Group #20273219 — subgroups

Subgroups — #20273219

Order 6784.

334 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16810587	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133124	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810360	C₂	1	2	yes	maximal, complemented
4	#16810357	C₂	1	2	yes	maximal, complemented
5	C₅₃×#525	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₅₃	—	2	1	no	—
7	C₂×#11719917	—	2	1	no	—
8	C₂²⋊#88322	—	2	1	no	—
9	C₂²⋊#88322	C₂²	1	2	yes	—
10	#13133124	—	2	1	no	—
11	#13133124	C₂²	1	2	yes	—
12	C₂×#41×C₅₃	C₄	1	1	yes	characteristic, complemented
13	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	#123×C₅₃	—	2	1	no	—
15	C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	#121×C₅₃	C₄	1	1	yes	characteristic, complemented
18	#121×C₅₃	C₂²	1	1	yes	characteristic
19	#120×C₅₃	—	2	1	no	—
20	C₄×D₄×C₅₃	—	2	1	no	—
21	C₄×D₄×C₅₃	—	2	1	no	—
22	#117×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
24	#41×C₅₃	—	4	2	no	—
25	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₄×C₅₃	—	2	1	no	—
28	C₂×D₄×C₅₃	—	2	1	no	—
29	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
30	C₂²×C₄×C₅₃	—	2	1	no	—
31	C₂²×C₄×C₅₃	—	4	2	no	—
32	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	2	4	no	—
39	C₂²×Dic₅₃	—	2	2	no	—
40	C₂²×Dic₅₃	—	4	2	no	—
41	C₂²×Dic₅₃	—	4	2	no	—
42	C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
45	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂²⋊C₄×C₅₃	—	2	1	no	—
47	C₂²⋊C₄×C₅₃	—	2	1	no	—
48	C₂²⋊C₄×C₅₃	—	2	1	no	—
49	C₂²⋊C₄×C₅₃	—	2	1	no	—
50	C₂²⋊C₄×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	—	2	1	no	—
52	#11719917	—	4	2	no	—
53	C₂×#88322	—	2	2	no	—
54	C₂³×C₅₃	—	2	2	no	—
55	C₂³×C₅₃	—	2	2	no	—
56	C₂³×C₅₃	—	2	1	no	—
57	C₂³×C₅₃	—	2	1	no	—
58	C₂³×C₅₃	—	2	1	no	—
59	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₅₃	—	4	1	no	—
63	D₄×C₅₃	—	4	1	no	—
64	D₄×C₅₃	—	4	2	no	—
65	D₄×C₅₃	—	4	2	no	—
66	D₄×C₅₃	—	4	1	no	—
67	D₄×C₅₃	—	4	1	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	4	1	no	—
74	C₂×C₄×C₅₃	—	4	1	no	—
75	C₂×C₄×C₅₃	—	4	1	no	—
76	C₂×C₄×C₅₃	—	4	1	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₅₃	—	2	4	no	—
83	C₂×Dic₅₃	—	4	8	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	4	2	no	—
86	#88322	—	4	2	no	—
87	C₂²×C₅₃	—	2	4	no	—
88	C₂²×C₅₃	—	2	4	no	—
89	C₂²×C₅₃	—	2	1	no	—
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	4	1	no	—
96	C₂²×C₅₃	—	4	1	no	—
97	C₂²×C₅₃	—	4	1	no	—
98	C₂²×C₅₃	#99	1	2	yes	—
99	C₂²×C₅₃	#104	1	2	yes	—
100	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂²×C₅₃	#137	1	1	yes	characteristic
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	4	1	no	—
108	C₄×C₅₃	—	4	1	no	—
109	C₄×C₅₃	—	4	1	no	—
110	Dic₅₃	—	4	4	no	—
111	#1504	—	53	1	no	maximal, Sylow
112	C₂×C₅₃	—	2	1	no	—
113	C₂×C₅₃	—	2	2	no	—
114	C₂×C₅₃	—	2	2	no	—
115	C₂×C₅₃	—	2	1	no	—
116	C₂×C₅₃	—	8	1	no	—
117	C₂×C₅₃	C₂×#99	1	1	yes	characteristic
118	C₂×C₅₃	#417	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
120	#525	D₅₃	1	1	yes	characteristic
121	#407	—	53	1	no	—
122	C₂×#95	—	53	1	no	—
123	#328	—	53	2	no	—
124	#327	—	53	2	no	—
125	C₅₃	#1504	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	Dic₅₃	1	1	yes	characteristic, complemented
127	C₂³×C₄	—	106	1	no	—
128	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
129	C₂×M₁₆	—	106	1	no	—
130	#123	—	2	1	no	—
131	C₂²⋊Q₈	Dic₅₃	1	1	yes	characteristic, complemented
132	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
133	#121	Dic₅₃	1	1	yes	characteristic, complemented
134	#121	C₂×D₅₃	1	1	yes	characteristic
135	#120	—	2	1	no	—
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	—	2	1	no	—
138	#117	Dic₅₃	1	1	yes	characteristic, complemented
139	C₂²⋊C₈	—	53	2	no	—
140	C₂²⋊C₈	—	106	1	no	—
141	#95	—	53	2	no	—
142	#95	—	106	1	no	—
143	C₂⁴	C₂²⋊D₅₃	1	1	yes	characteristic
144	#41	—	4	2	no	—
145	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic, complemented
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	1	no	—
149	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic, complemented
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	106	4	no	—
153	C₂²×C₄	—	106	2	no	—
154	C₂²×C₄	—	212	2	no	—
155	C₂²×C₄	—	212	2	no	—
156	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
162	M₁₆	—	212	2	no	—
163	C₂×C₈	—	106	2	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic, complemented
167	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic, complemented
168	C₂²⋊C₄	—	2	1	no	—
169	C₂²⋊C₄	—	2	1	no	—
170	C₂²⋊C₄	—	2	1	no	—
171	C₂²⋊C₄	—	2	1	no	—
172	C₂²⋊C₄	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	—	2	1	no	—
179	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
180	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
181	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
182	Q₈	—	4	1	no	—
183	D₄	—	4	1	no	—
184	D₄	—	4	2	no	—
185	D₄	—	4	2	no	—
186	D₄	—	4	1	no	—
187	D₄	—	4	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	106	4	no	—
199	C₂×C₄	—	212	8	no	—
200	C₂×C₄	—	212	2	no	—
201	C₂×C₄	—	212	2	no	—
202	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
206	C₈	—	212	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	1	no	—
217	C₂²	—	4	1	no	—
218	C₂²	#13133132	1	2	yes	—
219	C₂²	#13133144	1	2	yes	—
220	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
221	C₂²	#13133242	1	1	yes	characteristic
222	C₂²	#13133247	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	1	no	—
230	C₄	—	212	4	no	—
231	C₂	—	2	1	no	—
232	C₂	—	2	2	no	—
233	C₂	—	2	2	no	—
234	C₂	—	2	1	no	—
235	C₂	—	8	1	no	—
236	C₂	C₂×#13133132	1	1	yes	characteristic
237	C₂	#16810624	1	1	yes	characteristic
238	C₂	C₂×#13133144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810587

C₂

yes

maximal, characteristic, complemented

C₂×#13133124

C₂

yes

maximal, characteristic, complemented

#16810360

C₂

yes

maximal, complemented

#16810357

C₂

yes

maximal, complemented

C₅₃×#525

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

—

C₂×#11719917

—

C₂²⋊#88322

—

C₂²⋊#88322

C₂²

yes

—

#13133124

—

#13133124

C₂²

yes

—

C₂×#41×C₅₃

C₄

yes

characteristic, complemented

C₂³×C₄×C₅₃

C₂²

yes

characteristic

#123×C₅₃

—

C₂²⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

C₂²⋊Q₈×C₅₃

C₂²

yes

characteristic

#121×C₅₃

C₄

yes

characteristic, complemented

#121×C₅₃

C₂²

yes

characteristic

#120×C₅₃

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

#117×C₅₃

C₄

yes

characteristic, complemented

C₂⁴×C₅₃

D₄

yes

characteristic

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic, complemented

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic, complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic, complemented

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic, complemented

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#99

yes

—

C₂²×C₅₃

#104

yes

—

100

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

101

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

102

C₂²×C₅₃

#137

yes

characteristic

103

C₄×C₅₃

—

104

C₄×C₅₃

—

105

C₄×C₅₃

—

106

C₄×C₅₃

—

107

C₄×C₅₃

—

108

C₄×C₅₃

—

109

C₄×C₅₃

—

110

Dic₅₃

—

111

#1504

—

maximal, Sylow

112

C₂×C₅₃

—

113

C₂×C₅₃

—

114

C₂×C₅₃

—

115

C₂×C₅₃

—

116

C₂×C₅₃

—

117

C₂×C₅₃

C₂×#99

yes

characteristic

118

C₂×C₅₃

#417

yes

characteristic

119

C₂×C₅₃

C₂×#104

yes

characteristic

120

#525

D₅₃

yes

characteristic

121

#407

—

122

C₂×#95

—

123

#328

—

124

#327

—

125

C₅₃

#1504

yes

characteristic, complemented, Sylow

126

C₂×#41

Dic₅₃

yes

characteristic, complemented

127

C₂³×C₄

—

106

—

128

C₂³×C₄

C₂×D₅₃

yes

characteristic

129

C₂×M₁₆

—

106

—

130

#123

—

131

C₂²⋊Q₈

Dic₅₃

yes

characteristic, complemented

132

C₂²⋊Q₈

C₂×D₅₃

yes

characteristic

133

#121

Dic₅₃

yes

characteristic, complemented

134

#121

C₂×D₅₃

yes

characteristic

135

#120

—

136

C₄×D₄

—

137

C₄×D₄

—

138

#117

Dic₅₃

yes

characteristic, complemented

139

C₂²⋊C₈

—

140

C₂²⋊C₈

—

106

—

141

#95

—

142

#95

—

106

—

143

C₂⁴

C₂²⋊D₅₃

yes

characteristic

144

#41

—

145

C₂×Q₈

C₂×Dic₅₃

yes

characteristic, complemented

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

C₂×Dic₅₃

yes

characteristic, complemented

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

106

—

153

C₂²×C₄

—

106

—

154

C₂²×C₄

—

212

—

155

C₂²×C₄

—

212

—

156

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

157

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

158

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

159

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

160

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

161

C₂²×C₄

C₂²×D₅₃

yes

characteristic

162

M₁₆

—

212

—

163

C₂×C₈

—

106

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

C₂×Dic₅₃

yes

characteristic, complemented

167

C₄⋊C₄

C₂×Dic₅₃

yes

characteristic, complemented

168

C₂²⋊C₄

—

169

C₂²⋊C₄

—

170

C₂²⋊C₄

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₂³

—

175

C₂³

—

176

C₂³

—

177

C₂³

—

178

C₂³

—

179

C₂³

C₂²⋊Dic₅₃

yes

characteristic

180

C₂³

C₂²⋊Dic₅₃

yes

characteristic

181

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

182

Q₈

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

187

D₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

106

—

199

C₂×C₄

—

212

—

200

C₂×C₄

—

212

—

201

C₂×C₄

—

212

—

202

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

203

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

204

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

205

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

206

C₈

—

212

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

—

218

C₂²

#13133132

yes

—

219

C₂²

#13133144

yes

—

220

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

221

C₂²

#13133242

yes

characteristic

222

C₂²

#13133247

yes

characteristic

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

212

—

231

C₂

—

232

C₂

—

233

C₂

—

234

C₂

—

235

C₂

—

236

C₂

C₂×#13133132

yes

characteristic

237

C₂

#16810624

yes

characteristic

238

C₂

C₂×#13133144

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#525	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#525	D₅₃
hypocenter	C₅₃	#1504
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #525

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#525

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #525

N	H ≅ G/N	type
1. #16810587	C₂	semidirect
2. C₂×#13133124	C₂	semidirect
3. #16810360	C₂	semidirect
4. #16810357	C₂	semidirect
12. C₂×#41×C₅₃	C₄	semidirect
15. C₂²⋊Q₈×C₅₃	C₄	semidirect
17. #121×C₅₃	C₄	semidirect
22. #117×C₅₃	C₄	semidirect
25. C₂×Q₈×C₅₃	C₂×C₄	semidirect
29. C₂×D₄×C₅₃	C₂×C₄	semidirect
44. C₄⋊C₄×C₅₃	C₂×C₄	semidirect
45. C₄⋊C₄×C₅₃	C₂×C₄	semidirect
125. C₅₃	#1504	semidirect
126. C₂×#41	Dic₅₃	semidirect
131. C₂²⋊Q₈	Dic₅₃	semidirect
133. #121	Dic₅₃	semidirect
138. #117	Dic₅₃	semidirect
145. C₂×Q₈	C₂×Dic₅₃	semidirect
149. C₂×D₄	C₂×Dic₅₃	semidirect
166. C₄⋊C₄	C₂×Dic₅₃	semidirect
167. C₄⋊C₄	C₂×Dic₅₃	semidirect

H ≅ G/N

type

1. #16810587

C₂

semidirect

2. C₂×#13133124

C₂

semidirect

3. #16810360

C₂

semidirect

4. #16810357

C₂

semidirect

12. C₂×#41×C₅₃

C₄

semidirect

15. C₂²⋊Q₈×C₅₃

C₄

semidirect

17. #121×C₅₃

C₄

semidirect

22. #117×C₅₃

C₄

semidirect

25. C₂×Q₈×C₅₃

C₂×C₄

semidirect

29. C₂×D₄×C₅₃

C₂×C₄

semidirect

44. C₄⋊C₄×C₅₃

C₂×C₄

semidirect

45. C₄⋊C₄×C₅₃

C₂×C₄

semidirect

125. C₅₃

#1504

semidirect

126. C₂×#41

Dic₅₃

semidirect

131. C₂²⋊Q₈

Dic₅₃

semidirect

133. #121

Dic₅₃

semidirect

138. #117

Dic₅₃

semidirect

145. C₂×Q₈

C₂×Dic₅₃

semidirect

149. C₂×D₄

C₂×Dic₅₃

semidirect

166. C₄⋊C₄

C₂×Dic₅₃

semidirect

167. C₄⋊C₄

C₂×Dic₅₃

semidirect

Subgroups — #20273219

Conjugacy classes of subgroups

Complements