Group #27282992 — subgroups

Subgroups — #27282992

Order 13568.

482 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 147 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×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20275650	C₂	1	2	yes	maximal
3	C₂×#16810353	C₂	1	2	yes	maximal
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
8	C₄×Dic₂₁₂	—	2	2	no	—
9	C₄×Dic₂₁₂	—	2	2	no	—
10	C₂×#13133151	C₄	1	1	yes	characteristic
11	C₂×#13133142	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	#16810377	—	2	4	no	—
15	#16810353	—	2	4	no	—
16	#16810345	—	2	2	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#334	—	2	2	no	—
20	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	4	no	—
23	C₂²×#88322	—	2	1	no	—
24	C₂²×#88322	D₄	1	2	yes	—
25	C₄×Dic₁₀₆	—	2	2	no	—
26	C₄×Dic₁₀₆	—	2	2	no	—
27	C₄×Dic₁₀₆	—	2	2	no	—
28	C₄×Dic₁₀₆	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×#11719919	—	2	2	no	—
31	C₂×C₄×Dic₅₃	—	2	2	no	—
32	#13133151	—	2	2	no	—
33	#13133142	—	2	4	no	—
34	C₄×#88322	—	2	1	no	—
35	C₄×#88322	—	2	1	no	—
36	C₄×#88322	—	2	1	no	—
37	C₄×#88322	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	—	2	2	no	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	2	2	no	—
50	C₂²×Dic₅₃	—	2	2	no	—
51	C₂×Dic₁₀₆	—	2	2	no	—
52	C₂×Dic₁₀₆	—	2	2	no	—
53	C₂×Dic₁₀₆	—	2	2	no	—
54	C₂×Dic₁₀₆	—	2	2	no	—
55	C₂×Dic₁₀₆	—	2	2	no	—
56	C₂×Dic₁₀₆	M₁₆	1	4	yes	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	4	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	#41	1	1	yes	characteristic
67	C₄⋊Dic₅₃	—	2	2	no	—
68	C₄⋊Dic₅₃	—	2	2	no	—
69	#11719919	—	4	2	no	—
70	#11719919	—	4	2	no	—
71	C₄×Dic₅₃	—	4	2	no	—
72	C₄×Dic₅₃	—	4	2	no	—
73	C₂×#88322	—	2	4	no	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	—	2	2	no	—
76	C₂×#88322	—	2	4	no	—
77	Dic₂₁₂	—	2	8	no	—
78	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₅₃	#131	1	2	yes	—
88	C₂×C₄×C₅₃	#137	1	4	yes	—
89	C₂×Dic₅₃	—	2	4	no	—
90	C₂×Dic₅₃	—	4	2	no	—
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	—	4	4	no	—
94	Dic₁₀₆	—	2	4	no	—
95	Dic₁₀₆	—	2	4	no	—
96	Dic₁₀₆	—	4	2	no	—
97	Dic₁₀₆	—	4	2	no	—
98	C₈×C₅₃	—	2	2	no	—
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	—	8	2	no	—
101	#88322	—	2	4	no	—
102	#88322	—	4	2	no	—
103	#9664	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	#390	1	1	yes	characteristic
105	C₂²×C₅₃	#435	1	2	yes	—
106	C₂²×C₅₃	#441	1	2	yes	—
107	C₂²×C₅₃	#443	1	2	yes	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	#407	1	1	yes	characteristic
111	C₄×C₅₃	#408	1	1	yes	characteristic
112	C₄×C₅₃	#435	1	2	yes	—
113	C₄×C₅₃	#470	1	2	yes	—
114	C₄×C₅₃	#497	1	2	yes	—
115	Dic₅₃	—	4	4	no	—
116	Dic₅₃	—	8	2	no	—
117	C₂×C₄×Q₁₆	—	53	1	no	—
118	#1984	—	53	2	no	—
119	C₂×#334	D₅₃	1	1	yes	characteristic
120	C₂×Q₈⋊C₈	—	53	2	no	—
121	C₂×#322	—	53	1	no	—
122	C₂×C₅₃	#1587	1	4	yes	—
123	C₂×C₅₃	#1955	1	1	yes	characteristic
124	C₂×C₅₃	#1971	1	1	yes	characteristic
125	C₂×C₅₃	#1974	1	1	yes	characteristic
126	C₂²×Q₁₆	—	53	1	no	—
127	C₂×C₄×Q₈	—	53	2	no	—
128	C₄×Q₁₆	—	106	2	no	—
129	C₄×Q₁₆	—	106	2	no	—
130	C₂×#107	—	53	1	no	—
131	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
132	C₂×#103	—	53	2	no	—
133	C₂×C₄×C₈	—	53	1	no	—
134	C₂×C₄×C₈	—	53	1	no	—
135	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
136	#336	—	106	4	no	—
137	#334	—	2	2	no	—
138	Q₈⋊C₈	—	106	4	no	—
139	#322	—	106	2	no	—
140	C₅₃	#9664	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	53	2	no	—
142	C₂×Q₁₆	—	106	4	no	—
143	C₂×Q₁₆	—	106	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	53	2	no	—
146	C₂²×C₈	—	106	1	no	—
147	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
148	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
149	C₄×Q₈	—	106	2	no	—
150	C₄×Q₈	—	106	2	no	—
151	C₄×Q₈	—	106	2	no	—
152	C₄×Q₈	—	106	2	no	—
153	C₂×C₄⋊C₄	—	53	2	no	—
154	C₂×C₄⋊C₄	—	106	2	no	—
155	C₂×C₄²	—	106	2	no	—
156	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
157	#107	—	106	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	#103	—	106	4	no	—
161	C₄×C₈	—	53	2	no	—
162	C₄×C₈	—	106	1	no	—
163	C₄×C₈	—	106	1	no	—
164	C₄×C₈	—	106	1	no	—
165	C₄×C₈	D₂₁₂	1	2	yes	—
166	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
167	C₂×Q₈	—	53	4	no	—
168	C₂×Q₈	—	106	2	no	—
169	C₂×Q₈	—	106	2	no	—
170	C₂×Q₈	—	106	2	no	—
171	C₂×Q₈	—	106	2	no	—
172	C₂×Q₈	—	106	2	no	—
173	C₂²×C₄	—	106	2	no	—
174	C₂²×C₄	—	106	2	no	—
175	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
176	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
177	C₂²×C₄	#11719949	1	1	yes	characteristic
178	Q₁₆	—	106	8	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	106	4	no	—
184	C₂×C₈	—	106	2	no	—
185	C₂×C₈	—	106	2	no	—
186	C₂×C₈	—	106	4	no	—
187	C₂×C₈	#11719921	1	2	yes	—
188	C₂×C₈	#11719921	1	2	yes	—
189	C₄⋊C₄	—	106	2	no	—
190	C₄⋊C₄	—	106	2	no	—
191	C₄⋊C₄	—	212	2	no	—
192	C₄⋊C₄	—	212	2	no	—
193	C₄²	—	212	2	no	—
194	C₄²	—	212	2	no	—
195	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
196	C₄²	D₄×D₅₃	1	1	yes	characteristic
197	C₄²	#11719949	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂³	#13133179	1	1	yes	characteristic
200	Q₈	—	106	4	no	—
201	Q₈	—	106	4	no	—
202	Q₈	—	212	2	no	—
203	Q₈	—	212	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	106	4	no	—
206	C₂×C₄	—	212	2	no	—
207	C₂×C₄	—	212	4	no	—
208	C₂×C₄	—	212	2	no	—
209	C₂×C₄	—	212	2	no	—
210	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
211	C₂×C₄	#13133179	1	1	yes	characteristic
212	C₂×C₄	#13133192	1	1	yes	characteristic
213	C₂×C₄	#13133192	1	1	yes	characteristic
214	C₂×C₄	#13133206	1	1	yes	characteristic
215	C₂×C₄	#13133213	1	1	yes	characteristic
216	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
217	C₂×C₄	#13133223	1	2	yes	—
218	C₂×C₄	#13133246	1	4	yes	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	8	2	no	—
222	C₈	—	106	4	no	—
223	C₈	—	212	2	no	—
224	C₂²	#16810524	1	1	yes	characteristic
225	C₂²	#16810693	1	2	yes	—
226	C₂²	#16810729	1	2	yes	—
227	C₂²	#16810739	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	424	2	no	—
231	C₄	—	212	4	no	—
232	C₄	#16810586	1	1	yes	characteristic
233	C₄	#16810589	1	1	yes	characteristic
234	C₄	#16810693	1	2	yes	—
235	C₄	#16810882	1	2	yes	—
236	C₄	#16811014	1	2	yes	—
237	C₂	#20273585	1	4	yes	—
238	C₂	#20275512	1	1	yes	characteristic
239	C₂	#20275584	1	1	yes	characteristic
240	C₂	#20275601	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₁₂

C₂

yes

maximal, characteristic

#20275650

C₂

yes

maximal

C₂×#16810353

C₂

yes

maximal

C₂×#16810345

C₂

yes

maximal, characteristic

C₂×C₅₃×#334

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

C₄

yes

characteristic

C₂×C₄×Dic₁₀₆

C₂²

yes

—

C₄×Dic₂₁₂

—

C₄×Dic₂₁₂

—

C₂×#13133151

C₄

yes

characteristic

C₂×#13133142

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

—

—

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

C₂²×Dic₁₀₆

C₂×C₄

yes

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂²×#88322

—

C₂²×#88322

D₄

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₂×C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

—

—

—

—

—

D₄

yes

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

M₁₆

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

Dic₂₁₂

—

C₂³×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

100

C₈×C₅₃

—

101

#88322

—

102

#88322

—

103

#9664

—

maximal, Sylow

104

C₂²×C₅₃

#390

yes

characteristic

105

C₂²×C₅₃

#435

yes

—

106

C₂²×C₅₃

#441

yes

—

107

C₂²×C₅₃

#443

yes

—

108

C₄×C₅₃

—

109

C₄×C₅₃

—

110

C₄×C₅₃

#407

yes

characteristic

111

C₄×C₅₃

#408

yes

characteristic

112

C₄×C₅₃

#435

yes

—

113

C₄×C₅₃

#470

yes

—

114

C₄×C₅₃

#497

yes

—

115

Dic₅₃

—

116

Dic₅₃

—

117

C₂×C₄×Q₁₆

—

118

#1984

—

119

C₂×#334

D₅₃

yes

characteristic

120

C₂×Q₈⋊C₈

—

121

C₂×#322

—

122

C₂×C₅₃

#1587

yes

—

123

C₂×C₅₃

#1955

yes

characteristic

124

C₂×C₅₃

#1971

yes

characteristic

125

C₂×C₅₃

#1974

yes

characteristic

126

C₂²×Q₁₆

—

127

C₂×C₄×Q₈

—

128

C₄×Q₁₆

—

106

—

129

C₄×Q₁₆

—

106

—

130

C₂×#107

—

131

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

132

C₂×#103

—

133

C₂×C₄×C₈

—

134

C₂×C₄×C₈

—

135

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

136

#336

—

106

—

137

#334

—

138

Q₈⋊C₈

—

106

—

139

#322

—

106

—

140

C₅₃

#9664

yes

characteristic, complemented, Sylow

141

C₂²×Q₈

—

142

C₂×Q₁₆

—

106

—

143

C₂×Q₁₆

—

106

—

144

C₂²×C₈

—

145

C₂²×C₈

—

146

C₂²×C₈

—

106

—

147

C₂²×C₈

C₄×D₅₃

yes

characteristic

148

C₂²×C₈

C₄×D₅₃

yes

characteristic

149

C₄×Q₈

—

106

—

150

C₄×Q₈

—

106

—

151

C₄×Q₈

—

106

—

152

C₄×Q₈

—

106

—

153

C₂×C₄⋊C₄

—

154

C₂×C₄⋊C₄

—

106

—

155

C₂×C₄²

—

106

—

156

C₂×C₄²

C₂²×D₅₃

yes

characteristic

157

#107

—

106

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

#103

—

106

—

161

C₄×C₈

—

162

C₄×C₈

—

106

—

163

C₄×C₈

—

106

—

164

C₄×C₈

—

106

—

165

C₄×C₈

D₂₁₂

yes

—

166

C₄×C₈

C₂²⋊D₅₃

yes

—

167

C₂×Q₈

—

168

C₂×Q₈

—

106

—

169

C₂×Q₈

—

106

—

170

C₂×Q₈

—

106

—

171

C₂×Q₈

—

106

—

172

C₂×Q₈

—

106

—

173

C₂²×C₄

—

106

—

174

C₂²×C₄

—

106

—

175

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

176

C₂²×C₄

D₄×D₅₃

yes

characteristic

177

C₂²×C₄

#11719949

yes

characteristic

178

Q₁₆

—

106

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

106

—

184

C₂×C₈

—

106

—

185

C₂×C₈

—

106

—

186

C₂×C₈

—

106

—

187

C₂×C₈

#11719921

yes

—

188

C₂×C₈

#11719921

yes

—

189

C₄⋊C₄

—

106

—

190

C₄⋊C₄

—

106

—

191

C₄⋊C₄

—

212

—

192

C₄⋊C₄

—

212

—

193

C₄²

—

212

—

194

C₄²

—

212

—

195

C₄²

C₂×D₂₁₂

yes

characteristic

196

C₄²

D₄×D₅₃

yes

characteristic

197

C₄²

#11719949

yes

characteristic

198

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

199

C₂³

#13133179

yes

characteristic

200

Q₈

—

106

—

201

Q₈

—

106

—

202

Q₈

—

212

—

203

Q₈

—

212

—

204

C₂×C₄

—

205

C₂×C₄

—

106

—

206

C₂×C₄

—

212

—

207

C₂×C₄

—

212

—

208

C₂×C₄

—

212

—

209

C₂×C₄

—

212

—

210

C₂×C₄

C₂×#11719921

yes

characteristic

211

C₂×C₄

#13133179

yes

characteristic

212

C₂×C₄

#13133192

yes

characteristic

213

C₂×C₄

#13133192

yes

characteristic

214

C₂×C₄

#13133206

yes

characteristic

215

C₂×C₄

#13133213

yes

characteristic

216

C₂×C₄

M₁₆×D₅₃

yes

—

217

C₂×C₄

#13133223

yes

—

218

C₂×C₄

#13133246

yes

—

219

C₈

—

220

C₈

—

221

C₈

—

222

C₈

—

106

—

223

C₈

—

212

—

224

C₂²

#16810524

yes

characteristic

225

C₂²

#16810693

yes

—

226

C₂²

#16810729

yes

—

227

C₂²

#16810739

yes

—

228

C₄

—

229

C₄

—

230

C₄

—

424

—

231

C₄

—

212

—

232

C₄

#16810586

yes

characteristic

233

C₄

#16810589

yes

characteristic

234

C₄

#16810693

yes

—

235

C₄

#16810882

yes

—

236

C₄

#16811014

yes

—

237

C₂

#20273585

yes

—

238

C₂

#20275512

yes

characteristic

239

C₂

#20275584

yes

characteristic

240

C₂

#20275601

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#9664
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

subgroup

G/H

center

C₂²×C₄

D₄×D₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₅₃×#334

C₂

socle, prime radical

C₂³×C₅₃

C₄×D₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

N	H ≅ G/N	type
140. C₅₃	#9664	semidirect

H ≅ G/N

type

140. C₅₃

#9664

semidirect

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
#20275650	2
C₂×#16810353	2
C₂×#16810345	2
C₂×C₅₃×#334	2
#9664	53

subgroup

index

C₂×C₄×Dic₂₁₂

C₂×C₅₃×#334

Subgroups — #27282992

Conjugacy classes of subgroups

Complements

Maximal subgroups