Group #20285896 — subgroups

Subgroups — #20285896

Order 6784.

332 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 121 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	#16811180	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811157	C₂	1	1	yes	maximal, characteristic
3	#16811025	C₂	1	2	yes	maximal, complemented
4	#16810978	C₂	1	2	yes	maximal, complemented
5	#16810965	C₂	1	2	yes	maximal, complemented
6	#16810948	C₂	1	2	yes	maximal
7	#16810919	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810914	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810909	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810904	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	#13133217	C₂²	1	1	yes	characteristic
13	#13133217	C₂²	1	1	yes	characteristic
14	#13133211	—	2	1	no	—
15	#13133209	C₂²	1	1	yes	characteristic
16	#13133209	C₂²	1	2	yes	—
17	#13133205	C₂²	1	1	yes	characteristic
18	#13133200	—	2	1	no	—
19	#13133196	—	2	1	no	—
20	#13133196	—	2	1	no	—
21	#13133196	C₂²	1	1	yes	characteristic
22	#13133191	C₂²	1	1	yes	characteristic
23	C₄×Dic₁₀₆	—	2	1	no	—
24	C₄×Dic₁₀₆	—	2	1	no	—
25	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
26	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₅₃	—	2	1	no	—
28	#13133176	C₂²	1	2	yes	—
29	#13133174	—	2	1	no	—
30	#13133151	C₂²	1	2	yes	—
31	#13133151	C₂²	1	2	yes	—
32	#13133149	C₂²	1	2	yes	—
33	#13133149	C₂²	1	2	yes	—
34	#13133142	C₂²	1	2	yes	—
35	#13133142	C₂²	1	2	yes	—
36	#13133139	C₂²	1	2	yes	—
37	#13133139	C₂²	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
42	C₄×C₈×C₅₃	C₂²	1	2	yes	—
43	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₅₃	—	2	1	no	—
47	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₅₃	—	2	1	no	—
49	C₂×Dic₁₀₆	—	2	1	no	—
50	C₂×Dic₁₀₆	—	2	1	no	—
51	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂×C₈×C₅₃	C₂³	1	2	yes	—
54	C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₅₃	—	2	2	no	—
59	C₂²⋊Dic₅₃	—	2	1	no	—
60	#11719921	—	2	1	no	—
61	#11719921	—	2	1	no	—
62	#11719921	—	2	1	no	—
63	#11719921	—	2	1	no	—
64	C₄⋊Dic₅₃	—	2	1	no	—
65	C₄⋊Dic₅₃	—	2	1	no	—
66	C₄⋊Dic₅₃	C₂³	1	2	yes	—
67	C₄⋊Dic₅₃	C₂³	1	2	yes	—
68	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
70	#11719919	—	2	1	no	—
71	#11719919	—	2	1	no	—
72	#11719919	—	2	1	no	—
73	#11719919	—	2	2	no	—
74	#11719919	—	2	2	no	—
75	#11719919	—	2	1	no	—
76	C₄×Dic₅₃	—	2	1	no	—
77	C₄×Dic₅₃	—	2	2	no	—
78	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×D₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	4	1	no	—
84	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₅₃	—	4	2	no	—
91	C₂×Dic₅₃	—	2	2	no	—
92	C₂×Dic₅₃	—	2	2	no	—
93	C₂×Dic₅₃	—	2	1	no	—
94	C₂×Dic₅₃	—	2	1	no	—
95	C₂×Dic₅₃	—	2	1	no	—
96	D₂₁₂	—	2	1	no	—
97	D₂₁₂	—	2	1	no	—
98	C₄×D₅₃	—	4	2	no	—
99	Dic₁₀₆	—	2	1	no	—
100	Dic₁₀₆	—	2	1	no	—
101	Dic₁₀₆	—	4	2	no	—
102	Dic₁₀₆	—	4	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×D₅₃	—	4	1	no	—
110	C₂×D₅₃	—	4	1	no	—
111	C₂×D₅₃	—	4	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	#135	1	2	yes	—
117	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
118	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
119	Dic₅₃	—	4	1	no	—
120	Dic₅₃	—	4	2	no	—
121	Dic₅₃	—	4	1	no	—
122	Dic₅₃	—	4	2	no	—
123	Dic₅₃	—	4	1	no	—
124	#3233	—	53	1	no	maximal, Sylow
125	C₂×C₅₃	—	4	1	no	—
126	C₂×C₅₃	#536	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
128	C₂×C₅₃	#575	1	1	yes	characteristic
129	D₅₃	—	8	1	no	—
130	#533	—	53	1	no	—
131	#529	—	53	1	no	—
132	#499	—	53	2	no	—
133	#488	—	53	2	no	—
134	#485	—	53	2	no	—
135	#482	—	53	2	no	—
136	#477	—	53	1	no	—
137	#476	—	53	1	no	—
138	#475	—	53	1	no	—
139	#474	—	53	1	no	—
140	#431	D₅₃	1	1	yes	characteristic, complemented
141	C₅₃	#3233	1	1	yes	characteristic, complemented, Sylow
142	C₄⋊Q₈	—	53	1	no	—
143	C₄⋊Q₈	—	53	1	no	—
144	#126	—	106	1	no	—
145	#125	—	53	1	no	—
146	#125	—	53	2	no	—
147	#124	—	53	1	no	—
148	#123	—	106	1	no	—
149	C₂²⋊Q₈	—	53	1	no	—
150	C₂²⋊Q₈	—	106	1	no	—
151	C₂²⋊Q₈	—	106	1	no	—
152	#121	—	53	1	no	—
153	C₄×Q₈	—	53	1	no	—
154	C₄×Q₈	—	106	1	no	—
155	C₄×Q₈	—	106	1	no	—
156	C₄×D₄	—	53	1	no	—
157	C₄×D₄	—	106	1	no	—
158	#117	—	53	2	no	—
159	#117	—	106	1	no	—
160	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
161	#107	—	53	2	no	—
162	#107	—	53	2	no	—
163	#106	—	53	2	no	—
164	#106	—	53	2	no	—
165	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
166	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
167	#103	—	53	2	no	—
168	#103	—	53	2	no	—
169	#102	—	53	2	no	—
170	#102	—	53	2	no	—
171	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
172	C₄×C₈	C₂×D₅₃	1	2	yes	—
173	C₂×Q₈	—	53	1	no	—
174	C₂×Q₈	—	106	1	no	—
175	C₂×Q₈	—	106	1	no	—
176	C₂×D₄	—	53	1	no	—
177	C₂×D₄	—	106	1	no	—
178	C₂²×C₄	—	106	1	no	—
179	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
180	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
182	C₂×C₈	C₂²×D₅₃	1	2	yes	—
183	C₂×C₈	C₂²×D₅₃	1	2	yes	—
184	C₄⋊C₄	—	53	2	no	—
185	C₄⋊C₄	—	53	2	no	—
186	C₄⋊C₄	—	53	1	no	—
187	C₄⋊C₄	—	53	1	no	—
188	C₄⋊C₄	—	106	1	no	—
189	C₄⋊C₄	—	106	1	no	—
190	C₄⋊C₄	—	106	1	no	—
191	C₄⋊C₄	—	106	1	no	—
192	C₄⋊C₄	—	106	2	no	—
193	C₄⋊C₄	—	106	2	no	—
194	C₄⋊C₄	—	106	1	no	—
195	C₄⋊C₄	—	106	1	no	—
196	C₂²⋊C₄	—	106	2	no	—
197	C₂²⋊C₄	—	106	1	no	—
198	C₂²⋊C₄	—	106	1	no	—
199	C₂²⋊C₄	—	106	1	no	—
200	C₂²⋊C₄	—	106	1	no	—
201	C₂²⋊C₄	—	106	1	no	—
202	C₄²	—	106	2	no	—
203	C₄²	—	106	1	no	—
204	C₄²	D₂₁₂	1	1	yes	characteristic
205	C₄²	D₂₁₂	1	1	yes	characteristic
206	C₄²	C₂²×D₅₃	1	1	yes	characteristic
207	C₄²	C₂²×D₅₃	1	1	yes	characteristic
208	C₂³	—	106	1	no	—
209	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
210	Q₈	—	106	1	no	—
211	Q₈	—	106	1	no	—
212	Q₈	—	212	2	no	—
213	Q₈	—	212	2	no	—
214	D₄	—	106	1	no	—
215	D₄	—	106	1	no	—
216	D₄	—	212	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	106	1	no	—
222	C₂×C₄	—	106	1	no	—
223	C₂×C₄	—	106	2	no	—
224	C₂×C₄	—	106	2	no	—
225	C₂×C₄	—	106	1	no	—
226	C₂×C₄	—	212	2	no	—
227	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
228	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
229	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
230	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
231	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
232	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	212	1	no	—
239	C₂²	—	212	1	no	—
240	C₂²	—	212	1	no	—
241	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	212	1	no	—
247	C₄	—	212	2	no	—
248	C₄	—	212	1	no	—
249	C₄	—	212	2	no	—
250	C₄	—	212	1	no	—
251	C₄	#13133239	1	2	yes	—
252	C₄	#13133264	1	1	yes	characteristic
253	C₄	#13133264	1	1	yes	characteristic
254	C₂	—	4	1	no	—
255	C₂	—	424	1	no	—
256	C₂	#16811199	1	1	yes	characteristic
257	C₂	C₂×#13133239	1	1	yes	characteristic
258	C₂	#16811423	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic, complemented

#16811157

C₂

yes

maximal, characteristic

#16811025

C₂

yes

maximal, complemented

#16810978

C₂

yes

maximal, complemented

#16810965

C₂

yes

maximal, complemented

#16810948

C₂

yes

maximal

#16810919

C₂

yes

maximal, characteristic, complemented

#16810914

C₂

yes

maximal, characteristic, complemented

#16810909

C₂

yes

maximal, characteristic, complemented

#16810904

C₂

yes

maximal, characteristic, complemented

C₅₃×#431

C₂

yes

maximal, characteristic, complemented

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133211

—

#13133209

C₂²

yes

characteristic

#13133209

C₂²

yes

—

#13133205

C₂²

yes

—

—

—

C₂²

yes

characteristic

#13133191

C₂²

yes

characteristic

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×D₂₁₂

C₂²

yes

characteristic

C₄×C₂²⋊D₅₃

—

#13133176

C₂²

yes

—

#13133174

—

#13133151

C₂²

yes

—

#13133151

C₂²

yes

—

#13133149

C₂²

yes

—

#13133149

C₂²

yes

—

#13133142

C₂²

yes

—

#13133142

C₂²

yes

—

#13133139

C₂²

yes

—

#13133139

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

C₂³

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

—

—

—

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

—

—

—

—

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

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

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

100

Dic₁₀₆

—

101

Dic₁₀₆

—

102

Dic₁₀₆

—

103

C₈×C₅₃

—

104

C₈×C₅₃

—

105

C₂²×C₅₃

—

106

C₂²×C₅₃

—

107

C₂²×C₅₃

—

108

C₂²×C₅₃

C₂²×D₄

yes

characteristic

109

C₂×D₅₃

—

110

C₂×D₅₃

—

111

C₂×D₅₃

—

112

C₄×C₅₃

—

113

C₄×C₅₃

—

114

C₄×C₅₃

—

115

C₄×C₅₃

—

116

C₄×C₅₃

#135

yes

—

117

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

118

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

119

Dic₅₃

—

120

Dic₅₃

—

121

Dic₅₃

—

122

Dic₅₃

—

123

Dic₅₃

—

124

#3233

—

maximal, Sylow

125

C₂×C₅₃

—

126

C₂×C₅₃

#536

yes

characteristic

127

C₂×C₅₃

C₂×#135

yes

characteristic

128

C₂×C₅₃

#575

yes

characteristic

129

D₅₃

—

130

#533

—

131

#529

—

132

#499

—

133

#488

—

134

#485

—

135

#482

—

136

#477

—

137

#476

—

138

#475

—

139

#474

—

140

#431

D₅₃

yes

characteristic, complemented

141

C₅₃

#3233

yes

characteristic, complemented, Sylow

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

#126

—

106

—

145

#125

—

146

#125

—

147

#124

—

148

#123

—

106

—

149

C₂²⋊Q₈

—

150

C₂²⋊Q₈

—

106

—

151

C₂²⋊Q₈

—

106

—

152

#121

—

153

C₄×Q₈

—

154

C₄×Q₈

—

106

—

155

C₄×Q₈

—

106

—

156

C₄×D₄

—

157

C₄×D₄

—

106

—

158

#117

—

159

#117

—

106

—

160

C₂×C₄²

C₂×D₅₃

yes

characteristic

161

#107

—

162

#107

—

163

#106

—

164

#106

—

165

C₄⋊C₈

C₂×D₅₃

yes

characteristic

166

C₄⋊C₈

C₂×D₅₃

yes

characteristic

167

#103

—

168

#103

—

169

#102

—

170

#102

—

171

C₂²⋊C₈

C₂×D₅₃

yes

—

172

C₄×C₈

C₂×D₅₃

yes

—

173

C₂×Q₈

—

174

C₂×Q₈

—

106

—

175

C₂×Q₈

—

106

—

176

C₂×D₄

—

177

C₂×D₄

—

106

—

178

C₂²×C₄

—

106

—

179

C₂²×C₄

D₂₁₂

yes

characteristic

180

C₂²×C₄

D₂₁₂

yes

characteristic

181

C₂²×C₄

C₂²×D₅₃

yes

characteristic

182

C₂×C₈

C₂²×D₅₃

yes

—

183

C₂×C₈

C₂²×D₅₃

yes

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

106

—

189

C₄⋊C₄

—

106

—

190

C₄⋊C₄

—

106

—

191

C₄⋊C₄

—

106

—

192

C₄⋊C₄

—

106

—

193

C₄⋊C₄

—

106

—

194

C₄⋊C₄

—

106

—

195

C₄⋊C₄

—

106

—

196

C₂²⋊C₄

—

106

—

197

C₂²⋊C₄

—

106

—

198

C₂²⋊C₄

—

106

—

199

C₂²⋊C₄

—

106

—

200

C₂²⋊C₄

—

106

—

201

C₂²⋊C₄

—

106

—

202

C₄²

—

106

—

203

C₄²

—

106

—

204

C₄²

D₂₁₂

yes

characteristic

205

C₄²

D₂₁₂

yes

characteristic

206

C₄²

C₂²×D₅₃

yes

characteristic

207

C₄²

C₂²×D₅₃

yes

characteristic

208

C₂³

—

106

—

209

C₂³

C₂×D₂₁₂

yes

characteristic

210

Q₈

—

106

—

211

Q₈

—

106

—

212

Q₈

—

212

—

213

Q₈

—

212

—

214

D₄

—

106

—

215

D₄

—

106

—

216

D₄

—

212

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

106

—

222

C₂×C₄

—

106

—

223

C₂×C₄

—

106

—

224

C₂×C₄

—

106

—

225

C₂×C₄

—

106

—

226

C₂×C₄

—

212

—

227

C₂×C₄

C₂×D₂₁₂

yes

characteristic

228

C₂×C₄

C₂×D₂₁₂

yes

characteristic

229

C₂×C₄

C₂×D₂₁₂

yes

characteristic

230

C₂×C₄

C₂×D₂₁₂

yes

characteristic

231

C₂×C₄

C₂×D₂₁₂

yes

characteristic

232

C₂×C₄

C₂³×D₅₃

yes

characteristic

233

C₈

—

234

C₈

—

235

C₂²

—

236

C₂²

—

237

C₂²

—

238

C₂²

—

212

—

239

C₂²

—

212

—

240

C₂²

—

212

—

241

C₂²

C₂²×D₂₁₂

yes

characteristic

242

C₄

—

243

C₄

—

244

C₄

—

245

C₄

—

246

C₄

—

212

—

247

C₄

—

212

—

248

C₄

—

212

—

249

C₄

—

212

—

250

C₄

—

212

—

251

C₄

#13133239

yes

—

252

C₄

#13133264

yes

characteristic

253

C₄

#13133264

yes

characteristic

254

C₂

—

255

C₂

—

424

—

256

C₂

#16811199

yes

characteristic

257

C₂

C₂×#13133239

yes

characteristic

258

C₂

#16811423

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#431	D₅₃
hypocenter	C₅₃	#3233
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
1. #16811180	C₂	semidirect
3. #16811025	C₂	semidirect
4. #16810978	C₂	semidirect
5. #16810965	C₂	semidirect
7. #16810919	C₂	semidirect
8. #16810914	C₂	semidirect
9. #16810909	C₂	semidirect
10. #16810904	C₂	semidirect
11. C₅₃×#431	C₂	semidirect
140. #431	D₅₃	semidirect
141. C₅₃	#3233	semidirect

H ≅ G/N

type

1. #16811180

C₂

semidirect

3. #16811025

C₂

semidirect

4. #16810978

C₂

semidirect

5. #16810965

C₂

semidirect

7. #16810919

C₂

semidirect

8. #16810914

C₂

semidirect

9. #16810909

C₂

semidirect

10. #16810904

C₂

semidirect

11. C₅₃×#431

C₂

semidirect

140. #431

D₅₃

semidirect

141. C₅₃

#3233

semidirect

Subgroups — #20285896

Conjugacy classes of subgroups

Complements