Group #20286089 — subgroups

Subgroups — #20286089

Order 6784.

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

normal subgroups only

This group has 348 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	#16810931	C₂	1	2	yes	maximal, complemented
4	#16810919	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810914	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810881	C₂	1	2	yes	maximal, complemented
7	#16810869	C₂	1	2	yes	maximal
8	#16810846	C₂	1	2	yes	maximal, complemented
9	#16810834	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810829	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₁₂	C₂²	1	2	yes	—
13	C₂×#11719913	C₂²	1	2	yes	—
14	#13133217	C₂²	1	1	yes	characteristic
15	#13133217	C₂²	1	1	yes	characteristic
16	#13133211	—	2	1	no	—
17	#13133209	—	2	1	no	—
18	#13133209	C₂²	1	1	yes	characteristic
19	#13133205	C₂²	1	1	yes	characteristic
20	#13133200	—	2	1	no	—
21	#13133196	—	2	1	no	—
22	#13133196	C₂²	1	2	yes	—
23	#13133196	C₂²	1	1	yes	characteristic
24	#13133191	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₀₆	—	2	1	no	—
26	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₀₆	C₂²	1	2	yes	—
28	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₅₃	—	2	1	no	—
30	#13133176	—	2	1	no	—
31	#13133174	—	2	1	no	—
32	#13133151	C₂²	1	2	yes	—
33	#13133149	C₂²	1	2	yes	—
34	#13133142	C₂²	1	2	yes	—
35	#13133142	C₂²	1	2	yes	—
36	#13133142	C₂²	1	2	yes	—
37	#13133139	C₂²	1	2	yes	—
38	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₀₆	C₂³	1	2	yes	—
52	M₁₆×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂×C₈×C₅₃	C₂³	1	2	yes	—
55	C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₂²⋊Dic₅₃	—	2	2	no	—
60	C₂²⋊Dic₅₃	—	2	1	no	—
61	#11719921	—	2	1	no	—
62	#11719921	—	2	1	no	—
63	#11719921	—	2	1	no	—
64	#11719921	—	2	1	no	—
65	C₄⋊Dic₅₃	—	2	1	no	—
66	C₄⋊Dic₅₃	—	2	1	no	—
67	C₄⋊Dic₅₃	—	2	1	no	—
68	C₄⋊Dic₅₃	C₂³	1	2	yes	—
69	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
71	#11719919	—	2	1	no	—
72	#11719919	—	2	1	no	—
73	#11719919	—	2	1	no	—
74	#11719919	—	2	2	no	—
75	#11719919	—	2	2	no	—
76	#11719919	—	2	1	no	—
77	C₄×Dic₅₃	—	2	1	no	—
78	C₄×Dic₅₃	—	2	2	no	—
79	Dic₂₁₂	—	4	2	no	—
80	#11719913	—	4	2	no	—
81	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	4	1	no	—
87	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₅₃	—	4	2	no	—
94	C₂×Dic₅₃	—	2	2	no	—
95	C₂×Dic₅₃	—	2	2	no	—
96	C₂×Dic₅₃	—	2	1	no	—
97	C₂×Dic₅₃	—	2	1	no	—
98	C₂×Dic₅₃	—	2	1	no	—
99	D₂₁₂	—	2	1	no	—
100	D₂₁₂	—	2	1	no	—
101	C₄×D₅₃	—	4	2	no	—
102	Dic₁₀₆	—	2	2	no	—
103	Dic₁₀₆	—	2	1	no	—
104	Dic₁₀₆	—	2	1	no	—
105	Dic₁₀₆	—	2	2	no	—
106	Dic₁₀₆	—	4	2	no	—
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×D₅₃	—	4	1	no	—
114	C₂×D₅₃	—	4	1	no	—
115	C₂×D₅₃	—	4	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	#137	1	2	yes	—
121	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
123	Dic₅₃	—	4	2	no	—
124	Dic₅₃	—	4	1	no	—
125	Dic₅₃	—	4	1	no	—
126	Dic₅₃	—	4	2	no	—
127	Dic₅₃	—	4	1	no	—
128	#3252	—	53	1	no	maximal, Sylow
129	C₂×C₅₃	—	4	1	no	—
130	C₂×C₅₃	#537	1	1	yes	characteristic
131	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
132	C₂×C₅₃	#575	1	1	yes	characteristic
133	D₅₃	—	8	1	no	—
134	#533	—	53	1	no	—
135	#529	—	53	1	no	—
136	#479	—	53	2	no	—
137	#477	—	53	1	no	—
138	#476	—	53	1	no	—
139	#470	—	53	2	no	—
140	#468	—	53	2	no	—
141	#464	—	53	2	no	—
142	#462	—	53	1	no	—
143	#461	—	53	1	no	—
144	#423	D₅₃	1	1	yes	characteristic, complemented
145	C₅₃	#3252	1	1	yes	characteristic, complemented, Sylow
146	C₂×Q₁₆	—	53	2	no	—
147	C₂×SD₁₆	—	53	2	no	—
148	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
149	C₄⋊Q₈	—	53	1	no	—
150	C₄⋊Q₈	—	53	1	no	—
151	#126	—	106	1	no	—
152	#125	—	53	1	no	—
153	#125	—	106	1	no	—
154	#124	—	53	1	no	—
155	#123	—	106	1	no	—
156	C₂²⋊Q₈	—	53	1	no	—
157	C₂²⋊Q₈	—	53	2	no	—
158	C₂²⋊Q₈	—	106	1	no	—
159	#121	—	53	1	no	—
160	C₄×Q₈	—	53	1	no	—
161	C₄×Q₈	—	53	2	no	—
162	C₄×Q₈	—	106	1	no	—
163	C₄×D₄	—	53	1	no	—
164	C₄×D₄	—	106	1	no	—
165	#117	—	106	1	no	—
166	#117	—	106	1	no	—
167	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
168	#107	—	53	2	no	—
169	#106	—	53	2	no	—
170	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
171	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
172	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
173	#103	—	53	2	no	—
174	#103	—	53	2	no	—
175	#103	—	53	2	no	—
176	#102	—	53	2	no	—
177	C₂×Q₈	—	53	1	no	—
178	C₂×Q₈	—	53	2	no	—
179	C₂×Q₈	—	106	1	no	—
180	C₂×D₄	—	53	1	no	—
181	C₂×D₄	—	106	1	no	—
182	C₂²×C₄	—	106	1	no	—
183	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
184	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
186	Q₁₆	—	212	2	no	—
187	SD₁₆	—	212	2	no	—
188	M₁₆	—	4	2	no	—
189	C₂×C₈	C₂²×D₅₃	1	2	yes	—
190	C₂×C₈	C₂²×D₅₃	1	2	yes	—
191	C₄⋊C₄	—	53	2	no	—
192	C₄⋊C₄	—	53	1	no	—
193	C₄⋊C₄	—	53	1	no	—
194	C₄⋊C₄	—	106	1	no	—
195	C₄⋊C₄	—	106	1	no	—
196	C₄⋊C₄	—	106	1	no	—
197	C₄⋊C₄	—	106	1	no	—
198	C₄⋊C₄	—	106	1	no	—
199	C₄⋊C₄	—	106	2	no	—
200	C₄⋊C₄	—	106	1	no	—
201	C₄⋊C₄	—	106	2	no	—
202	C₄⋊C₄	—	106	1	no	—
203	C₂²⋊C₄	—	106	2	no	—
204	C₂²⋊C₄	—	106	1	no	—
205	C₂²⋊C₄	—	106	1	no	—
206	C₂²⋊C₄	—	106	1	no	—
207	C₂²⋊C₄	—	106	1	no	—
208	C₂²⋊C₄	—	106	1	no	—
209	C₄²	—	106	1	no	—
210	C₄²	—	106	2	no	—
211	C₄²	D₂₁₂	1	1	yes	characteristic
212	C₄²	D₂₁₂	1	1	yes	characteristic
213	C₄²	C₂²×D₅₃	1	1	yes	characteristic
214	C₄²	C₂²×D₅₃	1	1	yes	characteristic
215	C₂³	—	106	1	no	—
216	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
217	Q₈	—	106	2	no	—
218	Q₈	—	106	1	no	—
219	Q₈	—	106	1	no	—
220	Q₈	—	106	2	no	—
221	Q₈	—	212	2	no	—
222	D₄	—	106	1	no	—
223	D₄	—	106	1	no	—
224	D₄	—	212	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	106	1	no	—
230	C₂×C₄	—	106	1	no	—
231	C₂×C₄	—	106	2	no	—
232	C₂×C₄	—	106	2	no	—
233	C₂×C₄	—	106	1	no	—
234	C₂×C₄	—	212	2	no	—
235	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
236	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
237	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
238	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
239	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	212	1	no	—
247	C₂²	—	212	1	no	—
248	C₂²	—	212	1	no	—
249	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	212	1	no	—
255	C₄	—	212	1	no	—
256	C₄	—	212	2	no	—
257	C₄	—	212	1	no	—
258	C₄	—	212	2	no	—
259	C₄	#13133250	1	2	yes	—
260	C₄	#13133262	1	1	yes	characteristic
261	C₄	#13133265	1	1	yes	characteristic
262	C₂	—	4	1	no	—
263	C₂	—	424	1	no	—
264	C₂	#16811204	1	1	yes	characteristic
265	C₂	C₂×#13133250	1	1	yes	characteristic
266	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

#16810931

C₂

yes

maximal, complemented

#16810919

C₂

yes

maximal, characteristic, complemented

#16810914

C₂

yes

maximal, characteristic, complemented

#16810881

C₂

yes

maximal, complemented

#16810869

C₂

yes

maximal

#16810846

C₂

yes

maximal, complemented

#16810834

C₂

yes

maximal, characteristic, complemented

#16810829

C₂

yes

maximal, characteristic, complemented

C₅₃×#423

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

—

C₂×#11719913

C₂²

yes

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

—

C₂²

yes

characteristic

#13133205

C₂²

yes

—

—

C₂²

yes

—

#13133196

C₂²

yes

characteristic

#13133191

C₂²

yes

characteristic

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

characteristic

C₄×C₂²⋊D₅₃

—

#13133176

—

#13133174

—

#13133151

C₂²

yes

—

#13133149

C₂²

yes

—

#13133142

C₂²

yes

—

#13133142

C₂²

yes

—

#13133142

C₂²

yes

—

#13133139

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

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

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂³

yes

—

M₁₆×C₅₃

—

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

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

—

—

—

—

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

Dic₂₁₂

—

#11719913

—

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

—

100

D₂₁₂

—

101

C₄×D₅₃

—

102

Dic₁₀₆

—

103

Dic₁₀₆

—

104

Dic₁₀₆

—

105

Dic₁₀₆

—

106

Dic₁₀₆

—

107

C₈×C₅₃

—

108

C₈×C₅₃

—

109

C₂²×C₅₃

—

110

C₂²×C₅₃

—

111

C₂²×C₅₃

—

112

C₂²×C₅₃

C₂²×D₄

yes

characteristic

113

C₂×D₅₃

—

114

C₂×D₅₃

—

115

C₂×D₅₃

—

116

C₄×C₅₃

—

117

C₄×C₅₃

—

118

C₄×C₅₃

—

119

C₄×C₅₃

—

120

C₄×C₅₃

#137

yes

—

121

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

122

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

123

Dic₅₃

—

124

Dic₅₃

—

125

Dic₅₃

—

126

Dic₅₃

—

127

Dic₅₃

—

128

#3252

—

maximal, Sylow

129

C₂×C₅₃

—

130

C₂×C₅₃

#537

yes

characteristic

131

C₂×C₅₃

C₂×#137

yes

characteristic

132

C₂×C₅₃

#575

yes

characteristic

133

D₅₃

—

134

#533

—

135

#529

—

136

#479

—

137

#477

—

138

#476

—

139

#470

—

140

#468

—

141

#464

—

142

#462

—

143

#461

—

144

#423

D₅₃

yes

characteristic, complemented

145

C₅₃

#3252

yes

characteristic, complemented, Sylow

146

C₂×Q₁₆

—

147

C₂×SD₁₆

—

148

C₂×M₁₆

C₂×D₅₃

yes

—

149

C₄⋊Q₈

—

150

C₄⋊Q₈

—

151

#126

—

106

—

152

#125

—

153

#125

—

106

—

154

#124

—

155

#123

—

106

—

156

C₂²⋊Q₈

—

157

C₂²⋊Q₈

—

158

C₂²⋊Q₈

—

106

—

159

#121

—

160

C₄×Q₈

—

161

C₄×Q₈

—

162

C₄×Q₈

—

106

—

163

C₄×D₄

—

164

C₄×D₄

—

106

—

165

#117

—

106

—

166

#117

—

106

—

167

C₂×C₄²

C₂×D₅₃

yes

characteristic

168

#107

—

169

#106

—

170

C₄⋊C₈

C₂×D₅₃

yes

characteristic

171

C₄⋊C₈

C₂×D₅₃

yes

characteristic

172

C₄⋊C₈

C₂×D₅₃

yes

—

173

#103

—

174

#103

—

175

#103

—

176

#102

—

177

C₂×Q₈

—

178

C₂×Q₈

—

179

C₂×Q₈

—

106

—

180

C₂×D₄

—

181

C₂×D₄

—

106

—

182

C₂²×C₄

—

106

—

183

C₂²×C₄

D₂₁₂

yes

characteristic

184

C₂²×C₄

D₂₁₂

yes

characteristic

185

C₂²×C₄

C₂²×D₅₃

yes

characteristic

186

Q₁₆

—

212

—

187

SD₁₆

—

212

—

188

M₁₆

—

189

C₂×C₈

C₂²×D₅₃

yes

—

190

C₂×C₈

C₂²×D₅₃

yes

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₄⋊C₄

—

194

C₄⋊C₄

—

106

—

195

C₄⋊C₄

—

106

—

196

C₄⋊C₄

—

106

—

197

C₄⋊C₄

—

106

—

198

C₄⋊C₄

—

106

—

199

C₄⋊C₄

—

106

—

200

C₄⋊C₄

—

106

—

201

C₄⋊C₄

—

106

—

202

C₄⋊C₄

—

106

—

203

C₂²⋊C₄

—

106

—

204

C₂²⋊C₄

—

106

—

205

C₂²⋊C₄

—

106

—

206

C₂²⋊C₄

—

106

—

207

C₂²⋊C₄

—

106

—

208

C₂²⋊C₄

—

106

—

209

C₄²

—

106

—

210

C₄²

—

106

—

211

C₄²

D₂₁₂

yes

characteristic

212

C₄²

D₂₁₂

yes

characteristic

213

C₄²

C₂²×D₅₃

yes

characteristic

214

C₄²

C₂²×D₅₃

yes

characteristic

215

C₂³

—

106

—

216

C₂³

C₂×D₂₁₂

yes

characteristic

217

Q₈

—

106

—

218

Q₈

—

106

—

219

Q₈

—

106

—

220

Q₈

—

106

—

221

Q₈

—

212

—

222

D₄

—

106

—

223

D₄

—

106

—

224

D₄

—

212

—

225

C₂×C₄

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

—

229

C₂×C₄

—

106

—

230

C₂×C₄

—

106

—

231

C₂×C₄

—

106

—

232

C₂×C₄

—

106

—

233

C₂×C₄

—

106

—

234

C₂×C₄

—

212

—

235

C₂×C₄

C₂×D₂₁₂

yes

characteristic

236

C₂×C₄

C₂×D₂₁₂

yes

characteristic

237

C₂×C₄

C₂×D₂₁₂

yes

characteristic

238

C₂×C₄

C₂×D₂₁₂

yes

characteristic

239

C₂×C₄

C₂×D₂₁₂

yes

characteristic

240

C₂×C₄

C₂³×D₅₃

yes

characteristic

241

C₈

—

242

C₈

—

243

C₂²

—

244

C₂²

—

245

C₂²

—

246

C₂²

—

212

—

247

C₂²

—

212

—

248

C₂²

—

212

—

249

C₂²

C₂²×D₂₁₂

yes

characteristic

250

C₄

—

251

C₄

—

252

C₄

—

253

C₄

—

254

C₄

—

212

—

255

C₄

—

212

—

256

C₄

—

212

—

257

C₄

—

212

—

258

C₄

—

212

—

259

C₄

#13133250

yes

—

260

C₄

#13133262

yes

characteristic

261

C₄

#13133265

yes

characteristic

262

C₂

—

263

C₂

—

424

—

264

C₂

#16811204

yes

characteristic

265

C₂

C₂×#13133250

yes

characteristic

266

C₂

#16811423

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#3252
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#423

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #16811180	C₂	semidirect
3. #16810931	C₂	semidirect
4. #16810919	C₂	semidirect
5. #16810914	C₂	semidirect
6. #16810881	C₂	semidirect
8. #16810846	C₂	semidirect
9. #16810834	C₂	semidirect
10. #16810829	C₂	semidirect
11. C₅₃×#423	C₂	semidirect
144. #423	D₅₃	semidirect
145. C₅₃	#3252	semidirect

H ≅ G/N

type

1. #16811180

C₂

semidirect

3. #16810931

C₂

semidirect

4. #16810919

C₂

semidirect

5. #16810914

C₂

semidirect

6. #16810881

C₂

semidirect

8. #16810846

C₂

semidirect

9. #16810834

C₂

semidirect

10. #16810829

C₂

semidirect

11. C₅₃×#423

C₂

semidirect

144. #423

D₅₃

semidirect

145. C₅₃

#3252

semidirect

Subgroups — #20286089

Conjugacy classes of subgroups

Complements