	—	2	1	no	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	D₈	1	2	yes	—
66	Dic₂₁₂	—	2	4	no	—
67	Dic₂₁₂	—	2	4	no	—
68	Dic₂₁₂	—	4	2	no	—
69	Dic₂₁₂	—	4	2	no	—
70	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
71	#11719908	—	2	2	no	—
72	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	—	8	4	no	—
75	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
81	C₂×Dic₅₃	—	2	2	no	—
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	—	4	4	no	—
84	C₂×Dic₅₃	—	8	2	no	—
85	C₂×Dic₅₃	—	8	2	no	—
86	C₂×Dic₅₃	Q₃₂	1	4	yes	complemented
87	Dic₁₀₆	—	4	4	no	—
88	Dic₁₀₆	—	4	4	no	—
89	Dic₁₀₆	—	8	2	no	—
90	Dic₁₀₆	—	8	2	no	—
91	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₈×C₅₃	#121	1	1	yes	characteristic
94	C₈×C₅₃	#124	1	1	yes	characteristic
95	#88322	—	2	2	no	—
96	#13236	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
98	C₂²×C₅₃	#466	1	1	yes	characteristic
99	C₂²×C₅₃	#493	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
101	C₂²×C₅₃	#508	1	2	yes	—
102	C₄×C₅₃	—	16	2	no	—
103	C₄×C₅₃	#390	1	1	yes	characteristic
104	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
105	C₄×C₅₃	#466	1	1	yes	characteristic
106	C₄×C₅₃	#495	1	1	yes	characteristic
107	Dic₅₃	—	2	4	no	—
108	Dic₅₃	—	8	4	no	—
109	C₂²×Q₃₂	—	53	1	no	—
110	C₂×#366	D₅₃	1	1	yes	characteristic
111	C₂×#358	—	53	2	no	—
112	C₂×C₄×C₁₆	—	53	1	no	—
113	#1969	—	53	2	no	—
114	C₂×C₅₃	#1970	1	1	yes	characteristic
115	C₂×C₅₃	C₄×Q₃₂	1	2	yes	—
116	C₂×C₅₃	#2216	1	2	yes	—
117	C₂×C₅₃	#2247	1	1	yes	characteristic
118	C₂×C₅₃	#2249	1	1	yes	characteristic
119	C₂²×Q₁₆	—	53	2	no	—
120	C₂×Q₃₂	—	53	4	no	—
121	C₂×Q₃₂	—	106	2	no	—
122	C₂×Q₃₂	—	106	2	no	—
123	C₂²×C₁₆	—	53	1	no	—
124	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
125	C₂×#107	C₂×D₅₃	1	2	yes	—
126	C₂×#103	—	106	2	no	—
127	C₂×C₄×C₈	—	53	1	no	—
128	#391	—	106	2	no	—
129	#366	—	2	2	no	—
130	#358	—	106	4	no	—
131	C₄×C₁₆	—	106	2	no	—
132	C₅₃	#13236	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	106	2	no	—
134	C₂×Q₁₆	—	53	4	no	—
135	C₂×Q₁₆	—	106	2	no	—
136	C₂×Q₁₆	—	106	2	no	—
137	C₂×Q₁₆	—	106	2	no	—
138	C₂×Q₁₆	—	106	2	no	—
139	C₂×Q₁₆	—	106	2	no	—
140	C₂²×C₈	—	53	1	no	—
141	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄²	—	53	1	no	—
144	Q₃₂	—	106	8	no	—
145	C₂×C₁₆	—	53	2	no	—
146	C₂×C₁₆	—	106	1	no	—
147	C₂×C₁₆	—	106	1	no	—
148	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
149	C₂×C₁₆	D₂₁₂	1	2	yes	—
150	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
151	#107	—	2	2	no	—
152	#107	—	2	2	no	—
153	#103	—	212	4	no	—
154	C₄×C₈	—	106	1	no	—
155	C₄×C₈	—	106	1	no	—
156	#95	—	212	4	no	—
157	C₂×Q₈	—	106	4	no	—
158	C₂×Q₈	—	212	2	no	—
159	C₂×Q₈	—	212	2	no	—
160	C₂×Q₈	—	212	2	no	—
161	C₂×Q₈	—	212	2	no	—
162	C₂×Q₈	—	212	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	53	2	no	—
165	C₂²×C₄	—	212	2	no	—
166	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
167	Q₁₆	—	106	4	no	—
168	Q₁₆	—	106	4	no	—
169	Q₁₆	—	212	2	no	—
170	Q₁₆	—	212	2	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	106	1	no	—
173	C₂×C₈	—	106	1	no	—
174	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
176	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
177	C₂×C₈	#11719949	1	1	yes	characteristic
178	C₂×C₈	#11719949	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄²	—	106	1	no	—
183	C₄²	—	106	1	no	—
184	C₁₆	—	106	2	no	—
185	C₁₆	#11719921	1	4	yes	—
186	C₂³	D₈×D₅₃	1	1	yes	characteristic
187	Q₈	—	424	2	no	—
188	Q₈	—	424	2	no	—
189	Q₈	—	212	4	no	—
190	Q₈	—	212	4	no	—
191	C₂×C₄	—	8	4	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	53	4	no	—
194	C₂×C₄	—	106	2	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	—	424	2	no	—
197	C₂×C₄	—	424	2	no	—
198	C₂×C₄	—	212	4	no	—
199	C₂×C₄	#13133179	1	1	yes	characteristic
200	C₂×C₄	#13133192	1	1	yes	characteristic
201	C₂×C₄	#13133213	1	1	yes	characteristic
202	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
203	C₂×C₄	#13133240	1	1	yes	characteristic
204	C₂×C₄	#13133240	1	1	yes	characteristic
205	C₈	—	106	2	no	—
206	C₈	C₂×#11719921	1	1	yes	characteristic
207	C₈	#13133179	1	1	yes	characteristic
208	C₈	#13133192	1	1	yes	characteristic
209	C₈	#13133206	1	1	yes	characteristic
210	C₂²	#16810705	1	1	yes	characteristic
211	C₂²	#16810860	1	1	yes	characteristic
212	C₂²	#16810998	1	1	yes	characteristic
213	C₂²	Q₃₂×D₅₃	1	2	yes	—
214	C₂²	#16811052	1	2	yes	—
215	C₄	—	16	2	no	—
216	C₄	—	106	4	no	—
217	C₄	—	424	4	no	—
218	C₄	#16810524	1	1	yes	characteristic
219	C₄	#16810705	1	1	yes	characteristic
220	C₄	#16810860	1	1	yes	characteristic
221	C₄	#16811008	1	1	yes	characteristic
222	C₂	#20275579	1	1	yes	characteristic
223	C₂	#20276573	1	2	yes	—
224	C₂	#20276807	1	2	yes	—
225	C₂	#20276966	1	1	yes	characteristic
226	C₂	#20276973	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₂₄

C₂

yes

maximal, characteristic

C₂×#16810426

C₂

yes

maximal

C₂×C₁₆×Dic₅₃

C₂

yes

maximal, characteristic

#20275574

C₂

yes

maximal

C₂×C₅₃×#366

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

C₂²

yes

—

C₂×Dic₄₂₄

—

C₂×Dic₄₂₄

—

C₂×Dic₄₂₄

C₄

yes

complemented

C₂²×#11719908

C₂²

yes

characteristic

C₂×#13133140

—

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810528

—

#16810426

—

C₁₆×Dic₅₃

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#107×C₅₃

C₂²

yes

—

C₅₃×#366

—

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂×C₄

yes

—

C₂²×#88322

D₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

characteristic

Dic₄₂₄

—

C₂×#11719908

—

C₂×#11719908

—

C₂×#11719908

D₄

yes

—

#13133140

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

#13133124

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

D₄

yes

—

#107×C₅₃

—

#107×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₈

yes

—

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

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

D₈

yes

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

C₂²⋊C₄

yes

—

#11719908

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×D₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

Q₃₂

yes

complemented

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₄×D₄

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₄×D₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

100

C₂²×C₅₃

C₂×Q₃₂

yes

—

101

C₂²×C₅₃

#508

yes

—

102

C₄×C₅₃

—

103

C₄×C₅₃

#390

yes

characteristic

104

C₄×C₅₃

C₄×D₈

yes

characteristic

105

C₄×C₅₃

#466

yes

characteristic

106

C₄×C₅₃

#495

yes

characteristic

107

Dic₅₃

—

108

Dic₅₃

—

109

C₂²×Q₃₂

—

110

C₂×#366

D₅₃

yes

characteristic

111

C₂×#358

—

112

C₂×C₄×C₁₆

—

113

#1969

—

114

C₂×C₅₃

#1970

yes

characteristic

115

C₂×C₅₃

C₄×Q₃₂

yes

—

116

C₂×C₅₃

#2216

yes

—

117

C₂×C₅₃

#2247

yes

characteristic

118

C₂×C₅₃

#2249

yes

characteristic

119

C₂²×Q₁₆

—

120

C₂×Q₃₂

—

121

C₂×Q₃₂

—

106

—

122

C₂×Q₃₂

—

106

—

123

C₂²×C₁₆

—

124

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

125

C₂×#107

C₂×D₅₃

yes

—

126

C₂×#103

—

106

—

127

C₂×C₄×C₈

—

128

#391

—

106

—

129

#366

—

130

#358

—

106

—

131

C₄×C₁₆

—

106

—

132

C₅₃

#13236

yes

characteristic, complemented, Sylow

133

C₂²×Q₈

—

106

—

134

C₂×Q₁₆

—

135

C₂×Q₁₆

—

106

—

136

C₂×Q₁₆

—

106

—

137

C₂×Q₁₆

—

106

—

138

C₂×Q₁₆

—

106

—

139

C₂×Q₁₆

—

106

—

140

C₂²×C₈

—

141

C₂²×C₈

C₂²×D₅₃

yes

characteristic

142

C₂×C₄⋊C₄

—

143

C₂×C₄²

—

144

Q₃₂

—

106

—

145

C₂×C₁₆

—

146

C₂×C₁₆

—

106

—

147

C₂×C₁₆

—

106

—

148

C₂×C₁₆

C₄×D₅₃

yes

—

149

C₂×C₁₆

D₂₁₂

yes

—

150

C₂×C₁₆

C₂²⋊D₅₃

yes

—

151

#107

—

152

#107

—

153

#103

—

212

—

154

C₄×C₈

—

106

—

155

C₄×C₈

—

106

—

156

#95

—

212

—

157

C₂×Q₈

—

106

—

158

C₂×Q₈

—

212

—

159

C₂×Q₈

—

212

—

160

C₂×Q₈

—

212

—

161

C₂×Q₈

—

212

—

162

C₂×Q₈

—

212

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

—

212

—

166

C₂²×C₄

D₄×D₅₃

yes

characteristic

167

Q₁₆

—

106

—

168

Q₁₆

—

106

—

169

Q₁₆

—

212

—

170

Q₁₆

—

212

—

171

C₂×C₈

—

172

C₂×C₈

—

106

—

173

C₂×C₈

—

106

—

174

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

175

C₂×C₈

C₂×D₂₁₂

yes

characteristic

176

C₂×C₈

D₄×D₅₃

yes

characteristic

177

C₂×C₈

#11719949

yes

characteristic

178

C₂×C₈

#11719949

yes

characteristic

179

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄²

—

106

—

183

C₄²

—

106

—

184

C₁₆

—

106

—

185

C₁₆

#11719921

yes

—

186

C₂³

D₈×D₅₃

yes

characteristic

187

Q₈

—

424

—

188

Q₈

—

424

—

189

Q₈

—

212

—

190

Q₈

—

212

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

106

—

195

C₂×C₄

—

106

—

196

C₂×C₄

—

424

—

197

C₂×C₄

—

424

—

198

C₂×C₄

—

212

—

199

C₂×C₄

#13133179

yes

characteristic

200

C₂×C₄

#13133192

yes

characteristic

201

C₂×C₄

#13133213

yes

characteristic

202

C₂×C₄

D₈×D₅₃

yes

characteristic

203

C₂×C₄

#13133240

yes

characteristic

204

C₂×C₄

#13133240

yes

characteristic

205

C₈

—

106

—

206

C₈

C₂×#11719921

yes

characteristic

207

C₈

#13133179

yes

characteristic

208

C₈

#13133192

yes

characteristic

209

C₈

#13133206

yes

characteristic

210

C₂²

#16810705

yes

characteristic

211

C₂²

#16810860

yes

characteristic

212

C₂²

#16810998

yes

characteristic

213

C₂²

Q₃₂×D₅₃

yes

—

214

C₂²

#16811052

yes

—

215

C₄

—

216

C₄

—

106

—

217

C₄

—

424

—

218

C₄

#16810524

yes

characteristic

219

C₄

#16810705

yes

characteristic

220

C₄

#16810860

yes

characteristic

221

C₄

#16811008

yes

characteristic

222

C₂

#20275579

yes

characteristic

223

C₂

#20276573

yes

—

224

C₂

#20276807

yes

—

225

C₂

#20276966

yes

characteristic

226

C₂

#20276973

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#366	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#366	D₅₃
hypocenter	C₅₃	#13236
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#366

C₂

socle, prime radical

C₂³×C₅₃

C₂×D₈

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
9. C₂×Dic₄₂₄	C₄	semidirect
86. C₂×Dic₅₃	Q₃₂	semidirect
132. C₅₃	#13236	semidirect

H ≅ G/N

type

9. C₂×Dic₄₂₄

C₄

semidirect

86. C₂×Dic₅₃

Q₃₂

semidirect

132. C₅₃

#13236

semidirect

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

subgroup	index
C₂²×Dic₄₂₄	2
C₂×#16810426	2
C₂×C₁₆×Dic₅₃	2
#20275574	2
C₂×C₅₃×#366	2
#13236	53

subgroup

index

C₂²×Dic₄₂₄

C₂×#16810426

C₂×C₁₆×Dic₅₃

#20275574

C₂×C₅₃×#366

#13236

Subgroups — #27302465

Conjugacy classes of subgroups

Complements

Maximal subgroups