Group #27277232 — subgroups

Subgroups — #27277232

Order 13568.

378 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 123 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×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20274853	C₂	1	1	yes	maximal, characteristic
3	#20273145	C₂	1	2	yes	maximal
4	#20272611	C₂	1	2	yes	maximal
5	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₀₆	—	2	1	no	—
8	C₈×Dic₁₀₆	—	2	1	no	—
9	C₈×Dic₁₀₆	C₄	1	2	yes	—
10	C₈×Dic₁₀₆	C₄	1	2	yes	—
11	#16810647	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₂×#13133146	—	2	1	no	—
14	C₂×C₈×Dic₅₃	—	2	1	no	—
15	#16810388	—	2	1	no	—
16	#16810388	—	2	1	no	—
17	#16810355	—	2	4	no	—
18	C₅₃×#431	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#335	—	2	1	no	—
21	C₅₃×#334	—	2	1	no	—
22	C₂²×Dic₁₀₆	Q₈	1	1	yes	characteristic
23	C₂×#11719917	—	2	2	no	—
24	C₂²×#88322	—	2	1	no	—
25	C₄×Dic₁₀₆	—	2	1	no	—
26	C₄×Dic₁₀₆	—	2	1	no	—
27	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
28	C₄×Dic₁₀₆	D₄	1	2	yes	—
29	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
30	C₂×#11719919	—	2	1	no	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	—	2	2	no	—
34	C₄⋊#88322	—	2	2	no	—
35	C₄⋊#88322	C₂×C₄	1	2	yes	—
36	#13133146	—	2	2	no	—
37	#13133146	—	2	2	no	—
38	C₈×Dic₅₃	—	2	2	no	—
39	C₈×Dic₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₂²⋊C₈×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂²×Dic₅₃	—	2	1	no	—
54	C₂²×Dic₅₃	—	2	1	no	—
55	C₂×Dic₁₀₆	—	2	1	no	—
56	C₂×Dic₁₀₆	—	2	4	no	—
57	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₃	#41	1	1	yes	characteristic
69	C₄⋊Dic₅₃	—	2	1	no	—
70	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
71	#11719919	—	2	2	no	—
72	#11719919	—	2	2	no	—
73	C₄×Dic₅₃	—	2	2	no	—
74	C₄×Dic₅₃	—	2	2	no	—
75	#11719917	—	4	2	no	—
76	#11719917	—	4	2	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	4	1	no	—
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#135	1	2	yes	—
90	C₂×C₄×C₅₃	#137	1	2	yes	—
91	C₂×Dic₅₃	—	2	4	no	—
92	C₂×Dic₅₃	—	2	2	no	—
93	C₂×Dic₅₃	—	2	2	no	—
94	C₂×Dic₅₃	—	4	2	no	—
95	Dic₁₀₆	—	2	4	no	—
96	Dic₁₀₆	—	2	2	no	—
97	Dic₁₀₆	#108	1	4	yes	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	1	no	—
101	C₈×C₅₃	—	2	1	no	—
102	C₈×C₅₃	—	8	2	no	—
103	#88322	—	4	2	no	—
104	#88322	—	8	2	no	—
105	#8661	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	#380	1	1	yes	characteristic
109	C₂²×C₅₃	#479	1	2	yes	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	#416	1	1	yes	characteristic
114	C₄×C₅₃	#419	1	1	yes	characteristic
115	C₄×C₅₃	C₂×#108	1	2	yes	—
116	C₄×C₅₃	#464	1	2	yes	—
117	Dic₅₃	—	2	4	no	—
118	Dic₅₃	—	4	2	no	—
119	C₂×C₈×Q₈	—	53	1	no	—
120	#1837	—	53	1	no	—
121	#1564	D₅₃	1	1	yes	characteristic
122	#1487	—	53	2	no	—
123	#1301	—	53	2	no	—
124	C₂×C₅₃	—	2	2	no	—
125	C₂×C₅₃	#1805	1	1	yes	characteristic
126	C₂×C₅₃	#1810	1	1	yes	characteristic
127	C₂×C₅₃	#1814	1	1	yes	characteristic
128	C₂×C₄×Q₈	—	53	1	no	—
129	C₈×Q₈	—	53	2	no	—
130	C₈×Q₈	—	53	2	no	—
131	C₈×Q₈	—	106	1	no	—
132	C₈×Q₈	—	106	1	no	—
133	#431	C₂×D₅₃	1	2	yes	—
134	#423	—	53	2	no	—
135	C₂×C₄⋊C₈	—	53	1	no	—
136	C₂×C₄⋊C₈	—	106	1	no	—
137	C₂×C₄×C₈	—	106	1	no	—
138	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
139	#341	—	106	1	no	—
140	#341	—	106	1	no	—
141	#335	—	2	1	no	—
142	#334	—	2	1	no	—
143	Q₈⋊C₈	—	106	4	no	—
144	C₅₃	#8661	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	53	1	no	—
146	C₂×M₁₆	—	106	2	no	—
147	C₂²×C₈	—	106	1	no	—
148	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
149	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
150	C₄×Q₈	—	53	2	no	—
151	C₄×Q₈	—	53	2	no	—
152	C₄×Q₈	—	106	1	no	—
153	C₄×Q₈	—	106	1	no	—
154	C₂×C₄⋊C₄	—	53	1	no	—
155	C₂×C₄⋊C₄	—	106	1	no	—
156	C₂×C₄²	—	106	1	no	—
157	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	53	2	no	—
160	C₄⋊C₈	—	106	2	no	—
161	C₄⋊C₈	—	106	2	no	—
162	C₄⋊C₈	—	106	2	no	—
163	C₄⋊C₈	—	106	2	no	—
164	C₄⋊C₈	—	106	1	no	—
165	C₂²⋊C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	106	2	no	—
168	C₄×C₈	—	106	2	no	—
169	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
170	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×Q₈	—	53	2	no	—
174	C₂×Q₈	—	106	4	no	—
175	C₂×Q₈	—	106	1	no	—
176	C₂²×C₄	—	106	1	no	—
177	C₂²×C₄	—	106	1	no	—
178	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
179	C₂²×C₄	#11719947	1	1	yes	characteristic
180	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
181	M₁₆	—	212	2	no	—
182	M₁₆	—	212	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	106	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	#11719921	1	1	yes	characteristic
190	C₂×C₈	#11719921	1	1	yes	characteristic
191	C₂×C₈	#11719921	1	1	yes	characteristic
192	C₂×C₈	#11719921	1	1	yes	characteristic
193	C₄⋊C₄	—	53	2	no	—
194	C₄⋊C₄	—	106	2	no	—
195	C₄⋊C₄	—	106	2	no	—
196	C₄⋊C₄	—	106	1	no	—
197	C₄²	—	106	2	no	—
198	C₄²	—	106	2	no	—
199	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₄²	D₄×D₅₃	1	1	yes	characteristic
201	C₄²	#11719949	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂³	#13133197	1	1	yes	characteristic
204	Q₈	—	53	4	no	—
205	Q₈	—	106	4	no	—
206	Q₈	—	106	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	106	4	no	—
210	C₂×C₄	—	106	2	no	—
211	C₂×C₄	—	106	2	no	—
212	C₂×C₄	—	212	2	no	—
213	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
214	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
215	C₂×C₄	#13133175	1	1	yes	characteristic
216	C₂×C₄	#13133188	1	1	yes	characteristic
217	C₂×C₄	#13133192	1	1	yes	characteristic
218	C₂×C₄	#13133193	1	1	yes	characteristic
219	C₂×C₄	#13133239	1	2	yes	—
220	C₂×C₄	#13133250	1	2	yes	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	8	2	no	—
226	C₈	—	424	2	no	—
227	C₈	—	212	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#16810480	1	1	yes	characteristic
231	C₂²	#16810931	1	2	yes	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	106	4	no	—
236	C₄	—	212	2	no	—
237	C₄	#16810620	1	1	yes	characteristic
238	C₄	#16810633	1	1	yes	characteristic
239	C₄	#108×D₅₃	1	2	yes	—
240	C₄	#16810846	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#20274731	1	1	yes	characteristic
243	C₂	#20274750	1	1	yes	characteristic
244	C₂	#20274766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₀₆

C₂

yes

maximal, characteristic

#20274853

C₂

yes

maximal, characteristic

#20273145

C₂

yes

maximal

#20272611

C₂

yes

maximal

C₅₃×#1564

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₀₆

C₂²

yes

characteristic

C₈×Dic₁₀₆

—

C₈×Dic₁₀₆

—

C₈×Dic₁₀₆

C₄

yes

—

C₈×Dic₁₀₆

C₄

yes

—

#16810647

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

characteristic

C₂×#13133146

—

C₂×C₈×Dic₅₃

—

#16810388

—

#16810388

—

#16810355

—

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

—

C₅₃×#334

—

C₂²×Dic₁₀₆

Q₈

yes

characteristic

C₂×#11719917

—

C₂²×#88322

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂×C₄

yes

—

C₄×Dic₁₀₆

D₄

yes

—

C₂×C₄⋊Dic₅₃

Q₈

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

—

—

—

C₂×C₄

yes

—

#13133146

—

#13133146

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₄⋊C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

C₂³×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

#108

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

100

C₈×C₅₃

—

101

C₈×C₅₃

—

102

C₈×C₅₃

—

103

#88322

—

104

#88322

—

105

#8661

—

maximal, Sylow

106

C₂²×C₅₃

—

107

C₂²×C₅₃

—

108

C₂²×C₅₃

#380

yes

characteristic

109

C₂²×C₅₃

#479

yes

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

—

113

C₄×C₅₃

#416

yes

characteristic

114

C₄×C₅₃

#419

yes

characteristic

115

C₄×C₅₃

C₂×#108

yes

—

116

C₄×C₅₃

#464

yes

—

117

Dic₅₃

—

118

Dic₅₃

—

119

C₂×C₈×Q₈

—

120

#1837

—

121

#1564

D₅₃

yes

characteristic

122

#1487

—

123

#1301

—

124

C₂×C₅₃

—

125

C₂×C₅₃

#1805

yes

characteristic

126

C₂×C₅₃

#1810

yes

characteristic

127

C₂×C₅₃

#1814

yes

characteristic

128

C₂×C₄×Q₈

—

129

C₈×Q₈

—

130

C₈×Q₈

—

131

C₈×Q₈

—

106

—

132

C₈×Q₈

—

106

—

133

#431

C₂×D₅₃

yes

—

134

#423

—

135

C₂×C₄⋊C₈

—

136

C₂×C₄⋊C₈

—

106

—

137

C₂×C₄×C₈

—

106

—

138

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

139

#341

—

106

—

140

#341

—

106

—

141

#335

—

142

#334

—

143

Q₈⋊C₈

—

106

—

144

C₅₃

#8661

yes

characteristic, complemented, Sylow

145

C₂²×Q₈

—

146

C₂×M₁₆

—

106

—

147

C₂²×C₈

—

106

—

148

C₂²×C₈

D₂₁₂

yes

characteristic

149

C₂²×C₈

D₂₁₂

yes

characteristic

150

C₄×Q₈

—

151

C₄×Q₈

—

152

C₄×Q₈

—

106

—

153

C₄×Q₈

—

106

—

154

C₂×C₄⋊C₄

—

155

C₂×C₄⋊C₄

—

106

—

156

C₂×C₄²

—

106

—

157

C₂×C₄²

C₂²×D₅₃

yes

characteristic

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

C₄⋊C₈

—

106

—

161

C₄⋊C₈

—

106

—

162

C₄⋊C₈

—

106

—

163

C₄⋊C₈

—

106

—

164

C₄⋊C₈

—

106

—

165

C₂²⋊C₈

—

166

C₄×C₈

—

167

C₄×C₈

—

106

—

168

C₄×C₈

—

106

—

169

C₄×C₈

C₄×D₅₃

yes

characteristic

170

C₄×C₈

C₄×D₅₃

yes

characteristic

171

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

172

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

173

C₂×Q₈

—

174

C₂×Q₈

—

106

—

175

C₂×Q₈

—

106

—

176

C₂²×C₄

—

106

—

177

C₂²×C₄

—

106

—

178

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

179

C₂²×C₄

#11719947

yes

characteristic

180

C₂²×C₄

Q₈×D₅₃

yes

characteristic

181

M₁₆

—

212

—

182

M₁₆

—

212

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

—

106

—

187

C₂×C₈

—

212

—

188

C₂×C₈

—

212

—

189

C₂×C₈

#11719921

yes

characteristic

190

C₂×C₈

#11719921

yes

characteristic

191

C₂×C₈

#11719921

yes

characteristic

192

C₂×C₈

#11719921

yes

characteristic

193

C₄⋊C₄

—

194

C₄⋊C₄

—

106

—

195

C₄⋊C₄

—

106

—

196

C₄⋊C₄

—

106

—

197

C₄²

—

106

—

198

C₄²

—

106

—

199

C₄²

C₂×C₄×D₅₃

yes

characteristic

200

C₄²

D₄×D₅₃

yes

characteristic

201

C₄²

#11719949

yes

characteristic

202

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

203

C₂³

#13133197

yes

characteristic

204

Q₈

—

205

Q₈

—

106

—

206

Q₈

—

106

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

106

—

210

C₂×C₄

—

106

—

211

C₂×C₄

—

106

—

212

C₂×C₄

—

212

—

213

C₂×C₄

C₂×#11719921

yes

characteristic

214

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

215

C₂×C₄

#13133175

yes

characteristic

216

C₂×C₄

#13133188

yes

characteristic

217

C₂×C₄

#13133192

yes

characteristic

218

C₂×C₄

#13133193

yes

characteristic

219

C₂×C₄

#13133239

yes

—

220

C₂×C₄

#13133250

yes

—

221

C₈

—

222

C₈

—

223

C₈

—

224

C₈

—

225

C₈

—

226

C₈

—

424

—

227

C₈

—

212

—

228

C₂²

—

229

C₂²

—

230

C₂²

#16810480

yes

characteristic

231

C₂²

#16810931

yes

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

106

—

236

C₄

—

212

—

237

C₄

#16810620

yes

characteristic

238

C₄

#16810633

yes

characteristic

239

C₄

#108×D₅₃

yes

—

240

C₄

#16810846

yes

—

241

C₂

—

242

C₂

#20274731

yes

characteristic

243

C₂

#20274750

yes

characteristic

244

C₂

#20274766

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#8661
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

subgroup

G/H

center

C₂×C₄

#13133192

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1564

C₂

socle

C₂²×C₅₃

#380

prime radical

C₂³×C₅₃

C₂²⋊Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

N	H ≅ G/N	type
144. C₅₃	#8661	semidirect

H ≅ G/N

type

144. C₅₃

#8661

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₈×Dic₁₀₆	2
#20274853	2
#20273145	2
#20272611	2
C₅₃×#1564	2
#8661	53

subgroup

index

C₂×C₈×Dic₁₀₆

C₅₃×#1564

Subgroups — #27277232

Conjugacy classes of subgroups

Complements

Maximal subgroups