Group #20274899 — subgroups

Subgroups — #20274899

Order 6784.

324 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂²×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133146	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133129	C₂	1	1	yes	maximal, characteristic
5	C₄×#11719921	C₂	1	1	yes	maximal, characteristic
6	#16810377	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719912	—	2	2	no	—
10	C₂×#11719912	—	2	2	no	—
11	C₂×#11719912	—	2	2	no	—
12	C₂²×#88322	C₂²	1	1	yes	characteristic
13	C₂²×#88322	C₂²	1	1	yes	characteristic
14	C₂×#11719921	C₄	1	1	yes	characteristic
15	C₂×#11719921	C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊#88322	—	2	2	no	—
18	#13133146	—	2	2	no	—
19	#13133129	—	2	2	no	—
20	#13133124	C₄	1	1	yes	characteristic
21	#13133124	C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂×C₄	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	D₄	1	2	yes	—
37	C₂×C₄×D₅₃	Q₈	1	2	yes	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	D₄	1	2	yes	—
44	C₄²×C₅₃	—	2	1	no	—
45	C₄²×C₅₃	—	2	1	no	—
46	#11719921	—	2	2	no	—
47	#11719921	—	2	2	no	—
48	C₄×Dic₅₃	—	2	1	no	—
49	C₄×Dic₅₃	—	2	1	no	—
50	C₂×#88322	—	2	1	no	—
51	C₂×#88322	—	2	1	no	—
52	C₂×#88322	—	2	1	no	—
53	C₂×#88322	—	2	1	no	—
54	C₂×#88322	—	2	1	no	—
55	C₂×#88322	D₄	1	2	yes	—
56	#11719912	—	2	8	no	—
57	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×D₅₃	—	2	1	no	—
59	C₂²×D₅₃	—	2	1	no	—
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	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	M₁₆	1	2	yes	complemented
71	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
77	C₂×Dic₅₃	—	2	1	no	—
78	C₂×Dic₅₃	—	2	1	no	—
79	C₂×Dic₅₃	—	2	1	no	—
80	C₂×Dic₅₃	—	2	1	no	—
81	C₂×Dic₅₃	—	2	1	no	—
82	C₂×Dic₅₃	—	2	1	no	—
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
85	C₂×Dic₅₃	M₁₆	1	2	yes	complemented
86	C₄×D₅₃	—	2	2	no	—
87	C₄×D₅₃	—	2	2	no	—
88	C₄×D₅₃	—	2	2	no	—
89	C₄×D₅₃	—	2	2	no	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	—	4	1	no	—
92	#88322	—	2	2	no	—
93	#88322	—	4	1	no	—
94	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂²×C₅₃	#131	1	1	yes	characteristic
100	C₂²×C₅₃	#131	1	1	yes	characteristic
101	C₂×D₅₃	—	2	2	no	—
102	C₂×D₅₃	—	2	2	no	—
103	C₂×D₅₃	—	2	2	no	—
104	C₂×D₅₃	—	4	1	no	—
105	C₂×D₅₃	—	4	1	no	—
106	C₂×D₅₃	—	4	1	no	—
107	C₂×D₅₃	—	4	1	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	4	1	no	—
110	C₄×C₅₃	#120	1	1	yes	characteristic
111	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₄×C₅₃	#123	1	1	yes	characteristic
114	Dic₅₃	—	2	2	no	—
115	Dic₅₃	—	2	2	no	—
116	Dic₅₃	—	4	1	no	—
117	#1851	—	53	1	no	maximal, Sylow
118	C₂×C₅₃	#385	1	1	yes	characteristic
119	C₂×C₅₃	#423	1	1	yes	characteristic
120	C₂×C₅₃	#424	1	1	yes	characteristic
121	C₂×C₅₃	#435	1	1	yes	characteristic
122	C₂×C₅₃	#435	1	1	yes	characteristic
123	C₂×C₅₃	#436	1	1	yes	characteristic
124	C₂×C₅₃	#436	1	1	yes	characteristic
125	D₅₃	—	4	2	no	—
126	C₂²×M₁₆	—	53	1	no	—
127	C₂×C₄⋊C₈	—	53	1	no	—
128	C₂×C₄⋊C₈	—	53	1	no	—
129	C₂×C₂²⋊C₈	—	53	1	no	—
130	C₄×C₂²⋊C₄	—	53	1	no	—
131	#336	—	53	1	no	—
132	#336	D₅₃	1	1	yes	characteristic, complemented
133	C₅₃	#1851	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₄	—	53	1	no	—
135	C₂×M₁₆	—	106	2	no	—
136	C₂×M₁₆	—	106	2	no	—
137	C₂×M₁₆	—	106	2	no	—
138	C₂²×C₈	—	53	1	no	—
139	C₂²×C₈	—	53	1	no	—
140	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
141	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	—	53	1	no	—
143	C₂×C₂²⋊C₄	—	53	1	no	—
144	C₂×C₄²	—	53	1	no	—
145	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
146	C₄⋊C₈	—	106	2	no	—
147	C₄⋊C₈	—	106	2	no	—
148	C₂²⋊C₈	—	106	2	no	—
149	#95	—	53	1	no	—
150	#95	—	53	1	no	—
151	C₂⁴	—	53	1	no	—
152	C₂²×C₄	—	53	1	no	—
153	C₂²×C₄	—	53	1	no	—
154	C₂²×C₄	—	53	2	no	—
155	C₂²×C₄	—	53	2	no	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	106	1	no	—
158	C₂²×C₄	—	106	1	no	—
159	C₂²×C₄	—	106	1	no	—
160	C₂²×C₄	—	106	1	no	—
161	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
162	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
164	M₁₆	—	106	8	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	53	2	no	—
171	C₂×C₈	—	106	1	no	—
172	C₂×C₈	—	106	1	no	—
173	C₂×C₈	—	106	1	no	—
174	C₂×C₈	—	106	1	no	—
175	C₂×C₈	—	106	1	no	—
176	C₂×C₈	D₂₁₂	1	2	yes	—
177	C₂²⋊C₄	—	106	2	no	—
178	C₂²⋊C₄	—	106	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	106	1	no	—
182	C₄²	—	106	1	no	—
183	C₂³	—	53	2	no	—
184	C₂³	—	106	1	no	—
185	C₂³	—	106	1	no	—
186	C₂³	—	106	1	no	—
187	C₂³	—	106	1	no	—
188	C₂³	—	106	1	no	—
189	C₂³	—	106	1	no	—
190	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	2	no	—
198	C₂×C₄	—	106	1	no	—
199	C₂×C₄	—	106	1	no	—
200	C₂×C₄	—	106	1	no	—
201	C₂×C₄	—	106	1	no	—
202	C₂×C₄	—	106	2	no	—
203	C₂×C₄	—	106	2	no	—
204	C₂×C₄	—	106	2	no	—
205	C₂×C₄	—	106	2	no	—
206	C₂×C₄	—	106	1	no	—
207	C₂×C₄	—	106	1	no	—
208	C₂×C₄	—	106	1	no	—
209	C₂×C₄	#11719912	1	2	yes	complemented
210	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
211	C₂×C₄	#11719945	1	1	yes	characteristic
212	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
213	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
214	C₂×C₄	#11719947	1	1	yes	characteristic
215	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	4	1	no	—
218	C₈	—	106	2	no	—
219	C₈	—	212	1	no	—
220	C₂²	—	106	2	no	—
221	C₂²	—	106	2	no	—
222	C₂²	—	106	2	no	—
223	C₂²	—	212	1	no	—
224	C₂²	—	212	1	no	—
225	C₂²	—	212	1	no	—
226	C₂²	—	212	1	no	—
227	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
228	C₂²	#13133177	1	1	yes	characteristic
229	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
230	C₂²	M₁₆×D₅₃	1	1	yes	characteristic
231	C₂²	C₂×#11719912	1	1	yes	characteristic
232	C₂²	#13133223	1	1	yes	characteristic
233	C₂²	#13133225	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	106	2	no	—
237	C₄	—	106	2	no	—
238	C₄	—	212	1	no	—
239	C₄	#13133187	1	1	yes	characteristic
240	C₄	#13133195	1	1	yes	characteristic
241	C₄	#13133197	1	1	yes	characteristic
242	C₄	#13133199	1	1	yes	characteristic
243	C₂	—	212	2	no	—
244	C₂	#16810497	1	1	yes	characteristic
245	C₂	#16810645	1	1	yes	characteristic
246	C₂	#16810648	1	1	yes	characteristic
247	C₂	#16810691	1	1	yes	characteristic
248	C₂	#16810696	1	1	yes	characteristic
249	C₂	#16810698	1	1	yes	characteristic
250	C₂	#16810702	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719912

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic, complemented

C₂×#13133146

C₂

yes

maximal, characteristic, complemented

C₂×#13133129

C₂

yes

maximal, characteristic

C₄×#11719921

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal, characteristic, complemented

C₅₃×#336

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

—

—

—

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×#11719921

C₄

yes

characteristic

C₂×#11719921

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

C₄

yes

characteristic

#13133124

C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

#11719921

—

#11719921

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

D₄

yes

—

#11719912

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

M₁₆

yes

complemented

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×Q₈

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

C₄⋊C₄

yes

—

C₂×Dic₅₃

M₁₆

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×C₅₃

#131

yes

characteristic

100

C₂²×C₅₃

#131

yes

characteristic

101

C₂×D₅₃

—

102

C₂×D₅₃

—

103

C₂×D₅₃

—

104

C₂×D₅₃

—

105

C₂×D₅₃

—

106

C₂×D₅₃

—

107

C₂×D₅₃

—

108

C₄×C₅₃

—

109

C₄×C₅₃

—

110

C₄×C₅₃

#120

yes

characteristic

111

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

112

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

113

C₄×C₅₃

#123

yes

characteristic

114

Dic₅₃

—

115

Dic₅₃

—

116

Dic₅₃

—

117

#1851

—

maximal, Sylow

118

C₂×C₅₃

#385

yes

characteristic

119

C₂×C₅₃

#423

yes

characteristic

120

C₂×C₅₃

#424

yes

characteristic

121

C₂×C₅₃

#435

yes

characteristic

122

C₂×C₅₃

#435

yes

characteristic

123

C₂×C₅₃

#436

yes

characteristic

124

C₂×C₅₃

#436

yes

characteristic

125

D₅₃

—

126

C₂²×M₁₆

—

127

C₂×C₄⋊C₈

—

128

C₂×C₄⋊C₈

—

129

C₂×C₂²⋊C₈

—

130

C₄×C₂²⋊C₄

—

131

#336

—

132

#336

D₅₃

yes

characteristic, complemented

133

C₅₃

#1851

yes

characteristic, complemented, Sylow

134

C₂³×C₄

—

135

C₂×M₁₆

—

106

—

136

C₂×M₁₆

—

106

—

137

C₂×M₁₆

—

106

—

138

C₂²×C₈

—

139

C₂²×C₈

—

140

C₂²×C₈

C₂×D₅₃

yes

characteristic

141

C₂²×C₈

C₂×D₅₃

yes

characteristic

142

C₂×C₂²⋊C₄

—

143

C₂×C₂²⋊C₄

—

144

C₂×C₄²

—

145

C₂×C₄²

C₂×D₅₃

yes

characteristic

146

C₄⋊C₈

—

106

—

147

C₄⋊C₈

—

106

—

148

C₂²⋊C₈

—

106

—

149

#95

—

150

#95

—

151

C₂⁴

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

106

—

158

C₂²×C₄

—

106

—

159

C₂²×C₄

—

106

—

160

C₂²×C₄

—

106

—

161

C₂²×C₄

C₄×D₅₃

yes

characteristic

162

C₂²×C₄

C₄×D₅₃

yes

characteristic

163

C₂²×C₄

C₂²×D₅₃

yes

characteristic

164

M₁₆

—

106

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

106

—

172

C₂×C₈

—

106

—

173

C₂×C₈

—

106

—

174

C₂×C₈

—

106

—

175

C₂×C₈

—

106

—

176

C₂×C₈

D₂₁₂

yes

—

177

C₂²⋊C₄

—

106

—

178

C₂²⋊C₄

—

106

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

106

—

182

C₄²

—

106

—

183

C₂³

—

184

C₂³

—

106

—

185

C₂³

—

106

—

186

C₂³

—

106

—

187

C₂³

—

106

—

188

C₂³

—

106

—

189

C₂³

—

106

—

190

C₂³

C₂×C₄×D₅₃

yes

characteristic

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

—

106

—

200

C₂×C₄

—

106

—

201

C₂×C₄

—

106

—

202

C₂×C₄

—

106

—

203

C₂×C₄

—

106

—

204

C₂×C₄

—

106

—

205

C₂×C₄

—

106

—

206

C₂×C₄

—

106

—

207

C₂×C₄

—

106

—

208

C₂×C₄

—

106

—

209

C₂×C₄

#11719912

yes

complemented

210

C₂×C₄

C₂×D₂₁₂

yes

characteristic

211

C₂×C₄

#11719945

yes

characteristic

212

C₂×C₄

D₄×D₅₃

yes

characteristic

213

C₂×C₄

D₄×D₅₃

yes

characteristic

214

C₂×C₄

#11719947

yes

characteristic

215

C₂×C₄

Q₈×D₅₃

yes

characteristic

216

C₈

—

217

C₈

—

218

C₈

—

106

—

219

C₈

—

212

—

220

C₂²

—

106

—

221

C₂²

—

106

—

222

C₂²

—

106

—

223

C₂²

—

212

—

224

C₂²

—

212

—

225

C₂²

—

212

—

226

C₂²

—

212

—

227

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

228

C₂²

#13133177

yes

characteristic

229

C₂²

C₄×D₂₁₂

yes

characteristic

230

C₂²

M₁₆×D₅₃

yes

characteristic

231

C₂²

C₂×#11719912

yes

characteristic

232

C₂²

#13133223

yes

characteristic

233

C₂²

#13133225

yes

characteristic

234

C₄

—

235

C₄

—

236

C₄

—

106

—

237

C₄

—

106

—

238

C₄

—

212

—

239

C₄

#13133187

yes

characteristic

240

C₄

#13133195

yes

characteristic

241

C₄

#13133197

yes

characteristic

242

C₄

#13133199

yes

characteristic

243

C₂

—

212

—

244

C₂

#16810497

yes

characteristic

245

C₂

#16810645

yes

characteristic

246

C₂

#16810648

yes

characteristic

247

C₂

#16810691

yes

characteristic

248

C₂

#16810696

yes

characteristic

249

C₂

#16810698

yes

characteristic

250

C₂

#16810702

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₄
Fitting	C₅₃×#336	C₂
hypercenter	#336	D₅₃
hypocenter	C₅₃	#1851
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂²×C₄

Fitting

C₅₃×#336

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
2. C₂×C₄⋊#88322	C₂	semidirect
3. C₂×#13133146	C₂	semidirect
6. #16810377	C₂	semidirect
7. C₅₃×#336	C₂	semidirect
70. C₂×C₄×C₅₃	M₁₆	semidirect
85. C₂×Dic₅₃	M₁₆	semidirect
132. #336	D₅₃	semidirect
133. C₅₃	#1851	semidirect
209. C₂×C₄	#11719912	semidirect

H ≅ G/N

type

2. C₂×C₄⋊#88322

C₂

semidirect

3. C₂×#13133146

C₂

semidirect

6. #16810377

C₂

semidirect

7. C₅₃×#336

C₂

semidirect

70. C₂×C₄×C₅₃

M₁₆

semidirect

85. C₂×Dic₅₃

M₁₆

semidirect

132. #336

D₅₃

semidirect

133. C₅₃

#1851

semidirect

209. C₂×C₄

#11719912

semidirect

Subgroups — #20274899

Conjugacy classes of subgroups

Complements