Group #27287105 — subgroups

Subgroups — #27287105

Order 13568.

504 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20279744	C₂	1	1	yes	maximal, characteristic
2	#20275664	C₂	1	2	yes	maximal
3	#20272598	C₂	1	1	yes	maximal, characteristic
4	#20272555	C₂	1	2	yes	maximal
5	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133216	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	#16810543	—	2	2	no	—
9	#16810519	—	2	2	no	—
10	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	#16810385	—	2	2	no	—
12	#16810385	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#103×C₅₃	—	2	2	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#384	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₀₆	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	#13133216	—	4	4	no	—
20	#13133216	—	4	4	no	—
21	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
22	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
23	C₂×#11719919	—	2	4	no	—
24	C₂×C₄×Dic₅₃	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	#13133124	—	2	2	no	—
28	#13133124	—	2	2	no	—
29	C₂²×Q₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₄⋊Q₈×C₅₃	—	2	1	no	—
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₄⋊Q₈×C₅₃	Q₈	1	2	yes	complemented
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#103×C₅₃	—	4	4	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₂×Q₈×C₅₃	—	2	2	no	—
41	C₂×Q₈×C₅₃	—	4	2	no	—
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂×Q₈×C₅₃	—	4	1	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	4	1	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×Dic₅₃	—	2	2	no	—
49	C₂²×Dic₅₃	—	2	2	no	—
50	C₂×Dic₁₀₆	—	4	8	no	—
51	C₂×Dic₁₀₆	—	8	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	4	1	no	—
54	C₂×C₈×C₅₃	—	4	1	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₅₃	#41	1	1	yes	characteristic
62	C₄²×C₅₃	#41	1	2	yes	—
63	C₄⋊Dic₅₃	—	2	2	no	—
64	C₄⋊Dic₅₃	—	2	2	no	—
65	C₄⋊Dic₅₃	SD₁₆	1	4	yes	—
66	C₄⋊Dic₅₃	Q₁₆	1	4	yes	—
67	#11719919	—	4	8	no	—
68	C₄×Dic₅₃	—	4	2	no	—
69	C₄×Dic₅₃	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	—	4	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₂³×C₅₃	#120	1	1	yes	characteristic
74	Q₈×C₅₃	—	4	4	no	—
75	Q₈×C₅₃	—	8	2	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	4	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	8	1	no	—
82	C₂×C₄×C₅₃	—	8	1	no	—
83	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#124	1	2	yes	—
86	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₅₃	#137	1	2	yes	—
91	C₂×Dic₅₃	—	2	4	no	—
92	C₂×Dic₅₃	—	2	4	no	—
93	C₂×Dic₅₃	—	4	2	no	—
94	C₂×Dic₅₃	—	4	2	no	—
95	C₂×Dic₅₃	—	4	2	no	—
96	C₂×Dic₅₃	—	4	2	no	—
97	Dic₁₀₆	—	8	8	no	—
98	C₈×C₅₃	—	4	2	no	—
99	#88322	—	8	2	no	—
100	#10349	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	#393	1	1	yes	characteristic
102	C₂²×C₅₃	#450	1	2	yes	—
103	C₂²×C₅₃	#451	1	2	yes	—
104	C₂²×C₅₃	#453	1	1	yes	characteristic
105	C₂²×C₅₃	#454	1	1	yes	characteristic
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	8	2	no	—
109	C₄×C₅₃	—	8	2	no	—
110	C₄×C₅₃	#475	1	2	yes	—
111	C₄×C₅₃	#477	1	2	yes	—
112	C₄×C₅₃	#487	1	2	yes	—
113	C₄×C₅₃	#489	1	2	yes	—
114	Dic₅₃	—	4	4	no	—
115	Dic₅₃	—	4	4	no	—
116	#2551	—	53	1	no	—
117	#1987	—	53	2	no	—
118	#1956	D₅₃	1	1	yes	characteristic
119	#1296	—	53	1	no	—
120	#1279	—	53	2	no	—
121	C₂×C₅₃	#1629	1	1	yes	characteristic
122	C₂×C₅₃	#1633	1	1	yes	characteristic
123	C₂×C₅₃	#1637	1	2	yes	—
124	C₂×C₅₃	#2024	1	2	yes	—
125	C₂×C₅₃	#2027	1	1	yes	characteristic
126	C₂×C₄⋊Q₈	—	106	2	no	—
127	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	53	2	no	—
129	C₂×#103	—	2	2	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#395	—	106	2	no	—
132	#389	—	106	2	no	—
133	#384	C₂×D₅₃	1	1	yes	characteristic
134	C₄×C₄⋊C₄	—	53	2	no	—
135	#340	—	106	2	no	—
136	#340	—	106	2	no	—
137	C₅₃	#10349	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	2	1	no	—
139	C₂²×Q₈	—	106	2	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	106	2	no	—
142	C₄⋊Q₈	—	2	1	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	212	4	no	—
145	C₄⋊Q₈	—	212	4	no	—
146	C₄⋊Q₈	Dic₁₀₆	1	2	yes	complemented
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	53	2	no	—
150	C₂×C₄⋊C₄	—	53	2	no	—
151	C₂×C₄⋊C₄	—	106	4	no	—
152	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
153	C₂×C₄²	—	106	2	no	—
154	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
155	C₄⋊C₈	—	106	2	no	—
156	C₄⋊C₈	—	106	2	no	—
157	#103	—	4	4	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	106	2	no	—
160	#95	—	106	2	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	424	2	no	—
166	C₂×Q₈	—	212	8	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	—	106	2	no	—
170	C₂²×C₄	—	106	2	no	—
171	C₂²×C₄	D₄×D₅₃	1	2	yes	—
172	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	212	2	no	—
177	C₂×C₈	—	212	2	no	—
178	C₂×C₈	—	212	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	53	4	no	—
185	C₄⋊C₄	—	53	4	no	—
186	C₄⋊C₄	—	106	2	no	—
187	C₄⋊C₄	—	106	2	no	—
188	C₄⋊C₄	—	212	8	no	—
189	C₄²	—	212	2	no	—
190	C₄²	—	212	2	no	—
191	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
192	C₄²	#11719945	1	1	yes	characteristic
193	C₄²	#11719947	1	2	yes	—
194	C₂³	#13133185	1	1	yes	characteristic
195	Q₈	—	4	4	no	—
196	Q₈	—	8	2	no	—
197	Q₈	—	424	8	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	8	1	no	—
205	C₂×C₄	—	106	4	no	—
206	C₂×C₄	—	106	4	no	—
207	C₂×C₄	—	212	2	no	—
208	C₂×C₄	—	212	2	no	—
209	C₂×C₄	—	212	2	no	—
210	C₂×C₄	—	212	2	no	—
211	C₂×C₄	#13133194	1	2	yes	—
212	C₂×C₄	#13133196	1	1	yes	characteristic
213	C₂×C₄	#13133203	1	2	yes	—
214	C₂×C₄	#13133204	1	1	yes	characteristic
215	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
216	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
217	C₂×C₄	#13133244	1	2	yes	—
218	C₂×C₄	#13133249	1	2	yes	—
219	C₈	—	4	2	no	—
220	C₈	—	424	2	no	—
221	C₂²	#16810535	1	1	yes	characteristic
222	C₂²	#16810772	1	2	yes	—
223	C₂²	#16810777	1	2	yes	—
224	C₂²	#16810790	1	1	yes	characteristic
225	C₂²	#16810795	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	8	2	no	—
229	C₄	—	8	2	no	—
230	C₄	—	212	4	no	—
231	C₄	—	212	4	no	—
232	C₄	#16810906	1	2	yes	—
233	C₄	#16810916	1	2	yes	—
234	C₄	#16810972	1	2	yes	—
235	C₄	#16810984	1	2	yes	—
236	C₂	#20273829	1	1	yes	characteristic
237	C₂	#20273853	1	1	yes	characteristic
238	C₂	#20273877	1	2	yes	—
239	C₂	#20275854	1	2	yes	—
240	C₂	#20275871	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20279744

C₂

yes

maximal, characteristic

#20275664

C₂

yes

maximal

#20272598

C₂

yes

maximal, characteristic

#20272555

C₂

yes

maximal

C₅₃×#1956

C₂

yes

maximal, characteristic

C₂×#13133216

—

C₂×C₄⋊#88322

C₂²

yes

—

#16810543

—

#16810519

—

C₄×C₄⋊Dic₅₃

C₂²

yes

—

#16810385

—

#16810385

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×#103×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

C₂²×#88322

—

#13133216

—

#13133216

—

C₂×C₄⋊Dic₅₃

D₄

yes

—

C₂×C₄⋊Dic₅₃

D₄

yes

—

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

—

—

—

—

C₂²×Q₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

Q₈

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#103×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

SD₁₆

yes

—

C₄⋊Dic₅₃

Q₁₆

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

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

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

100

#10349

—

maximal, Sylow

101

C₂²×C₅₃

#393

yes

characteristic

102

C₂²×C₅₃

#450

yes

—

103

C₂²×C₅₃

#451

yes

—

104

C₂²×C₅₃

#453

yes

characteristic

105

C₂²×C₅₃

#454

yes

characteristic

106

C₄×C₅₃

—

107

C₄×C₅₃

—

108

C₄×C₅₃

—

109

C₄×C₅₃

—

110

C₄×C₅₃

#475

yes

—

111

C₄×C₅₃

#477

yes

—

112

C₄×C₅₃

#487

yes

—

113

C₄×C₅₃

#489

yes

—

114

Dic₅₃

—

115

Dic₅₃

—

116

#2551

—

117

#1987

—

118

#1956

D₅₃

yes

characteristic

119

#1296

—

120

#1279

—

121

C₂×C₅₃

#1629

yes

characteristic

122

C₂×C₅₃

#1633

yes

characteristic

123

C₂×C₅₃

#1637

yes

—

124

C₂×C₅₃

#2024

yes

—

125

C₂×C₅₃

#2027

yes

characteristic

126

C₂×C₄⋊Q₈

—

106

—

127

C₂×C₄⋊Q₈

C₂×D₅₃

yes

characteristic

128

C₂×C₄⋊C₈

—

129

C₂×#103

—

130

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

131

#395

—

106

—

132

#389

—

106

—

133

#384

C₂×D₅₃

yes

characteristic

134

C₄×C₄⋊C₄

—

135

#340

—

106

—

136

#340

—

106

—

137

C₅₃

#10349

yes

characteristic, complemented, Sylow

138

C₂²×Q₈

—

139

C₂²×Q₈

—

106

—

140

C₂²×C₈

—

141

C₂²×C₈

—

106

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

212

—

145

C₄⋊Q₈

—

212

—

146

C₄⋊Q₈

Dic₁₀₆

yes

complemented

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄⋊C₄

—

106

—

152

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

153

C₂×C₄²

—

106

—

154

C₂×C₄²

C₂²×D₅₃

yes

characteristic

155

C₄⋊C₈

—

106

—

156

C₄⋊C₈

—

106

—

157

#103

—

158

C₄×C₈

—

159

#95

—

106

—

160

#95

—

106

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

424

—

166

C₂×Q₈

—

212

—

167

C₂²×C₄

—

168

C₂²×C₄

—

169

C₂²×C₄

—

106

—

170

C₂²×C₄

—

106

—

171

C₂²×C₄

D₄×D₅₃

yes

—

172

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

212

—

177

C₂×C₈

—

212

—

178

C₂×C₈

—

212

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

106

—

187

C₄⋊C₄

—

106

—

188

C₄⋊C₄

—

212

—

189

C₄²

—

212

—

190

C₄²

—

212

—

191

C₄²

C₂×Dic₁₀₆

yes

characteristic

192

C₄²

#11719945

yes

characteristic

193

C₄²

#11719947

yes

—

194

C₂³

#13133185

yes

characteristic

195

Q₈

—

196

Q₈

—

197

Q₈

—

424

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

106

—

206

C₂×C₄

—

106

—

207

C₂×C₄

—

212

—

208

C₂×C₄

—

212

—

209

C₂×C₄

—

212

—

210

C₂×C₄

—

212

—

211

C₂×C₄

#13133194

yes

—

212

C₂×C₄

#13133196

yes

characteristic

213

C₂×C₄

#13133203

yes

—

214

C₂×C₄

#13133204

yes

characteristic

215

C₂×C₄

SD₁₆×D₅₃

yes

—

216

C₂×C₄

Q₁₆×D₅₃

yes

—

217

C₂×C₄

#13133244

yes

—

218

C₂×C₄

#13133249

yes

—

219

C₈

—

220

C₈

—

424

—

221

C₂²

#16810535

yes

characteristic

222

C₂²

#16810772

yes

—

223

C₂²

#16810777

yes

—

224

C₂²

#16810790

yes

characteristic

225

C₂²

#16810795

yes

characteristic

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

212

—

231

C₄

—

212

—

232

C₄

#16810906

yes

—

233

C₄

#16810916

yes

—

234

C₄

#16810972

yes

—

235

C₄

#16810984

yes

—

236

C₂

#20273829

yes

characteristic

237

C₂

#20273853

yes

characteristic

238

C₂

#20273877

yes

—

239

C₂

#20275854

yes

—

240

C₂

#20275871

yes

characteristic

	subgroup	G/H
center	C₂³	#13133185
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1956	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#1956	D₅₃
hypocenter	C₅₃	#10349
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#13133185

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1956

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
33. C₄⋊Q₈×C₅₃	Q₈	semidirect
137. C₅₃	#10349	semidirect
146. C₄⋊Q₈	Dic₁₀₆	semidirect

H ≅ G/N

type

33. C₄⋊Q₈×C₅₃

Q₈

semidirect

137. C₅₃

#10349

semidirect

146. C₄⋊Q₈

Dic₁₀₆

semidirect

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

subgroup	index
#20279744	2
#20275664	2
#20272598	2
#20272555	2
C₅₃×#1956	2
#10349	53

subgroup

index

C₅₃×#1956

Subgroups — #27287105

Conjugacy classes of subgroups

Complements

Maximal subgroups