Group #27288531 — subgroups

Subgroups — #27288531

Order 13568.

494 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 145 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	#20275478	C₂	1	2	yes	maximal
2	#20274871	C₂	1	1	yes	maximal, characteristic
3	#20272598	C₂	1	1	yes	maximal, characteristic
4	#20272556	C₂	1	2	yes	maximal
5	C₅₃×#2551	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133148	—	2	2	no	—
7	C₂×#13133148	—	2	2	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810493	C₂²	1	1	yes	characteristic
11	#16810385	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
13	C₅₃×#395	—	2	2	no	—
14	C₅₃×#389	—	2	2	no	—
15	C₅₃×#384	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂²×#88322	—	2	1	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	1	no	—
21	#13133148	—	4	4	no	—
22	#13133148	—	4	4	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	Q₈	1	4	yes	complemented
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	Q₈	1	2	yes	complemented
27	C₂²×Q₈×C₅₃	—	2	2	no	—
28	C₄⋊Q₈×C₅₃	—	4	4	no	—
29	C₄⋊Q₈×C₅₃	—	4	4	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#95×C₅₃	—	2	2	no	—
38	#95×C₅₃	—	2	2	no	—
39	C₂×Q₈×C₅₃	—	4	8	no	—
40	C₂×Q₈×C₅₃	—	8	2	no	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂²×Dic₅₃	—	4	1	no	—
47	C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₄⋊C₄×C₅₃	—	4	8	no	—
50	C₄⋊C₄×C₅₃	—	4	2	no	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₄⋊C₄×C₅₃	SD₁₆	1	4	yes	—
54	C₄⋊C₄×C₅₃	SD₁₆	1	4	yes	—
55	C₄²×C₅₃	—	4	2	no	—
56	C₄²×C₅₃	—	4	2	no	—
57	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
59	C₄²×C₅₃	#41	1	1	yes	characteristic
60	C₄⋊Dic₅₃	—	4	2	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	2	4	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	4	2	no	—
69	C₂³×C₅₃	#120	1	1	yes	characteristic
70	Q₈×C₅₃	—	8	8	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	4	no	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	4	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×C₄×C₅₃	#137	1	2	yes	—
89	C₂×Dic₅₃	—	8	2	no	—
90	C₂×Dic₅₃	—	8	1	no	—
91	#88322	—	2	4	no	—
92	#88322	—	4	4	no	—
93	#10574	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#395	1	1	yes	characteristic
95	C₂²×C₅₃	#448	1	2	yes	—
96	C₂²×C₅₃	#450	1	2	yes	—
97	C₂²×C₅₃	#453	1	1	yes	characteristic
98	C₂²×C₅₃	#456	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	4	4	no	—
102	C₄×C₅₃	—	4	4	no	—
103	C₄×C₅₃	—	8	2	no	—
104	C₄×C₅₃	#475	1	2	yes	—
105	C₄×C₅₃	#476	1	2	yes	—
106	C₄×C₅₃	#498	1	2	yes	—
107	C₄×C₅₃	#501	1	2	yes	—
108	Dic₅₃	—	16	1	no	—
109	#2551	D₅₃	1	1	yes	characteristic
110	#1948	—	53	2	no	—
111	#1844	—	53	1	no	—
112	#1296	—	53	1	no	—
113	#1279	—	53	2	no	—
114	C₂×C₅₃	#1623	1	1	yes	characteristic
115	C₂×C₅₃	#1629	1	1	yes	characteristic
116	C₂×C₅₃	#1651	1	2	yes	—
117	C₂×C₅₃	#2058	1	2	yes	—
118	C₂×C₅₃	#2060	1	1	yes	characteristic
119	C₂×C₄⋊Q₈	—	2	2	no	—
120	C₂×#106	—	106	2	no	—
121	C₂×#106	—	106	2	no	—
122	C₂×C₄⋊C₈	—	53	2	no	—
123	C₂×C₄×C₈	—	53	1	no	—
124	#395	—	2	2	no	—
125	#389	—	2	2	no	—
126	#384	—	53	1	no	—
127	#384	C₂×D₅₃	1	1	yes	characteristic
128	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
129	#340	—	106	2	no	—
130	C₅₃	#10574	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₈	—	2	2	no	—
132	C₂²×C₈	—	106	1	no	—
133	C₂²×C₈	—	106	2	no	—
134	C₄⋊Q₈	—	4	4	no	—
135	C₄⋊Q₈	—	4	4	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	106	2	no	—
139	C₂×C₄⋊C₄	—	106	1	no	—
140	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
141	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
142	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
145	#106	—	212	4	no	—
146	#106	—	212	4	no	—
147	C₄⋊C₈	—	53	4	no	—
148	C₄⋊C₈	—	106	2	no	—
149	C₄×C₈	—	53	2	no	—
150	C₄×C₈	—	106	1	no	—
151	#95	—	2	2	no	—
152	#95	—	2	2	no	—
153	C₂×Q₈	—	4	8	no	—
154	C₂×Q₈	—	8	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	212	1	no	—
159	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
161	C₂×C₈	—	106	2	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	2	no	—
164	C₂×C₈	—	106	4	no	—
165	C₂×C₈	—	212	2	no	—
166	C₂×C₈	—	212	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	8	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	212	2	no	—
174	C₄⋊C₄	—	212	2	no	—
175	C₄⋊C₄	—	212	2	no	—
176	C₄⋊C₄	#11719923	1	4	yes	—
177	C₄⋊C₄	#11719924	1	4	yes	—
178	C₄²	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	Q₈×D₅₃	1	1	yes	characteristic
181	C₄²	Q₈×D₅₃	1	2	yes	—
182	C₄²	#11719949	1	1	yes	characteristic
183	C₂³	#13133186	1	1	yes	characteristic
184	Q₈	—	8	8	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	424	2	no	—
196	C₂×C₄	—	424	1	no	—
197	C₂×C₄	#13133193	1	1	yes	characteristic
198	C₂×C₄	#13133193	1	2	yes	—
199	C₂×C₄	#13133215	1	1	yes	characteristic
200	C₂×C₄	#13133215	1	2	yes	—
201	C₂×C₄	C₂×#11719924	1	2	yes	—
202	C₂×C₄	C₂×#11719923	1	2	yes	—
203	C₂×C₄	#13133243	1	2	yes	—
204	C₂×C₄	#13133248	1	2	yes	—
205	C₈	—	106	4	no	—
206	C₈	—	212	4	no	—
207	C₂²	#16810542	1	1	yes	characteristic
208	C₂²	#16810762	1	2	yes	—
209	C₂²	#16810774	1	2	yes	—
210	C₂²	#16810791	1	1	yes	characteristic
211	C₂²	#16810806	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	4	4	no	—
216	C₄	—	8	2	no	—
217	C₄	—	848	1	no	—
218	C₄	#16810905	1	2	yes	—
219	C₄	#16810910	1	2	yes	—
220	C₄	#16811019	1	2	yes	—
221	C₄	#16811029	1	2	yes	—
222	C₂	#20273793	1	1	yes	characteristic
223	C₂	#20273831	1	1	yes	characteristic
224	C₂	#20273978	1	2	yes	—
225	C₂	#20276045	1	2	yes	—
226	C₂	#20276057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275478

C₂

yes

maximal

#20274871

C₂

yes

maximal, characteristic

#20272598

C₂

yes

maximal, characteristic

#20272556

C₂

yes

maximal

C₅₃×#2551

C₂

yes

maximal, characteristic

C₂×#13133148

—

C₂×#13133148

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

#16810493

C₂²

yes

characteristic

#16810385

—

C₂×C₄⋊Q₈×C₅₃

—

C₅₃×#395

—

C₅₃×#389

—

C₅₃×#384

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂²×#88322

—

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

Q₈

yes

complemented

C₄×#88322

—

C₄×#88322

Q₈

yes

complemented

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

—

—

—

—

—

—

C₂³×C₅₃

#120

yes

characteristic

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#450

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

100

C₄×C₅₃

—

101

C₄×C₅₃

—

102

C₄×C₅₃

—

103

C₄×C₅₃

—

104

C₄×C₅₃

#475

yes

—

105

C₄×C₅₃

#476

yes

—

106

C₄×C₅₃

#498

yes

—

107

C₄×C₅₃

#501

yes

—

108

Dic₅₃

—

109

#2551

D₅₃

yes

characteristic

110

#1948

—

111

#1844

—

112

#1296

—

113

#1279

—

114

C₂×C₅₃

#1623

yes

characteristic

115

C₂×C₅₃

#1629

yes

characteristic

116

C₂×C₅₃

#1651

yes

—

117

C₂×C₅₃

#2058

yes

—

118

C₂×C₅₃

#2060

yes

characteristic

119

C₂×C₄⋊Q₈

—

120

C₂×#106

—

106

—

121

C₂×#106

—

106

—

122

C₂×C₄⋊C₈

—

123

C₂×C₄×C₈

—

124

#395

—

125

#389

—

126

#384

—

127

#384

C₂×D₅₃

yes

characteristic

128

C₄×C₄⋊C₄

C₂×D₅₃

yes

—

129

#340

—

106

—

130

C₅₃

#10574

yes

characteristic, complemented, Sylow

131

C₂²×Q₈

—

132

C₂²×C₈

—

106

—

133

C₂²×C₈

—

106

—

134

C₄⋊Q₈

—

135

C₄⋊Q₈

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

106

—

139

C₂×C₄⋊C₄

—

106

—

140

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

141

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

142

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

143

C₂×C₄²

—

144

C₂×C₄²

C₂²×D₅₃

yes

characteristic

145

#106

—

212

—

146

#106

—

212

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

106

—

149

C₄×C₈

—

150

C₄×C₈

—

106

—

151

#95

—

152

#95

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

212

—

159

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

160

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

161

C₂×C₈

—

106

—

162

C₂×C₈

—

106

—

163

C₂×C₈

—

106

—

164

C₂×C₈

—

106

—

165

C₂×C₈

—

212

—

166

C₂×C₈

—

212

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

212

—

174

C₄⋊C₄

—

212

—

175

C₄⋊C₄

—

212

—

176

C₄⋊C₄

#11719923

yes

—

177

C₄⋊C₄

#11719924

yes

—

178

C₄²

—

179

C₄²

—

180

C₄²

Q₈×D₅₃

yes

characteristic

181

C₄²

Q₈×D₅₃

yes

—

182

C₄²

#11719949

yes

characteristic

183

C₂³

#13133186

yes

characteristic

184

Q₈

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

424

—

196

C₂×C₄

—

424

—

197

C₂×C₄

#13133193

yes

characteristic

198

C₂×C₄

#13133193

yes

—

199

C₂×C₄

#13133215

yes

characteristic

200

C₂×C₄

#13133215

yes

—

201

C₂×C₄

C₂×#11719924

yes

—

202

C₂×C₄

C₂×#11719923

yes

—

203

C₂×C₄

#13133243

yes

—

204

C₂×C₄

#13133248

yes

—

205

C₈

—

106

—

206

C₈

—

212

—

207

C₂²

#16810542

yes

characteristic

208

C₂²

#16810762

yes

—

209

C₂²

#16810774

yes

—

210

C₂²

#16810791

yes

characteristic

211

C₂²

#16810806

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

848

—

218

C₄

#16810905

yes

—

219

C₄

#16810910

yes

—

220

C₄

#16811019

yes

—

221

C₄

#16811029

yes

—

222

C₂

#20273793

yes

characteristic

223

C₂

#20273831

yes

characteristic

224

C₂

#20273978

yes

—

225

C₂

#20276045

yes

—

226

C₂

#20276057

yes

characteristic

	subgroup	G/H
center	C₂³	#13133186
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2551	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#2551	D₅₃
hypocenter	C₅₃	#10574
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2551

subgroup

G/H

center

C₂³

#13133186

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2551

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2551

N	H ≅ G/N	type
24. C₄⋊#88322	Q₈	semidirect
26. C₄×#88322	Q₈	semidirect
130. C₅₃	#10574	semidirect

H ≅ G/N

type

24. C₄⋊#88322

Q₈

semidirect

26. C₄×#88322

Q₈

semidirect

130. C₅₃

#10574

semidirect

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

subgroup	index
#20275478	2
#20274871	2
#20272598	2
#20272556	2
C₅₃×#2551	2
#10574	53

subgroup

index

C₅₃×#2551

Subgroups — #27288531

Conjugacy classes of subgroups

Complements

Maximal subgroups