Group #20275921 — subgroups

Subgroups — #20275921

Order 6784.

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

normal subgroups only

This group has 316 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	C₂×#13133197	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133142	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133140	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133129	C₂	1	1	yes	maximal, characteristic
6	#16810480	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	#13133197	—	2	2	no	—
12	#13133197	—	2	2	no	—
13	C₂×C₄⋊Dic₅₃	—	2	1	no	—
14	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719921	—	2	1	no	—
16	C₂×#11719921	—	2	1	no	—
17	#13133150	—	2	2	no	—
18	#13133142	—	2	2	no	—
19	#13133140	—	2	2	no	—
20	#13133129	—	2	2	no	—
21	#13133124	—	2	1	no	—
22	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂³×D₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	—	2	1	no	—
27	C₂²×C₄×C₅₃	—	2	1	no	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	—	2	1	no	—
30	C₂²×Dic₅₃	—	2	1	no	—
31	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	—	2	1	no	—
33	C₂×C₄×D₅₃	—	2	1	no	—
34	C₂×C₄×D₅₃	—	2	1	no	—
35	C₂×C₄×D₅₃	—	2	1	no	—
36	C₂×C₄×D₅₃	—	2	1	no	—
37	C₂×C₄×D₅₃	—	2	1	no	—
38	C₂×Dic₁₀₆	—	2	1	no	—
39	C₂×Dic₁₀₆	—	2	1	no	—
40	C₂×Dic₁₀₆	—	2	1	no	—
41	C₂×Dic₁₀₆	—	2	1	no	—
42	C₂×Dic₁₀₆	—	2	1	no	—
43	C₂×Dic₁₀₆	D₄	1	2	yes	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₄⋊C₄×C₅₃	—	2	1	no	—
48	C₄⋊C₄×C₅₃	—	2	1	no	—
49	C₄⋊C₄×C₅₃	—	2	1	no	—
50	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
51	#11719921	—	2	4	no	—
52	#11719921	—	4	2	no	—
53	C₄⋊Dic₅₃	—	2	4	no	—
54	C₄⋊Dic₅₃	—	2	1	no	—
55	C₄⋊Dic₅₃	—	2	1	no	—
56	C₂×#88322	—	2	1	no	—
57	C₂×#88322	—	2	1	no	—
58	C₂×#88322	D₄	1	2	yes	—
59	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×D₅₃	—	2	1	no	—
61	C₂²×D₅₃	—	2	1	no	—
62	C₂²×D₅₃	—	2	1	no	—
63	C₂²×D₅₃	—	2	1	no	—
64	C₂²×D₅₃	—	2	1	no	—
65	C₂²×D₅₃	—	2	1	no	—
66	C₂²×D₅₃	Q₁₆	1	2	yes	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	1	no	—
69	C₂×C₄×C₅₃	—	4	1	no	—
70	C₂×C₄×C₅₃	—	4	1	no	—
71	C₂×C₄×C₅₃	—	4	1	no	—
72	C₂×C₄×C₅₃	—	4	1	no	—
73	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
79	C₂×Dic₅₃	—	2	2	no	—
80	C₂×Dic₅₃	—	2	1	no	—
81	C₂×Dic₅₃	—	2	1	no	—
82	C₂×Dic₅₃	—	2	1	no	—
83	C₂×Dic₅₃	—	4	1	no	—
84	C₂×Dic₅₃	—	4	1	no	—
85	C₂×Dic₅₃	—	4	1	no	—
86	C₂×Dic₅₃	—	4	1	no	—
87	C₂×Dic₅₃	—	4	1	no	—
88	C₄×D₅₃	—	4	1	no	—
89	C₄×D₅₃	—	4	1	no	—
90	C₄×D₅₃	—	4	1	no	—
91	C₄×D₅₃	—	4	1	no	—
92	Dic₁₀₆	—	2	2	no	—
93	Dic₁₀₆	—	2	2	no	—
94	Dic₁₀₆	—	4	1	no	—
95	Dic₁₀₆	—	4	1	no	—
96	C₈×C₅₃	—	4	1	no	—
97	#88322	—	2	2	no	—
98	C₂²×C₅₃	#120	1	1	yes	characteristic
99	C₂²×C₅₃	#121	1	1	yes	characteristic
100	C₂²×C₅₃	#123	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
102	C₂²×C₅₃	#135	1	1	yes	characteristic
103	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂²×C₅₃	#137	1	1	yes	characteristic
105	C₂×D₅₃	—	2	2	no	—
106	C₂×D₅₃	—	2	2	no	—
107	C₂×D₅₃	—	2	2	no	—
108	C₂×D₅₃	—	4	1	no	—
109	C₂×D₅₃	—	4	1	no	—
110	C₂×D₅₃	—	4	1	no	—
111	C₂×D₅₃	—	4	1	no	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	—	8	1	no	—
114	C₄×C₅₃	#121	1	1	yes	characteristic
115	C₄×C₅₃	#121	1	1	yes	characteristic
116	C₄×C₅₃	#123	1	1	yes	characteristic
117	C₄×C₅₃	#124	1	1	yes	characteristic
118	Dic₅₃	—	4	2	no	—
119	Dic₅₃	—	4	1	no	—
120	Dic₅₃	—	8	1	no	—
121	#2036	—	53	1	no	maximal, Sylow
122	C₂×C₅₃	#394	1	1	yes	characteristic
123	C₂×C₅₃	#449	1	1	yes	characteristic
124	C₂×C₅₃	#451	1	1	yes	characteristic
125	C₂×C₅₃	#467	1	1	yes	characteristic
126	C₂×C₅₃	#468	1	1	yes	characteristic
127	C₂×C₅₃	#484	1	1	yes	characteristic
128	C₂×C₅₃	#485	1	1	yes	characteristic
129	D₅₃	—	4	2	no	—
130	C₂×C₂²⋊Q₈	—	53	1	no	—
131	C₂×#107	—	53	1	no	—
132	C₂×#103	—	53	1	no	—
133	C₂×#103	—	53	1	no	—
134	C₂×C₂²⋊C₈	—	53	1	no	—
135	#380	—	53	1	no	—
136	#340	D₅₃	1	1	yes	characteristic, complemented
137	C₅₃	#2036	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	53	1	no	—
139	C₂³×C₄	—	53	1	no	—
140	C₂²×C₈	—	53	1	no	—
141	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂²⋊Q₈	—	106	2	no	—
143	C₂²⋊Q₈	—	106	2	no	—
144	C₂×C₄⋊C₄	—	53	1	no	—
145	C₂×C₄⋊C₄	—	106	1	no	—
146	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	—	106	1	no	—
149	C₂×C₂²⋊C₄	—	106	1	no	—
150	#107	—	106	2	no	—
151	#103	—	106	2	no	—
152	#103	—	106	2	no	—
153	C₂²⋊C₈	—	106	2	no	—
154	#95	—	106	1	no	—
155	C₂⁴	—	53	1	no	—
156	C₂×Q₈	—	53	2	no	—
157	C₂×Q₈	—	106	1	no	—
158	C₂×Q₈	—	106	1	no	—
159	C₂×Q₈	—	106	1	no	—
160	C₂×Q₈	—	106	1	no	—
161	C₂×Q₈	—	106	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	53	1	no	—
165	C₂²×C₄	—	106	1	no	—
166	C₂²×C₄	—	106	1	no	—
167	C₂²×C₄	—	106	1	no	—
168	C₂²×C₄	—	106	1	no	—
169	C₂²×C₄	—	106	1	no	—
170	C₂²×C₄	—	106	1	no	—
171	C₂²×C₄	—	106	1	no	—
172	C₂²×C₄	—	106	1	no	—
173	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	53	2	no	—
178	C₂×C₈	—	106	1	no	—
179	C₂×C₈	—	106	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	106	4	no	—
184	C₄⋊C₄	—	106	1	no	—
185	C₄⋊C₄	—	106	1	no	—
186	C₄⋊C₄	D₂₁₂	1	2	yes	—
187	C₂²⋊C₄	—	106	4	no	—
188	C₂²⋊C₄	—	212	2	no	—
189	C₂³	—	53	2	no	—
190	C₂³	—	106	1	no	—
191	C₂³	—	106	1	no	—
192	C₂³	—	106	1	no	—
193	C₂³	—	106	1	no	—
194	C₂³	—	106	1	no	—
195	C₂³	—	106	1	no	—
196	C₂³	D₄×D₅₃	1	1	yes	characteristic
197	Q₈	—	106	2	no	—
198	Q₈	—	106	2	no	—
199	Q₈	—	212	1	no	—
200	Q₈	—	212	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	106	2	no	—
208	C₂×C₄	—	106	1	no	—
209	C₂×C₄	—	106	1	no	—
210	C₂×C₄	—	106	1	no	—
211	C₂×C₄	—	212	1	no	—
212	C₂×C₄	—	212	1	no	—
213	C₂×C₄	—	212	1	no	—
214	C₂×C₄	—	212	1	no	—
215	C₂×C₄	—	212	1	no	—
216	C₂×C₄	—	212	1	no	—
217	C₂×C₄	—	212	1	no	—
218	C₂×C₄	—	212	1	no	—
219	C₂×C₄	—	212	1	no	—
220	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
221	C₂×C₄	#11719945	1	1	yes	characteristic
222	C₂×C₄	#11719945	1	1	yes	characteristic
223	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
224	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
225	C₂×C₄	#11719949	1	1	yes	characteristic
226	C₈	—	4	1	no	—
227	C₈	—	106	2	no	—
228	C₂²	—	106	2	no	—
229	C₂²	—	106	2	no	—
230	C₂²	—	106	2	no	—
231	C₂²	—	212	1	no	—
232	C₂²	—	212	1	no	—
233	C₂²	—	212	1	no	—
234	C₂²	—	212	1	no	—
235	C₂²	#13133187	1	1	yes	characteristic
236	C₂²	#13133190	1	1	yes	characteristic
237	C₂²	#13133202	1	1	yes	characteristic
238	C₂²	Q₁₆×D₅₃	1	1	yes	characteristic
239	C₂²	#13133238	1	1	yes	characteristic
240	C₂²	#13133241	1	1	yes	characteristic
241	C₂²	#13133246	1	1	yes	characteristic
242	C₄	—	4	2	no	—
243	C₄	—	8	1	no	—
244	C₄	—	424	1	no	—
245	C₄	—	212	1	no	—
246	C₄	—	212	2	no	—
247	C₄	#13133190	1	1	yes	characteristic
248	C₄	#13133192	1	1	yes	characteristic
249	C₄	#13133202	1	1	yes	characteristic
250	C₄	#13133205	1	1	yes	characteristic
251	C₂	—	212	2	no	—
252	C₂	#16810541	1	1	yes	characteristic
253	C₂	#16810769	1	1	yes	characteristic
254	C₂	#16810780	1	1	yes	characteristic
255	C₂	#16810863	1	1	yes	characteristic
256	C₂	#16810868	1	1	yes	characteristic
257	C₂	#16810960	1	1	yes	characteristic
258	C₂	#16810967	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133197

C₂

yes

maximal, characteristic

C₂×#13133150

C₂

yes

maximal, characteristic, complemented

C₂×#13133142

C₂

yes

maximal, characteristic, complemented

C₂×#13133140

C₂

yes

maximal, characteristic, complemented

C₂×#13133129

C₂

yes

maximal, characteristic

#16810480

C₂

yes

maximal, characteristic

C₅₃×#340

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133197

—

#13133197

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

100

C₂²×C₅₃

#123

yes

characteristic

101

C₂²×C₅₃

C₂×Q₁₆

yes

characteristic

102

C₂²×C₅₃

#135

yes

characteristic

103

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

104

C₂²×C₅₃

#137

yes

characteristic

105

C₂×D₅₃

—

106

C₂×D₅₃

—

107

C₂×D₅₃

—

108

C₂×D₅₃

—

109

C₂×D₅₃

—

110

C₂×D₅₃

—

111

C₂×D₅₃

—

112

C₄×C₅₃

—

113

C₄×C₅₃

—

114

C₄×C₅₃

#121

yes

characteristic

115

C₄×C₅₃

#121

yes

characteristic

116

C₄×C₅₃

#123

yes

characteristic

117

C₄×C₅₃

#124

yes

characteristic

118

Dic₅₃

—

119

Dic₅₃

—

120

Dic₅₃

—

121

#2036

—

maximal, Sylow

122

C₂×C₅₃

#394

yes

characteristic

123

C₂×C₅₃

#449

yes

characteristic

124

C₂×C₅₃

#451

yes

characteristic

125

C₂×C₅₃

#467

yes

characteristic

126

C₂×C₅₃

#468

yes

characteristic

127

C₂×C₅₃

#484

yes

characteristic

128

C₂×C₅₃

#485

yes

characteristic

129

D₅₃

—

130

C₂×C₂²⋊Q₈

—

131

C₂×#107

—

132

C₂×#103

—

133

C₂×#103

—

134

C₂×C₂²⋊C₈

—

135

#380

—

136

#340

D₅₃

yes

characteristic, complemented

137

C₅₃

#2036

yes

characteristic, complemented, Sylow

138

C₂²×Q₈

—

139

C₂³×C₄

—

140

C₂²×C₈

—

141

C₂²×C₈

C₂×D₅₃

yes

characteristic

142

C₂²⋊Q₈

—

106

—

143

C₂²⋊Q₈

—

106

—

144

C₂×C₄⋊C₄

—

145

C₂×C₄⋊C₄

—

106

—

146

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

147

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

148

C₂×C₂²⋊C₄

—

106

—

149

C₂×C₂²⋊C₄

—

106

—

150

#107

—

106

—

151

#103

—

106

—

152

#103

—

106

—

153

C₂²⋊C₈

—

106

—

154

#95

—

106

—

155

C₂⁴

—

156

C₂×Q₈

—

157

C₂×Q₈

—

106

—

158

C₂×Q₈

—

106

—

159

C₂×Q₈

—

106

—

160

C₂×Q₈

—

106

—

161

C₂×Q₈

—

106

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

—

106

—

166

C₂²×C₄

—

106

—

167

C₂²×C₄

—

106

—

168

C₂²×C₄

—

106

—

169

C₂²×C₄

—

106

—

170

C₂²×C₄

—

106

—

171

C₂²×C₄

—

106

—

172

C₂²×C₄

—

106

—

173

C₂²×C₄

C₂²×D₅₃

yes

characteristic

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₂×C₈

—

106

—

179

C₂×C₈

—

106

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

106

—

184

C₄⋊C₄

—

106

—

185

C₄⋊C₄

—

106

—

186

C₄⋊C₄

D₂₁₂

yes

—

187

C₂²⋊C₄

—

106

—

188

C₂²⋊C₄

—

212

—

189

C₂³

—

190

C₂³

—

106

—

191

C₂³

—

106

—

192

C₂³

—

106

—

193

C₂³

—

106

—

194

C₂³

—

106

—

195

C₂³

—

106

—

196

C₂³

D₄×D₅₃

yes

characteristic

197

Q₈

—

106

—

198

Q₈

—

106

—

199

Q₈

—

212

—

200

Q₈

—

212

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

106

—

208

C₂×C₄

—

106

—

209

C₂×C₄

—

106

—

210

C₂×C₄

—

106

—

211

C₂×C₄

—

212

—

212

C₂×C₄

—

212

—

213

C₂×C₄

—

212

—

214

C₂×C₄

—

212

—

215

C₂×C₄

—

212

—

216

C₂×C₄

—

212

—

217

C₂×C₄

—

212

—

218

C₂×C₄

—

212

—

219

C₂×C₄

—

212

—

220

C₂×C₄

C₂×D₂₁₂

yes

characteristic

221

C₂×C₄

#11719945

yes

characteristic

222

C₂×C₄

#11719945

yes

characteristic

223

C₂×C₄

D₄×D₅₃

yes

characteristic

224

C₂×C₄

D₄×D₅₃

yes

characteristic

225

C₂×C₄

#11719949

yes

characteristic

226

C₈

—

227

C₈

—

106

—

228

C₂²

—

106

—

229

C₂²

—

106

—

230

C₂²

—

106

—

231

C₂²

—

212

—

232

C₂²

—

212

—

233

C₂²

—

212

—

234

C₂²

—

212

—

235

C₂²

#13133187

yes

characteristic

236

C₂²

#13133190

yes

characteristic

237

C₂²

#13133202

yes

characteristic

238

C₂²

Q₁₆×D₅₃

yes

characteristic

239

C₂²

#13133238

yes

characteristic

240

C₂²

#13133241

yes

characteristic

241

C₂²

#13133246

yes

characteristic

242

C₄

—

243

C₄

—

244

C₄

—

424

—

245

C₄

—

212

—

246

C₄

—

212

—

247

C₄

#13133190

yes

characteristic

248

C₄

#13133192

yes

characteristic

249

C₄

#13133202

yes

characteristic

250

C₄

#13133205

yes

characteristic

251

C₂

—

212

—

252

C₂

#16810541

yes

characteristic

253

C₂

#16810769

yes

characteristic

254

C₂

#16810780

yes

characteristic

255

C₂

#16810863

yes

characteristic

256

C₂

#16810868

yes

characteristic

257

C₂

#16810960

yes

characteristic

258

C₂

#16810967

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#340	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#340	D₅₃
hypocenter	C₅₃	#2036
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#340

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
2. C₂×#13133150	C₂	semidirect
3. C₂×#13133142	C₂	semidirect
4. C₂×#13133140	C₂	semidirect
7. C₅₃×#340	C₂	semidirect
136. #340	D₅₃	semidirect
137. C₅₃	#2036	semidirect

H ≅ G/N

type

2. C₂×#13133150

C₂

semidirect

3. C₂×#13133142

C₂

semidirect

4. C₂×#13133140

C₂

semidirect

7. C₅₃×#340

C₂

semidirect

136. #340

D₅₃

semidirect

137. C₅₃

#2036

semidirect

Subgroups — #20275921

Conjugacy classes of subgroups

Complements