Group #27286700 — subgroups

Subgroups — #27286700

Order 13568.

574 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 139 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×#16811003	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810908	C₂	1	2	yes	maximal
3	C₂×#16810370	C₂	1	2	yes	maximal
4	#20272550	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	—	2	2	no	—
7	C₂×#13133217	C₂²	1	2	yes	—
8	#16811003	—	2	4	no	—
9	#16810908	—	2	4	no	—
10	#16810908	—	2	4	no	—
11	C₂×#13133148	—	2	2	no	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×#13133141	—	2	2	no	—
14	#16810370	—	2	4	no	—
15	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	2	no	—
19	C₂²×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₂₁₂	—	2	8	no	—
21	C₂×Dic₂₁₂	—	4	4	no	—
22	C₂×Dic₂₁₂	—	4	4	no	—
23	C₂²×#88322	—	2	2	no	—
24	#13133217	—	2	2	no	—
25	#13133217	—	2	2	no	—
26	#13133217	—	2	2	no	—
27	#13133217	D₄	1	4	yes	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	#13133148	—	2	8	no	—
31	C₄⋊#88322	—	2	2	no	—
32	C₄⋊#88322	D₄	1	4	yes	—
33	#13133141	—	2	8	no	—
34	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	—	2	1	no	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×Q₈×C₅₃	—	2	2	no	—
38	C₄×Q₈×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	—	2	1	no	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	—	2	1	no	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	Q₁₆	1	2	yes	complemented
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂²×Dic₅₃	—	4	2	no	—
55	C₂×Dic₁₀₆	—	2	4	no	—
56	C₂×Dic₁₀₆	—	4	4	no	—
57	C₂×Dic₁₀₆	—	4	4	no	—
58	C₂×Dic₁₀₆	—	4	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	4	1	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₄²×C₅₃	—	4	1	no	—
67	C₄²×C₅₃	—	4	1	no	—
68	C₄²×C₅₃	C₂×D₄	1	2	yes	—
69	C₄²×C₅₃	C₂×D₄	1	2	yes	—
70	C₄⋊Dic₅₃	—	2	4	no	—
71	C₄⋊Dic₅₃	—	2	4	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₂×#88322	—	2	4	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	—	4	2	no	—
77	Dic₂₁₂	—	4	16	no	—
78	C₂³×C₅₃	#121	1	1	yes	characteristic
79	Q₈×C₅₃	—	2	4	no	—
80	Q₈×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	4	1	no	—
84	C₂×C₄×C₅₃	—	4	1	no	—
85	C₂×C₄×C₅₃	—	4	1	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#120	1	2	yes	—
89	C₂×C₄×C₅₃	#121	1	2	yes	—
90	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
94	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×C₄×C₅₃	#137	1	2	yes	—
96	C₂×Dic₅₃	—	4	4	no	—
97	C₂×Dic₅₃	—	8	4	no	—
98	Dic₁₀₆	—	4	8	no	—
99	Dic₁₀₆	—	8	2	no	—
100	Dic₁₀₆	—	8	2	no	—
101	C₈×C₅₃	—	2	4	no	—
102	C₈×C₅₃	—	8	1	no	—
103	#88322	—	4	4	no	—
104	#10276	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	#392	1	1	yes	characteristic
106	C₂²×C₅₃	#460	1	2	yes	—
107	C₂²×C₅₃	#467	1	1	yes	characteristic
108	C₂²×C₅₃	#470	1	1	yes	characteristic
109	C₂²×C₅₃	#473	1	2	yes	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	4	2	no	—
112	C₄×C₅₃	—	8	1	no	—
113	C₄×C₅₃	#450	1	2	yes	—
114	C₄×C₅₃	#451	1	2	yes	—
115	C₄×C₅₃	#452	1	2	yes	—
116	C₄×C₅₃	#492	1	1	yes	characteristic
117	C₄×C₅₃	#496	1	1	yes	characteristic
118	Dic₅₃	—	8	4	no	—
119	C₂×#494	—	53	1	no	—
120	C₂×#475	—	53	2	no	—
121	#1984	D₅₃	1	1	yes	characteristic
122	C₂×#332	—	53	2	no	—
123	#1277	—	53	1	no	—
124	C₂×C₅₃	#1670	1	2	yes	—
125	C₂×C₅₃	#1690	1	2	yes	—
126	C₂×C₅₃	#2001	1	1	yes	characteristic
127	C₂×C₅₃	#2014	1	1	yes	characteristic
128	C₂×C₅₃	#2020	1	1	yes	characteristic
129	C₂²×Q₁₆	—	106	2	no	—
130	C₂×C₄⋊Q₈	—	53	2	no	—
131	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
132	#494	—	106	4	no	—
133	#475	—	106	4	no	—
134	#475	—	106	4	no	—
135	C₂×#106	—	106	2	no	—
136	C₂×C₄⋊C₈	—	53	2	no	—
137	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
138	C₂×#103	—	106	2	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	#336	—	2	2	no	—
141	#332	—	106	4	no	—
142	C₅₃	#10276	1	1	yes	characteristic, complemented, Sylow
143	C₂²×Q₈	—	106	2	no	—
144	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
145	C₂×Q₁₆	—	106	8	no	—
146	C₂×Q₁₆	—	212	4	no	—
147	C₂×Q₁₆	—	212	4	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	106	2	no	—
150	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
151	C₄⋊Q₈	—	53	4	no	—
152	C₄⋊Q₈	—	106	2	no	—
153	C₄⋊Q₈	—	106	2	no	—
154	C₄⋊Q₈	—	106	2	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₄×Q₈	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	106	2	no	—
159	C₂×C₄⋊C₄	—	106	2	no	—
160	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
163	#106	—	106	8	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	53	4	no	—
166	C₄⋊C₈	—	106	2	no	—
167	#103	—	106	8	no	—
168	C₄×C₈	—	2	2	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	106	4	no	—
173	C₂×Q₈	—	212	4	no	—
174	C₂×Q₈	—	212	4	no	—
175	C₂×Q₈	—	212	2	no	—
176	C₂×Q₈	Dic₂₁₂	1	2	yes	complemented
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	212	2	no	—
180	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
181	C₂²×C₄	#11719945	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
183	Q₁₆	—	212	16	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	106	4	no	—
189	C₂×C₈	—	212	2	no	—
190	C₂×C₈	—	212	2	no	—
191	C₂×C₈	#11719924	1	4	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	106	4	no	—
195	C₄⋊C₄	—	106	4	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₄⋊C₄	—	212	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	D₄×D₅₃	1	2	yes	—
201	C₄²	D₄×D₅₃	1	2	yes	—
202	C₂³	#13133191	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	424	2	no	—
206	Q₈	—	424	2	no	—
207	Q₈	—	212	8	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	424	4	no	—
215	C₂×C₄	—	212	4	no	—
216	C₂×C₄	#13133185	1	1	yes	characteristic
217	C₂×C₄	#13133187	1	2	yes	—
218	C₂×C₄	#13133190	1	2	yes	—
219	C₂×C₄	#13133213	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719924	1	2	yes	—
221	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
222	C₂×C₄	#13133239	1	1	yes	characteristic
223	C₂×C₄	#13133242	1	2	yes	—
224	C₂×C₄	#13133250	1	2	yes	—
225	C₈	—	2	4	no	—
226	C₈	—	8	1	no	—
227	C₈	—	212	4	no	—
228	C₂²	#16810532	1	1	yes	characteristic
229	C₂²	#16810823	1	2	yes	—
230	C₂²	#16810864	1	1	yes	characteristic
231	C₂²	#16810881	1	1	yes	characteristic
232	C₂²	#16810898	1	2	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	1	no	—
236	C₄	—	424	4	no	—
237	C₄	#16810773	1	2	yes	—
238	C₄	#16810781	1	2	yes	—
239	C₄	#16810787	1	2	yes	—
240	C₄	#16810997	1	1	yes	characteristic
241	C₄	#16811013	1	1	yes	characteristic
242	C₂	#20274085	1	2	yes	—
243	C₂	#20274209	1	2	yes	—
244	C₂	#20275725	1	1	yes	characteristic
245	C₂	#20275802	1	1	yes	characteristic
246	C₂	#20275836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811003

C₂

yes

maximal, characteristic

C₂×#16810908

C₂

yes

maximal

C₂×#16810370

C₂

yes

maximal

#20272550

C₂

yes

maximal, characteristic

C₅₃×#1984

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

—

C₂×#13133217

C₂²

yes

—

—

—

—

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×#13133141

—

#16810370

—

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂²×#88322

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133148

—

C₄⋊#88322

—

C₄⋊#88322

D₄

yes

—

#13133141

—

C₂²×Q₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

Q₁₆

yes

complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

Dic₂₁₂

—

C₂³×C₅₃

#121

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

100

Dic₁₀₆

—

101

C₈×C₅₃

—

102

C₈×C₅₃

—

103

#88322

—

104

#10276

—

maximal, Sylow

105

C₂²×C₅₃

#392

yes

characteristic

106

C₂²×C₅₃

#460

yes

—

107

C₂²×C₅₃

#467

yes

characteristic

108

C₂²×C₅₃

#470

yes

characteristic

109

C₂²×C₅₃

#473

yes

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

—

113

C₄×C₅₃

#450

yes

—

114

C₄×C₅₃

#451

yes

—

115

C₄×C₅₃

#452

yes

—

116

C₄×C₅₃

#492

yes

characteristic

117

C₄×C₅₃

#496

yes

characteristic

118

Dic₅₃

—

119

C₂×#494

—

120

C₂×#475

—

121

#1984

D₅₃

yes

characteristic

122

C₂×#332

—

123

#1277

—

124

C₂×C₅₃

#1670

yes

—

125

C₂×C₅₃

#1690

yes

—

126

C₂×C₅₃

#2001

yes

characteristic

127

C₂×C₅₃

#2014

yes

characteristic

128

C₂×C₅₃

#2020

yes

characteristic

129

C₂²×Q₁₆

—

106

—

130

C₂×C₄⋊Q₈

—

131

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

132

#494

—

106

—

133

#475

—

106

—

134

#475

—

106

—

135

C₂×#106

—

106

—

136

C₂×C₄⋊C₈

—

137

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

138

C₂×#103

—

106

—

139

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

140

#336

—

141

#332

—

106

—

142

C₅₃

#10276

yes

characteristic, complemented, Sylow

143

C₂²×Q₈

—

106

—

144

C₂²×Q₈

D₂₁₂

yes

characteristic

145

C₂×Q₁₆

—

106

—

146

C₂×Q₁₆

—

212

—

147

C₂×Q₁₆

—

212

—

148

C₂²×C₈

—

149

C₂²×C₈

—

106

—

150

C₂²×C₈

C₂²⋊D₅₃

yes

—

151

C₄⋊Q₈

—

152

C₄⋊Q₈

—

106

—

153

C₄⋊Q₈

—

106

—

154

C₄⋊Q₈

—

106

—

155

C₄×Q₈

—

156

C₄×Q₈

—

157

C₂×C₄⋊C₄

—

158

C₂×C₄⋊C₄

—

106

—

159

C₂×C₄⋊C₄

—

106

—

160

C₂×C₄⋊C₄

D₂₁₂

yes

characteristic

161

C₂×C₄²

—

162

C₂×C₄²

C₂²×D₅₃

yes

characteristic

163

#106

—

106

—

164

C₄⋊C₈

—

165

C₄⋊C₈

—

166

C₄⋊C₈

—

106

—

167

#103

—

106

—

168

C₄×C₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

106

—

173

C₂×Q₈

—

212

—

174

C₂×Q₈

—

212

—

175

C₂×Q₈

—

212

—

176

C₂×Q₈

Dic₂₁₂

yes

complemented

177

C₂²×C₄

—

178

C₂²×C₄

—

179

C₂²×C₄

—

212

—

180

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

181

C₂²×C₄

#11719945

yes

characteristic

182

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

183

Q₁₆

—

212

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

—

187

C₂×C₈

—

188

C₂×C₈

—

106

—

189

C₂×C₈

—

212

—

190

C₂×C₈

—

212

—

191

C₂×C₈

#11719924

yes

—

192

C₄⋊C₄

—

193

C₄⋊C₄

—

194

C₄⋊C₄

—

106

—

195

C₄⋊C₄

—

106

—

196

C₄⋊C₄

—

212

—

197

C₄⋊C₄

—

212

—

198

C₄²

—

199

C₄²

—

200

C₄²

D₄×D₅₃

yes

—

201

C₄²

D₄×D₅₃

yes

—

202

C₂³

#13133191

yes

characteristic

203

Q₈

—

204

Q₈

—

205

Q₈

—

424

—

206

Q₈

—

424

—

207

Q₈

—

212

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

424

—

215

C₂×C₄

—

212

—

216

C₂×C₄

#13133185

yes

characteristic

217

C₂×C₄

#13133187

yes

—

218

C₂×C₄

#13133190

yes

—

219

C₂×C₄

#13133213

yes

characteristic

220

C₂×C₄

C₂×#11719924

yes

—

221

C₂×C₄

C₂×Dic₂₁₂

yes

characteristic

222

C₂×C₄

#13133239

yes

characteristic

223

C₂×C₄

#13133242

yes

—

224

C₂×C₄

#13133250

yes

—

225

C₈

—

226

C₈

—

227

C₈

—

212

—

228

C₂²

#16810532

yes

characteristic

229

C₂²

#16810823

yes

—

230

C₂²

#16810864

yes

characteristic

231

C₂²

#16810881

yes

characteristic

232

C₂²

#16810898

yes

—

233

C₄

—

234

C₄

—

235

C₄

—

236

C₄

—

424

—

237

C₄

#16810773

yes

—

238

C₄

#16810781

yes

—

239

C₄

#16810787

yes

—

240

C₄

#16810997

yes

characteristic

241

C₄

#16811013

yes

characteristic

242

C₂

#20274085

yes

—

243

C₂

#20274209

yes

—

244

C₂

#20275725

yes

characteristic

245

C₂

#20275802

yes

characteristic

246

C₂

#20275836

yes

characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1984	C₂
socle, prime radical	C₂³×C₅₃	#121
hypercenter	#1984	D₅₃
hypocenter	C₅₃	#10276
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

subgroup

G/H

center

C₂³

#13133191

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1984

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

N	H ≅ G/N	type
48. C₂×Q₈×C₅₃	Q₁₆	semidirect
142. C₅₃	#10276	semidirect
176. C₂×Q₈	Dic₂₁₂	semidirect

H ≅ G/N

type

48. C₂×Q₈×C₅₃

Q₁₆

semidirect

142. C₅₃

#10276

semidirect

176. C₂×Q₈

Dic₂₁₂

semidirect

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

subgroup	index
C₂×#16811003	2
C₂×#16810908	2
C₂×#16810370	2
#20272550	2
C₅₃×#1984	2
#10276	53

subgroup

index

C₅₃×#1984

Subgroups — #27286700

Conjugacy classes of subgroups

Complements

Maximal subgroups