Group #27287122 — subgroups

Subgroups — #27287122

Order 13568.

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

normal subgroups only

This group has 416 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	#20279957	C₂	1	1	yes	maximal, characteristic
2	#20275664	C₂	1	2	yes	maximal
3	#20272598	C₂	1	1	yes	maximal, characteristic
4	#20272555	C₂	1	2	yes	maximal
5	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	#16810562	—	2	2	no	—
8	#16810522	—	2	2	no	—
9	#16810488	—	2	2	no	—
10	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	#16810385	—	2	2	no	—
12	#16810385	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#103×C₅₃	—	2	2	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#384	C₂²	1	1	yes	characteristic
17	C₂²×#88322	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
21	C₂×C₄×Dic₅₃	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	#13133124	—	2	4	no	—
25	#13133124	—	2	2	no	—
26	#13133124	—	2	2	no	—
27	C₂²×Q₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₄⋊Q₈×C₅₃	—	2	1	no	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	Q₈	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#103×C₅₃	—	4	4	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₂×Q₈×C₅₃	—	2	2	no	—
39	C₂×Q₈×C₅₃	—	4	2	no	—
40	C₂×Q₈×C₅₃	—	4	2	no	—
41	C₂×Q₈×C₅₃	—	4	1	no	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	4	1	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂²×Dic₅₃	—	2	2	no	—
47	C₂²×Dic₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	4	1	no	—
50	C₂×C₈×C₅₃	—	4	1	no	—
51	C₄⋊C₄×C₅₃	—	2	4	no	—
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₄⋊C₄×C₅₃	—	4	2	no	—
56	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₅₃	#41	1	1	yes	characteristic
58	C₄²×C₅₃	#41	1	2	yes	—
59	C₄⋊Dic₅₃	—	2	2	no	—
60	C₄⋊Dic₅₃	—	2	2	no	—
61	C₄⋊Dic₅₃	—	2	2	no	—
62	C₄⋊Dic₅₃	—	2	2	no	—
63	C₄⋊Dic₅₃	—	8	2	no	—
64	C₄×Dic₅₃	—	4	2	no	—
65	C₄×Dic₅₃	—	4	2	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	4	2	no	—
69	C₂³×C₅₃	#120	1	1	yes	characteristic
70	Q₈×C₅₃	—	4	4	no	—
71	Q₈×C₅₃	—	8	2	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	4	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	8	1	no	—
78	C₂×C₄×C₅₃	—	8	1	no	—
79	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#124	1	2	yes	—
82	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#135	1	2	yes	—
84	C₂×C₄×C₅₃	#135	1	2	yes	—
85	C₂×C₄×C₅₃	#137	1	2	yes	—
86	C₂×C₄×C₅₃	#137	1	2	yes	—
87	C₂×Dic₅₃	—	4	2	no	—
88	C₂×Dic₅₃	—	4	2	no	—
89	C₂×Dic₅₃	—	4	2	no	—
90	C₂×Dic₅₃	—	4	2	no	—
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₈×C₅₃	—	4	2	no	—
94	#88322	—	8	2	no	—
95	#10352	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	#393	1	1	yes	characteristic
97	C₂²×C₅₃	#452	1	2	yes	—
98	C₂²×C₅₃	#452	1	2	yes	—
99	C₂²×C₅₃	#453	1	1	yes	characteristic
100	C₂²×C₅₃	#454	1	1	yes	characteristic
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	8	2	no	—
104	C₄×C₅₃	—	8	2	no	—
105	C₄×C₅₃	#479	1	2	yes	—
106	C₄×C₅₃	#479	1	2	yes	—
107	C₄×C₅₃	#488	1	2	yes	—
108	C₄×C₅₃	#491	1	2	yes	—
109	Dic₅₃	—	8	2	no	—
110	Dic₅₃	—	8	2	no	—
111	#2571	—	53	1	no	—
112	#1987	—	53	2	no	—
113	#1956	D₅₃	1	1	yes	characteristic
114	#1296	—	53	1	no	—
115	#1279	—	53	2	no	—
116	C₂×C₅₃	#1630	1	1	yes	characteristic
117	C₂×C₅₃	#1634	1	1	yes	characteristic
118	C₂×C₅₃	#1638	1	2	yes	—
119	C₂×C₅₃	#2026	1	2	yes	—
120	C₂×C₅₃	#2027	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	53	2	no	—
123	C₂×#103	—	2	2	no	—
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	#400	—	106	2	no	—
126	#389	—	106	2	no	—
127	#384	C₂×D₅₃	1	1	yes	characteristic
128	#382	—	106	2	no	—
129	C₄×C₄⋊C₄	—	53	2	no	—
130	#340	—	106	2	no	—
131	#340	—	106	2	no	—
132	C₅₃	#10352	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	2	1	no	—
134	C₂²×C₈	—	2	1	no	—
135	C₂²×C₈	—	106	2	no	—
136	C₄⋊Q₈	—	2	1	no	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	Dic₁₀₆	1	2	yes	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	53	2	no	—
142	C₂×C₄⋊C₄	—	53	2	no	—
143	C₂×C₄⋊C₄	—	106	2	no	—
144	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
145	C₂×C₄²	—	106	2	no	—
146	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
147	C₄⋊C₈	—	106	2	no	—
148	C₄⋊C₈	—	106	2	no	—
149	#103	—	4	4	no	—
150	C₄×C₈	—	2	2	no	—
151	#95	—	106	2	no	—
152	#95	—	106	4	no	—
153	#95	—	106	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	106	2	no	—
161	C₂²×C₄	—	106	2	no	—
162	C₂²×C₄	D₄×D₅₃	1	2	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	212	2	no	—
168	C₂×C₈	—	212	2	no	—
169	C₂×C₈	—	212	2	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	106	2	no	—
176	C₄⋊C₄	—	106	2	no	—
177	C₄⋊C₄	—	106	2	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	424	2	no	—
180	C₄²	—	212	2	no	—
181	C₄²	—	212	2	no	—
182	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
183	C₄²	#11719945	1	1	yes	characteristic
184	C₄²	#11719947	1	2	yes	—
185	C₂³	#13133185	1	1	yes	characteristic
186	Q₈	—	4	4	no	—
187	Q₈	—	8	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	212	2	no	—
196	C₂×C₄	—	212	2	no	—
197	C₂×C₄	—	212	2	no	—
198	C₂×C₄	—	212	2	no	—
199	C₂×C₄	—	212	2	no	—
200	C₂×C₄	—	212	2	no	—
201	C₂×C₄	#13133194	1	2	yes	—
202	C₂×C₄	#13133196	1	1	yes	characteristic
203	C₂×C₄	#13133203	1	2	yes	—
204	C₂×C₄	#13133204	1	1	yes	characteristic
205	C₂×C₄	#13133238	1	2	yes	—
206	C₂×C₄	#13133240	1	2	yes	—
207	C₂×C₄	#13133246	1	2	yes	—
208	C₂×C₄	#13133249	1	2	yes	—
209	C₈	—	4	2	no	—
210	C₈	—	424	2	no	—
211	C₂²	#16810535	1	1	yes	characteristic
212	C₂²	#16810782	1	2	yes	—
213	C₂²	#16810784	1	2	yes	—
214	C₂²	#16810790	1	1	yes	characteristic
215	C₂²	#16810795	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	2	no	—
220	C₄	—	424	2	no	—
221	C₄	—	424	2	no	—
222	C₄	#16810928	1	2	yes	—
223	C₄	#16810932	1	2	yes	—
224	C₄	#16810979	1	2	yes	—
225	C₄	#16810990	1	2	yes	—
226	C₂	#20273834	1	1	yes	characteristic
227	C₂	#20273858	1	1	yes	characteristic
228	C₂	#20273884	1	2	yes	—
229	C₂	#20275864	1	2	yes	—
230	C₂	#20275871	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20279957

C₂

yes

maximal, characteristic

#20275664

C₂

yes

maximal

#20272598

C₂

yes

maximal, characteristic

#20272555

C₂

yes

maximal

C₅₃×#1956

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

—

#16810562

—

#16810522

—

#16810488

—

C₄×C₄⋊Dic₅₃

C₂²

yes

—

#16810385

—

#16810385

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×#103×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

D₄

yes

—

C₂×C₄⋊Dic₅₃

D₄

yes

—

C₂×C₄×Dic₅₃

—

—

—

—

—

—

C₂²×Q₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

Q₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#103×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#120

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

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#452

yes

—

C₂²×C₅₃

#453

yes

characteristic

100

C₂²×C₅₃

#454

yes

characteristic

101

C₄×C₅₃

—

102

C₄×C₅₃

—

103

C₄×C₅₃

—

104

C₄×C₅₃

—

105

C₄×C₅₃

#479

yes

—

106

C₄×C₅₃

#479

yes

—

107

C₄×C₅₃

#488

yes

—

108

C₄×C₅₃

#491

yes

—

109

Dic₅₃

—

110

Dic₅₃

—

111

#2571

—

112

#1987

—

113

#1956

D₅₃

yes

characteristic

114

#1296

—

115

#1279

—

116

C₂×C₅₃

#1630

yes

characteristic

117

C₂×C₅₃

#1634

yes

characteristic

118

C₂×C₅₃

#1638

yes

—

119

C₂×C₅₃

#2026

yes

—

120

C₂×C₅₃

#2027

yes

characteristic

121

C₂×C₄⋊Q₈

C₂×D₅₃

yes

characteristic

122

C₂×C₄⋊C₈

—

123

C₂×#103

—

124

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

125

#400

—

106

—

126

#389

—

106

—

127

#384

C₂×D₅₃

yes

characteristic

128

#382

—

106

—

129

C₄×C₄⋊C₄

—

130

#340

—

106

—

131

#340

—

106

—

132

C₅₃

#10352

yes

characteristic, complemented, Sylow

133

C₂²×Q₈

—

134

C₂²×C₈

—

135

C₂²×C₈

—

106

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

Dic₁₀₆

yes

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄⋊C₄

—

143

C₂×C₄⋊C₄

—

106

—

144

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

145

C₂×C₄²

—

106

—

146

C₂×C₄²

C₂²×D₅₃

yes

characteristic

147

C₄⋊C₈

—

106

—

148

C₄⋊C₈

—

106

—

149

#103

—

150

C₄×C₈

—

151

#95

—

106

—

152

#95

—

106

—

153

#95

—

106

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

106

—

161

C₂²×C₄

—

106

—

162

C₂²×C₄

D₄×D₅₃

yes

—

163

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

212

—

168

C₂×C₈

—

212

—

169

C₂×C₈

—

212

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

106

—

176

C₄⋊C₄

—

106

—

177

C₄⋊C₄

—

106

—

178

C₄⋊C₄

—

106

—

179

C₄⋊C₄

—

424

—

180

C₄²

—

212

—

181

C₄²

—

212

—

182

C₄²

C₂×Dic₁₀₆

yes

characteristic

183

C₄²

#11719945

yes

characteristic

184

C₄²

#11719947

yes

—

185

C₂³

#13133185

yes

characteristic

186

Q₈

—

187

Q₈

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

212

—

196

C₂×C₄

—

212

—

197

C₂×C₄

—

212

—

198

C₂×C₄

—

212

—

199

C₂×C₄

—

212

—

200

C₂×C₄

—

212

—

201

C₂×C₄

#13133194

yes

—

202

C₂×C₄

#13133196

yes

characteristic

203

C₂×C₄

#13133203

yes

—

204

C₂×C₄

#13133204

yes

characteristic

205

C₂×C₄

#13133238

yes

—

206

C₂×C₄

#13133240

yes

—

207

C₂×C₄

#13133246

yes

—

208

C₂×C₄

#13133249

yes

—

209

C₈

—

210

C₈

—

424

—

211

C₂²

#16810535

yes

characteristic

212

C₂²

#16810782

yes

—

213

C₂²

#16810784

yes

—

214

C₂²

#16810790

yes

characteristic

215

C₂²

#16810795

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

424

—

221

C₄

—

424

—

222

C₄

#16810928

yes

—

223

C₄

#16810932

yes

—

224

C₄

#16810979

yes

—

225

C₄

#16810990

yes

—

226

C₂

#20273834

yes

characteristic

227

C₂

#20273858

yes

characteristic

228

C₂

#20273884

yes

—

229

C₂

#20275864

yes

—

230

C₂

#20275871

yes

characteristic

	subgroup	G/H
center	C₂³	#13133185
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1956	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#1956	D₅₃
hypocenter	C₅₃	#10352
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#13133185

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1956

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
132. C₅₃	#10352	semidirect

H ≅ G/N

type

132. C₅₃

#10352

semidirect

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

subgroup	index
#20279957	2
#20275664	2
#20272598	2
#20272555	2
C₅₃×#1956	2
#10352	53

subgroup

index

C₅₃×#1956

Subgroups — #27287122

Conjugacy classes of subgroups

Complements

Maximal subgroups