Group #27301542 — subgroups

Subgroups — #27301542

Order 13568.

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

normal subgroups only

This group has 490 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	#20287818	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276431	C₂	1	2	yes	maximal, complemented
3	#20274761	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272816	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3058	C₂	1	1	yes	maximal, characteristic
6	#16811040	—	2	4	no	—
7	C₂×#13133157	—	2	2	no	—
8	C₂×#13133157	C₂²	1	2	yes	—
9	#16810661	C₂²	1	2	yes	—
10	#16810597	—	2	1	no	—
11	#16810424	—	2	4	no	—
12	#16810380	—	2	1	no	—
13	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#535	—	2	1	no	—
16	C₅₃×#469	C₄	1	4	yes	complemented
17	C₅₃×#466	—	2	2	no	—
18	C₅₃×#427	C₂²	1	1	yes	characteristic
19	C₅₃×#418	—	2	1	no	—
20	#13133176	—	2	2	no	—
21	#13133157	—	2	4	no	—
22	#13133157	—	2	2	no	—
23	#13133157	—	2	2	no	—
24	C₂×#11719908	—	2	4	no	—
25	C₂×#11719908	D₄	1	4	yes	complemented
26	#13133151	—	2	2	no	—
27	#13133149	—	2	2	no	—
28	#13133132	—	4	2	no	—
29	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×D₄×C₅₃	—	2	1	no	—
31	C₂×D₈×C₅₃	—	2	1	no	—
32	C₂×D₈×C₅₃	—	2	2	no	—
33	C₂×D₈×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
35	#131×C₅₃	—	2	2	no	—
36	#131×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	#127×C₅₃	—	4	1	no	—
42	#124×C₅₃	—	4	1	no	—
43	#121×C₅₃	—	2	4	no	—
44	#120×C₅₃	—	4	2	no	—
45	#117×C₅₃	—	2	1	no	—
46	#107×C₅₃	—	2	1	no	—
47	#106×C₅₃	C₂×C₄	1	2	yes	—
48	#102×C₅₃	—	2	4	no	—
49	C₂⁴×C₅₃	—	4	1	no	—
50	#41×C₅₃	—	2	2	no	—
51	#41×C₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	SD₁₆	1	1	yes	characteristic
53	C₂×D₄×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	—	4	4	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	4	1	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	D₈	1	2	yes	—
59	C₂×D₄×C₅₃	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	D₈×C₅₃	—	2	4	no	—
63	D₈×C₅₃	—	4	1	no	—
64	D₈×C₅₃	—	4	1	no	—
65	M₁₆×C₅₃	—	2	2	no	—
66	M₁₆×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	#41	1	2	yes	—
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₂²⋊C₄×C₅₃	—	4	1	no	—
76	C₂²⋊C₄×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	—	4	1	no	—
78	C₂²⋊Dic₅₃	—	4	2	no	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₄×Dic₅₃	—	4	2	no	—
81	#11719908	—	2	4	no	—
82	C₂³×C₅₃	—	4	2	no	—
83	C₂³×C₅₃	—	8	2	no	—
84	C₂³×C₅₃	—	16	2	no	—
85	C₂³×C₅₃	#102	1	2	yes	—
86	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
87	Q₈×C₅₃	—	2	2	no	—
88	D₄×C₅₃	—	2	4	no	—
89	D₄×C₅₃	—	2	2	no	—
90	D₄×C₅₃	—	4	2	no	—
91	D₄×C₅₃	—	4	2	no	—
92	D₄×C₅₃	—	8	2	no	—
93	D₄×C₅₃	—	16	4	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	#102	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#121	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#135	1	2	yes	—
102	C₂×Dic₅₃	—	8	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	C₄×D₄	1	2	yes	—
105	C₈×C₅₃	#120	1	1	yes	characteristic
106	C₈×C₅₃	#123	1	1	yes	characteristic
107	#13074	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	—	4	1	no	—
110	C₂²×C₅₃	—	4	2	no	—
111	C₂²×C₅₃	—	8	4	no	—
112	C₂²×C₅₃	—	8	2	no	—
113	C₂²×C₅₃	—	16	4	no	—
114	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
115	C₂²×C₅₃	#466	1	2	yes	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	8	2	no	—
118	C₄×C₅₃	#386	1	1	yes	characteristic
119	C₄×C₅₃	#416	1	1	yes	characteristic
120	C₄×C₅₃	#465	1	2	yes	—
121	Dic₅₃	—	16	2	no	—
122	#3407	—	53	1	no	—
123	#3058	D₅₃	1	1	yes	characteristic
124	#2140	—	53	2	no	—
125	#1812	—	53	1	no	—
126	#1384	—	53	2	no	—
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	—	2	1	no	—
129	C₂×C₅₃	—	4	2	no	—
130	C₂×C₅₃	—	16	2	no	—
131	C₂×C₅₃	#1893	1	1	yes	characteristic
132	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
133	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
134	#535	—	2	1	no	—
135	#504	—	106	4	no	—
136	C₂×M₃₂	—	53	2	no	—
137	C₂×M₃₂	—	106	2	no	—
138	#469	Dic₅₃	1	4	yes	complemented
139	#466	—	2	2	no	—
140	#427	—	53	2	no	—
141	#427	C₂×D₅₃	1	1	yes	characteristic
142	#418	—	2	1	no	—
143	#410	—	106	1	no	—
144	#357	—	106	4	no	—
145	H(3,ℤ/4)	—	106	1	no	—
146	C₅₃	#13074	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂²×D₄	—	2	1	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	—	2	2	no	—
151	C₂×D₈	—	2	1	no	—
152	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
153	#131	—	2	2	no	—
154	#131	—	2	2	no	—
155	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
156	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
158	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	#127	—	4	1	no	—
160	#124	—	4	1	no	—
161	#121	—	2	4	no	—
162	#120	—	4	2	no	—
163	#117	—	2	1	no	—
164	#117	—	106	2	no	—
165	M₃₂	—	106	4	no	—
166	M₃₂	—	106	2	no	—
167	M₃₂	—	106	2	no	—
168	C₂×C₁₆	—	53	4	no	—
169	C₂×C₁₆	—	106	4	no	—
170	#107	—	2	1	no	—
171	#107	—	106	2	no	—
172	#106	—	106	2	no	—
173	#106	C₂×Dic₅₃	1	2	yes	—
174	#102	—	2	4	no	—
175	#99	—	212	2	no	—
176	C₂⁴	—	4	1	no	—
177	#41	—	2	2	no	—
178	#41	—	2	2	no	—
179	C₂×Q₈	#11719923	1	1	yes	characteristic
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	1	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	#11719922	1	2	yes	—
186	C₂×D₄	#11719923	1	1	yes	characteristic
187	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
188	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
189	D₈	—	2	4	no	—
190	D₈	—	4	1	no	—
191	D₈	—	4	1	no	—
192	M₁₆	—	2	2	no	—
193	M₁₆	—	2	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
197	C₂×C₈	D₄×D₅₃	1	2	yes	—
198	C₂×C₈	#11719947	1	2	yes	—
199	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	212	2	no	—
203	C₂²⋊C₄	—	4	1	no	—
204	C₂²⋊C₄	—	4	4	no	—
205	C₂²⋊C₄	—	212	2	no	—
206	C₄²	—	4	1	no	—
207	C₄²	—	212	2	no	—
208	C₁₆	—	106	4	no	—
209	C₂³	—	4	2	no	—
210	C₂³	—	8	2	no	—
211	C₂³	—	16	2	no	—
212	C₂³	#13133138	1	2	yes	—
213	C₂³	C₂×#11719922	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	D₄	—	2	4	no	—
216	D₄	—	2	2	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	2	no	—
219	D₄	—	8	2	no	—
220	D₄	—	16	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	424	2	no	—
225	C₂×C₄	#13133138	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
227	C₂×C₄	#13133189	1	2	yes	—
228	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
229	C₂×C₄	#13133237	1	2	yes	—
230	C₈	—	2	2	no	—
231	C₈	D₄×Dic₅₃	1	2	yes	—
232	C₈	#13133185	1	1	yes	characteristic
233	C₈	#13133198	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	8	4	no	—
238	C₂²	—	8	2	no	—
239	C₂²	—	16	4	no	—
240	C₂²	C₂×#13133138	1	1	yes	characteristic
241	C₂²	#16810857	1	2	yes	—
242	C₄	—	2	2	no	—
243	C₄	—	8	2	no	—
244	C₄	—	848	2	no	—
245	C₄	#16810503	1	1	yes	characteristic
246	C₄	#16810619	1	1	yes	characteristic
247	C₄	#16810850	1	2	yes	—
248	C₂	—	2	1	no	—
249	C₂	—	2	2	no	—
250	C₂	—	4	2	no	—
251	C₂	—	16	2	no	—
252	C₂	#20275144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20287818

C₂

yes

maximal, characteristic, complemented

#20276431

C₂

yes

maximal, complemented

#20274761

C₂

yes

maximal, characteristic, complemented

#20272816

C₂

yes

maximal, complemented

C₅₃×#3058

C₂

yes

maximal, characteristic

#16811040

—

C₂×#13133157

—

C₂×#13133157

C₂²

yes

—

#16810661

C₂²

yes

—

#16810597

—

#16810424

—

#16810380

—

C₂²×D₈×C₅₃

C₂²

yes

characteristic

C₂×#131×C₅₃

C₂²

yes

characteristic

C₅₃×#535

—

C₅₃×#469

C₄

yes

complemented

C₅₃×#466

—

C₅₃×#427

C₂²

yes

characteristic

C₅₃×#418

—

—

—

—

—

—

D₄

yes

—

—

—

C₂×#41×C₅₃

D₄

yes

characteristic

C₂²×D₄×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

C₂×C₄

yes

—

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#127×C₅₃

—

#124×C₅₃

—

#121×C₅₃

—

#120×C₅₃

—

#117×C₅₃

—

#107×C₅₃

—

#106×C₅₃

C₂×C₄

yes

—

#102×C₅₃

—

C₂⁴×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

SD₁₆

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₈

yes

—

C₂×D₄×C₅₃

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#102

yes

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

100

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

101

C₂×C₄×C₅₃

#135

yes

—

102

C₂×Dic₅₃

—

103

C₈×C₅₃

—

104

C₈×C₅₃

C₄×D₄

yes

—

105

C₈×C₅₃

#120

yes

characteristic

106

C₈×C₅₃

yes

107

—

108

C₂²×C₅₃

—

109

C₂²×C₅₃

—

110

C₂²×C₅₃

—

111

C₂²×C₅₃

—

112

C₂²×C₅₃

—

113

C₂²×C₅₃

—

114

C₂²×C₅₃

C₂×#102

yes

characteristic

115

C₂²×C₅₃

#466

yes

—

116

C₄×C₅₃

—

117

C₄×C₅₃

—

118

C₄×C₅₃

#386

yes

characteristic

119

C₄×C₅₃

#416

yes

characteristic

120

C₄×C₅₃

#465

yes

—

121

Dic₅₃

—

122

#3407

—

123

#3058

D₅₃

yes

characteristic

124

#2140

—

125

#1812

—

126

#1384

—

127

C₂×C₅₃

—

128

C₂×C₅₃

—

129

C₂×C₅₃

—

130

C₂×C₅₃

—

131

C₂×C₅₃

#1893

yes

characteristic

132

C₂²×D₈

C₂×D₅₃

yes

characteristic

133

C₂×#131

C₂×D₅₃

yes

characteristic

134

#535

—

135

#504

—

106

—

136

C₂×M₃₂

—

137

C₂×M₃₂

—

106

—

138

#469

Dic₅₃

yes

complemented

139

#466

—

140

#427

—

141

#427

C₂×D₅₃

yes

characteristic

142

#418

—

143

#410

—

106

—

144

#357

—

106

—

145

H(3,ℤ/4)

—

106

—

146

C₅₃

#13074

yes

characteristic, complemented, Sylow

147

C₂×#41

C₂²⋊D₅₃

yes

characteristic

148

C₂²×D₄

—

149

C₂×D₈

—

150

C₂×D₈

—

151

C₂×D₈

—

152

C₂×D₈

C₂×Dic₅₃

yes

—

153

#131

—

154

#131

—

155

C₂×M₁₆

C₂×Dic₅₃

yes

—

156

C₂×M₁₆

C₂²⋊D₅₃

yes

characteristic

157

C₂²×C₈

C₂²⋊D₅₃

yes

—

158

C₂²×C₈

C₂²×D₅₃

yes

characteristic

159

#127

—

160

#124

—

161

#121

—

162

#120

—

163

#117

—

164

#117

—

106

—

165

M₃₂

—

106

—

166

M₃₂

—

106

—

167

M₃₂

—

106

—

168

C₂×C₁₆

—

169

C₂×C₁₆

—

106

—

170

#107

—

171

#107

—

106

—

172

#106

—

106

—

173

#106

C₂×Dic₅₃

yes

—

174

#102

—

175

#99

—

212

—

176

C₂⁴

—

177

#41

—

178

#41

—

179

C₂×Q₈

#11719923

yes

characteristic

180

C₂×D₄

—

181

C₂×D₄

—

182

C₂×D₄

—

183

C₂×D₄

—

184

C₂×D₄

—

185

C₂×D₄

#11719922

yes

—

186

C₂×D₄

#11719923

yes

characteristic

187

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

188

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

189

D₈

—

190

D₈

—

191

D₈

—

192

M₁₆

—

193

M₁₆

—

194

C₂×C₈

—

195

C₂×C₈

—

196

C₂×C₈

C₂²⋊Dic₅₃

yes

—

197

C₂×C₈

D₄×D₅₃

yes

—

198

C₂×C₈

#11719947

yes

—

199

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

200

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

201

C₄⋊C₄

—

202

C₄⋊C₄

—

212

—

203

C₂²⋊C₄

—

204

C₂²⋊C₄

—

205

C₂²⋊C₄

—

212

—

206

C₄²

—

207

C₄²

—

212

—

208

C₁₆

—

106

—

209

C₂³

—

210

C₂³

—

211

C₂³

—

212

C₂³

#13133138

yes

—

213

C₂³

C₂×#11719922

yes

characteristic

214

Q₈

—

215

D₄

—

216

D₄

—

217

D₄

—

218

D₄

—

219

D₄

—

220

D₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

—

424

—

225

C₂×C₄

#13133138

yes

—

226

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

227

C₂×C₄

#13133189

yes

—

228

C₂×C₄

C₂×#11719923

yes

characteristic

229

C₂×C₄

#13133237

yes

—

230

C₈

—

231

C₈

D₄×Dic₅₃

yes

—

232

C₈

#13133185

yes

characteristic

233

C₈

#13133198

yes

characteristic

234

C₂²

—

235

C₂²

—

236

C₂²

—

237

C₂²

—

238

C₂²

—

239

C₂²

—

240

C₂²

C₂×#13133138

yes

characteristic

241

C₂²

#16810857

yes

—

242

C₄

—

243

C₄

—

244

C₄

—

848

—

245

C₄

#16810503

yes

characteristic

246

C₄

#16810619

yes

characteristic

247

C₄

#16810850

yes

—

248

C₂

—

249

C₂

—

250

C₂

—

251

C₂

—

252

C₂

#20275144

yes

characteristic

	subgroup	G/H
center	C₂	#20275144
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3058	C₂
socle	C₂×C₅₃	#1893
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3058	D₅₃
hypocenter	C₅₃	#13074
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3058

subgroup

G/H

center

C₂

#20275144

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting, prime radical

C₅₃×#3058

C₂

socle

C₂×C₅₃

#1893

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3058

N	H ≅ G/N	type
1. #20287818	C₂	semidirect
2. #20276431	C₂	semidirect
3. #20274761	C₂	semidirect
4. #20272816	C₂	semidirect
16. C₅₃×#469	C₄	semidirect
25. C₂×#11719908	D₄	semidirect
138. #469	Dic₅₃	semidirect
146. C₅₃	#13074	semidirect

H ≅ G/N

type

1. #20287818

C₂

semidirect

2. #20276431

C₂

semidirect

3. #20274761

C₂

semidirect

4. #20272816

C₂

semidirect

16. C₅₃×#469

C₄

semidirect

25. C₂×#11719908

D₄

semidirect

138. #469

Dic₅₃

semidirect

146. C₅₃

#13074

semidirect

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

subgroup	index
#20287818	2
#20276431	2
#20274761	2
#20272816	2
C₅₃×#3058	2
#13074	53

subgroup

index

C₅₃×#3058

Subgroups — #27301542

Conjugacy classes of subgroups

Complements

Maximal subgroups