Group #27283256 — subgroups

Subgroups — #27283256

Order 13568.

422 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20275658	C₂	1	2	yes	maximal
3	#20273213	C₂	1	2	yes	maximal
4	#20273004	C₂	1	1	yes	maximal, characteristic
5	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	2	yes	—
7	#16811028	C₄	1	1	yes	characteristic, complemented
8	#16811003	C₄	1	1	yes	characteristic, complemented
9	#16810974	C₄	1	2	yes	complemented
10	#16810864	—	2	1	no	—
11	#16810864	—	2	1	no	—
12	C₄×Dic₂₁₂	—	2	1	no	—
13	C₄×Dic₂₁₂	—	2	1	no	—
14	#16810676	C₂²	1	1	yes	characteristic
15	C₄×#11719917	C₂²	1	1	yes	characteristic
16	#16810383	—	2	4	no	—
17	#16810359	—	2	4	no	—
18	#16810345	—	2	1	no	—
19	#16810344	—	2	1	no	—
20	C₂×#106×C₅₃	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#340	—	2	1	no	—
24	C₂×Dic₂₁₂	—	2	1	no	—
25	C₂×Dic₂₁₂	—	2	1	no	—
26	C₂×#11719917	D₄	1	2	yes	—
27	#13133217	C₂×C₄	1	2	yes	—
28	#13133217	C₂×C₄	1	2	yes	—
29	#13133209	—	2	2	no	—
30	#13133196	—	2	2	no	—
31	#13133196	—	2	2	no	—
32	C₄×Dic₁₀₆	—	2	2	no	—
33	C₄×Dic₁₀₆	—	2	2	no	—
34	#13133176	—	2	2	no	—
35	#13133151	—	2	1	no	—
36	#13133151	—	2	1	no	—
37	C₄⋊#88322	—	2	1	no	—
38	#13133142	—	2	2	no	—
39	#13133142	—	2	2	no	—
40	C₂²⋊#88322	D₄	1	2	yes	—
41	#13133128	—	2	1	no	—
42	C₄×#88322	—	2	1	no	—
43	C₄×#88322	—	2	1	no	—
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	#106×C₅₃	—	2	4	no	—
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	#95×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₁₀₆	—	2	2	no	—
60	C₂×Dic₁₀₆	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	4	1	no	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	—	4	4	no	—
70	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	#41	1	1	yes	characteristic
73	C₄²×C₅₃	#41	1	1	yes	characteristic
74	C₂²⋊Dic₅₃	—	4	2	no	—
75	C₄⋊Dic₅₃	—	2	2	no	—
76	C₄⋊Dic₅₃	—	2	2	no	—
77	C₄⋊Dic₅₃	—	2	2	no	—
78	C₄⋊Dic₅₃	—	2	2	no	—
79	#11719919	—	4	2	no	—
80	#11719919	—	4	2	no	—
81	C₄×Dic₅₃	—	4	2	no	—
82	#11719917	—	2	4	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	2	2	no	—
85	Dic₂₁₂	—	4	2	no	—
86	Dic₂₁₂	—	4	2	no	—
87	C₂³×C₅₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	4	1	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	—	8	2	no	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#137	1	4	yes	—
101	C₂×Dic₅₃	—	4	2	no	—
102	C₂×Dic₅₃	—	4	2	no	—
103	Dic₁₀₆	—	4	2	no	—
104	Dic₁₀₆	—	4	2	no	—
105	Dic₁₀₆	—	8	2	no	—
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	—	2	1	no	—
109	C₈×C₅₃	#104	1	2	yes	—
110	#88322	—	4	2	no	—
111	#88322	—	4	2	no	—
112	#9720	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	#390	1	1	yes	characteristic
116	C₂²×C₅₃	#497	1	2	yes	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	8	4	no	—
121	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
122	C₄×C₅₃	#421	1	1	yes	characteristic
123	C₄×C₅₃	#441	1	2	yes	—
124	C₄×C₅₃	#470	1	2	yes	—
125	Dic₅₃	—	8	2	no	—
126	Dic₅₃	—	8	2	no	—
127	#3170	—	53	1	no	—
128	#1986	—	53	2	no	—
129	C₄×#106	D₅₃	1	1	yes	characteristic
130	#1503	—	53	2	no	—
131	#1451	—	53	1	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	#1971	1	1	yes	characteristic
134	C₂×C₅₃	#1976	1	1	yes	characteristic
135	C₂×C₅₃	#1980	1	1	yes	characteristic
136	#533	—	53	2	no	—
137	#500	—	53	1	no	—
138	#494	—	53	1	no	—
139	#487	—	53	2	no	—
140	#467	—	106	1	no	—
141	#467	—	106	1	no	—
142	C₄×Q₁₆	—	106	1	no	—
143	C₄×Q₁₆	—	106	1	no	—
144	#431	—	53	1	no	—
145	C₂×#106	—	2	1	no	—
146	C₄×M₁₆	—	53	1	no	—
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
149	#340	—	2	1	no	—
150	#339	—	106	4	no	—
151	#328	—	106	4	no	—
152	#322	—	106	1	no	—
153	#322	—	106	1	no	—
154	C₅₃	#9720	1	1	yes	characteristic, complemented, Sylow
155	C₂×Q₁₆	—	106	1	no	—
156	C₂×Q₁₆	—	106	1	no	—
157	C₂×M₁₆	—	53	2	no	—
158	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄⋊Q₈	—	53	2	no	—
161	C₄⋊Q₈	—	53	2	no	—
162	#125	—	106	2	no	—
163	C₂²⋊Q₈	—	106	2	no	—
164	C₂²⋊Q₈	—	106	2	no	—
165	C₄×Q₈	—	106	2	no	—
166	C₄×Q₈	—	106	2	no	—
167	#117	—	106	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
171	#107	—	106	1	no	—
172	#107	—	106	1	no	—
173	#106	—	2	4	no	—
174	C₄⋊C₈	—	106	1	no	—
175	#103	—	106	2	no	—
176	#103	—	106	2	no	—
177	C₂²⋊C₈	—	53	2	no	—
178	#97	—	106	1	no	—
179	C₄×C₈	—	106	1	no	—
180	C₄×C₈	—	106	1	no	—
181	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
182	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
183	C₄×C₈	D₂₁₂	1	1	yes	characteristic
184	C₄×C₈	D₂₁₂	1	1	yes	characteristic
185	#95	—	2	2	no	—
186	C₂×Q₈	—	106	2	no	—
187	C₂×Q₈	—	106	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
190	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	Q₁₆	—	212	2	no	—
193	Q₁₆	—	212	2	no	—
194	M₁₆	—	106	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	106	2	no	—
198	C₂×C₈	—	106	2	no	—
199	C₂×C₈	#11719921	1	1	yes	characteristic
200	C₂×C₈	#11719921	1	1	yes	characteristic
201	C₂×C₈	#11719921	1	1	yes	characteristic
202	C₂×C₈	#11719921	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	106	2	no	—
206	C₄⋊C₄	—	106	2	no	—
207	C₄⋊C₄	—	106	2	no	—
208	C₄⋊C₄	—	106	2	no	—
209	C₄⋊C₄	—	212	2	no	—
210	C₄⋊C₄	—	212	2	no	—
211	C₂²⋊C₄	—	212	2	no	—
212	C₄²	—	4	4	no	—
213	C₄²	—	212	2	no	—
214	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
215	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
216	C₄²	#11719949	1	1	yes	characteristic
217	C₄²	#11719949	1	1	yes	characteristic
218	C₂³	#13133213	1	1	yes	characteristic
219	Q₈	—	424	2	no	—
220	Q₈	—	212	2	no	—
221	Q₈	—	212	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	212	2	no	—
229	C₂×C₄	—	212	2	no	—
230	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
231	C₂×C₄	#13133179	1	1	yes	characteristic
232	C₂×C₄	#13133179	1	1	yes	characteristic
233	C₂×C₄	#13133192	1	1	yes	characteristic
234	C₂×C₄	#13133192	1	1	yes	characteristic
235	C₂×C₄	#13133206	1	1	yes	characteristic
236	C₂×C₄	#13133246	1	4	yes	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	212	2	no	—
241	C₈	—	212	2	no	—
242	C₈	#13133145	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	#16810524	1	1	yes	characteristic
246	C₂²	#16811014	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	8	4	no	—
251	C₄	—	424	2	no	—
252	C₄	—	424	2	no	—
253	C₄	C₂×#13133145	1	1	yes	characteristic
254	C₄	#16810641	1	1	yes	characteristic
255	C₄	#16810729	1	2	yes	—
256	C₄	#16810882	1	2	yes	—
257	C₂	—	2	2	no	—
258	C₂	#20275584	1	1	yes	characteristic
259	C₂	#20275611	1	1	yes	characteristic
260	C₂	#20275633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20285301

C₂

yes

maximal, characteristic

#20275658

C₂

yes

maximal

#20273213

C₂

yes

maximal

#20273004

C₂

yes

maximal, characteristic

C₄×#106×C₅₃

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

—

#16811028

C₄

yes

characteristic, complemented

#16811003

C₄

yes

characteristic, complemented

#16810974

C₄

yes

complemented

#16810864

—

#16810864

—

C₄×Dic₂₁₂

—

C₄×Dic₂₁₂

—

#16810676

C₂²

yes

characteristic

C₄×#11719917

C₂²

yes

—

—

—

—

C₂×#106×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₅₃

C₂²

yes

—

C₅₃×#340

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×#11719917

D₄

yes

—

#13133217

C₂×C₄

yes

—

#13133217

C₂×C₄

yes

—

#13133209

—

#13133196

—

#13133196

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

—

—

—

—

—

—

D₄

yes

—

#13133128

—

C₄×#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#106×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

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

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

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

C₂³×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

100

C₂×C₄×C₅₃

#137

yes

—

101

C₂×Dic₅₃

—

102

C₂×Dic₅₃

—

103

Dic₁₀₆

—

104

Dic₁₀₆

—

105

Dic₁₀₆

—

106

C₈×C₅₃

—

107

C₈×C₅₃

—

108

C₈×C₅₃

—

109

C₈×C₅₃

#104

yes

—

110

#88322

—

111

#88322

—

112

#9720

—

maximal, Sylow

113

C₂²×C₅₃

—

114

C₂²×C₅₃

—

115

C₂²×C₅₃

#390

yes

characteristic

116

C₂²×C₅₃

#497

yes

—

117

C₄×C₅₃

—

118

C₄×C₅₃

—

119

C₄×C₅₃

—

120

C₄×C₅₃

—

121

C₄×C₅₃

C₂×#104

yes

characteristic

122

C₄×C₅₃

#421

yes

characteristic

123

C₄×C₅₃

#441

yes

—

124

C₄×C₅₃

#470

yes

—

125

Dic₅₃

—

126

Dic₅₃

—

127

#3170

—

128

#1986

—

129

C₄×#106

D₅₃

yes

characteristic

130

#1503

—

131

#1451

—

132

C₂×C₅₃

—

133

C₂×C₅₃

#1971

yes

characteristic

134

C₂×C₅₃

#1976

yes

characteristic

135

C₂×C₅₃

#1980

yes

characteristic

136

#533

—

137

#500

—

138

#494

—

139

#487

—

140

#467

—

106

—

141

#467

—

106

—

142

C₄×Q₁₆

—

106

—

143

C₄×Q₁₆

—

106

—

144

#431

—

145

C₂×#106

—

146

C₄×M₁₆

—

147

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

148

C₄×C₄⋊C₄

C₂×D₅₃

yes

—

149

#340

—

150

#339

—

106

—

151

#328

—

106

—

152

#322

—

106

—

153

#322

—

106

—

154

C₅₃

#9720

yes

characteristic, complemented, Sylow

155

C₂×Q₁₆

—

106

—

156

C₂×Q₁₆

—

106

—

157

C₂×M₁₆

—

158

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

159

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

160

C₄⋊Q₈

—

161

C₄⋊Q₈

—

162

#125

—

106

—

163

C₂²⋊Q₈

—

106

—

164

C₂²⋊Q₈

—

106

—

165

C₄×Q₈

—

106

—

166

C₄×Q₈

—

106

—

167

#117

—

106

—

168

C₂×C₄⋊C₄

—

169

C₂×C₄²

—

170

C₂×C₄²

C₂²×D₅₃

yes

characteristic

171

#107

—

106

—

172

#107

—

106

—

173

#106

—

174

C₄⋊C₈

—

106

—

175

#103

—

106

—

176

#103

—

106

—

177

C₂²⋊C₈

—

178

#97

—

106

—

179

C₄×C₈

—

106

—

180

C₄×C₈

—

106

—

181

C₄×C₈

C₄×D₅₃

yes

characteristic

182

C₄×C₈

C₄×D₅₃

yes

characteristic

183

C₄×C₈

D₂₁₂

yes

characteristic

184

C₄×C₈

D₂₁₂

yes

characteristic

185

#95

—

186

C₂×Q₈

—

106

—

187

C₂×Q₈

—

106

—

188

C₂²×C₄

—

189

C₂²×C₄

D₄×D₅₃

yes

characteristic

190

C₂²×C₄

D₄×D₅₃

yes

characteristic

191

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

192

Q₁₆

—

212

—

193

Q₁₆

—

212

—

194

M₁₆

—

106

—

195

C₂×C₈

—

196

C₂×C₈

—

197

C₂×C₈

—

106

—

198

C₂×C₈

—

106

—

199

C₂×C₈

#11719921

yes

characteristic

200

C₂×C₈

#11719921

yes

characteristic

201

C₂×C₈

#11719921

yes

characteristic

202

C₂×C₈

#11719921

yes

characteristic

203

C₄⋊C₄

—

204

C₄⋊C₄

—

205

C₄⋊C₄

—

106

—

206

C₄⋊C₄

—

106

—

207

C₄⋊C₄

—

106

—

208

C₄⋊C₄

—

106

—

209

C₄⋊C₄

—

212

—

210

C₄⋊C₄

—

212

—

211

C₂²⋊C₄

—

212

—

212

C₄²

—

213

C₄²

—

212

—

214

C₄²

C₂×C₄×D₅₃

yes

characteristic

215

C₄²

C₂×D₂₁₂

yes

characteristic

216

C₄²

#11719949

yes

characteristic

217

C₄²

#11719949

yes

characteristic

218

C₂³

#13133213

yes

characteristic

219

Q₈

—

424

—

220

Q₈

—

212

—

221

Q₈

—

212

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

—

225

C₂×C₄

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

—

212

—

229

C₂×C₄

—

212

—

230

C₂×C₄

C₂×#11719921

yes

characteristic

231

C₂×C₄

#13133179

yes

characteristic

232

C₂×C₄

#13133179

yes

characteristic

233

C₂×C₄

#13133192

yes

characteristic

234

C₂×C₄

#13133192

yes

characteristic

235

C₂×C₄

#13133206

yes

characteristic

236

C₂×C₄

#13133246

yes

—

237

C₈

—

238

C₈

—

239

C₈

—

240

C₈

—

212

—

241

C₈

—

212

—

242

C₈

#13133145

yes

—

243

C₂²

—

244

C₂²

—

245

C₂²

#16810524

yes

characteristic

246

C₂²

#16811014

yes

—

247

C₄

—

248

C₄

—

249

C₄

—

250

C₄

—

251

C₄

—

424

—

252

C₄

—

424

—

253

C₄

C₂×#13133145

yes

characteristic

254

C₄

#16810641

yes

characteristic

255

C₄

#16810729

yes

—

256

C₄

#16810882

yes

—

257

C₂

—

258

C₂

#20275584

yes

characteristic

259

C₂

#20275611

yes

characteristic

260

C₂

#20275633

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#106×C₅₃	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#106	D₅₃
hypocenter	C₅₃	#9720
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#106

subgroup

G/H

center

C₂×C₄

#13133192

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₄×#106×C₅₃

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#106

N	H ≅ G/N	type
7. #16811028	C₄	semidirect
8. #16811003	C₄	semidirect
9. #16810974	C₄	semidirect
154. C₅₃	#9720	semidirect

H ≅ G/N

type

7. #16811028

C₄

semidirect

8. #16811003

C₄

semidirect

9. #16810974

C₄

semidirect

154. C₅₃

#9720

semidirect

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

subgroup	index
#20285301	2
#20275658	2
#20273213	2
#20273004	2
C₄×#106×C₅₃	2
#9720	53

subgroup

index

C₄×#106×C₅₃

Subgroups — #27283256

Conjugacy classes of subgroups

Complements

Maximal subgroups