Group #27289108 — subgroups

Subgroups — #27289108

Order 13568.

354 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20275547	C₂	1	1	yes	maximal, characteristic
2	#20274869	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273490	C₂	1	2	yes	maximal, complemented
4	#20273115	C₂	1	2	yes	maximal
5	C₅₃×#1274	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811173	C₂²	1	1	yes	characteristic
7	#16810647	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#88322	—	2	2	no	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	C₂×#13133145	—	2	2	no	—
11	#16810376	—	2	4	no	—
12	#16810352	—	2	4	no	—
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₅₃×#376	C₂²	1	1	yes	characteristic
15	C₂×#11719945	—	2	1	no	—
16	C₂×#11719917	—	2	1	no	—
17	C₂²×#88322	—	2	2	no	—
18	#13133217	Q₈	1	1	yes	characteristic
19	#13133214	Q₈	1	1	yes	characteristic
20	#13133205	—	2	1	no	—
21	#13133191	—	2	2	no	—
22	C₄×D₂₁₂	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	4	4	no	—
27	#13133145	—	4	4	no	—
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	2	no	—
31	C₂²⋊C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	D₄	1	4	yes	—
33	#95×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	#41	1	2	yes	—
37	C₂×C₂²⋊D₅₃	—	4	1	no	—
38	#11719945	—	4	2	no	—
39	#11719945	—	4	2	no	—
40	C₂×D₂₁₂	—	2	2	no	—
41	C₂×D₂₁₂	—	2	1	no	—
42	C₂×C₄×D₅₃	—	4	1	no	—
43	C₂×Dic₁₀₆	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₄²×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	D₈	1	4	yes	—
47	C₄²×C₅₃	C₂×D₄	1	2	yes	—
48	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₃	#41	1	1	yes	characteristic
50	#11719921	—	4	2	no	—
51	C₄⋊Dic₅₃	—	2	2	no	—
52	#11719917	—	4	2	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	—	4	1	no	—
55	C₂×#88322	—	4	4	no	—
56	C₂³×C₅₃	#123	1	1	yes	characteristic
57	C₂²×D₅₃	—	4	1	no	—
58	C₂×C₄×C₅₃	—	2	4	no	—
59	C₂×C₄×C₅₃	—	4	2	no	—
60	C₂×C₄×C₅₃	—	4	4	no	—
61	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#123	1	2	yes	—
64	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₅₃	#137	1	2	yes	—
67	C₂²⋊D₅₃	—	8	2	no	—
68	C₂×Dic₅₃	—	4	1	no	—
69	D₂₁₂	—	4	4	no	—
70	D₂₁₂	—	4	2	no	—
71	C₄×D₅₃	—	8	1	no	—
72	C₄×D₅₃	—	8	1	no	—
73	Dic₁₀₆	—	4	2	no	—
74	C₈×C₅₃	—	4	4	no	—
75	#88322	—	8	2	no	—
76	#88322	—	8	1	no	—
77	#10667	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	—	2	2	no	—
79	C₂²×C₅₃	#399	1	1	yes	characteristic
80	C₂²×C₅₃	#480	1	2	yes	—
81	C₂×D₅₃	—	8	1	no	—
82	C₂×D₅₃	—	8	2	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	4	4	no	—
86	C₄×C₅₃	#474	1	2	yes	—
87	C₄×C₅₃	#493	1	1	yes	characteristic
88	C₄×C₅₃	#497	1	1	yes	characteristic
89	Dic₅₃	—	8	1	no	—
90	#1963	—	53	1	no	—
91	#1843	—	53	1	no	—
92	#1571	—	53	2	no	—
93	#1480	—	53	2	no	—
94	#1274	D₅₃	1	1	yes	characteristic, complemented
95	C₂×C₅₃	—	2	2	no	—
96	C₂×C₅₃	#2093	1	1	yes	characteristic
97	C₂×C₅₃	#2098	1	2	yes	—
98	D₅₃	—	16	1	no	—
99	#532	—	53	1	no	—
100	#431	C₂×D₅₃	1	2	yes	—
101	#423	—	53	1	no	—
102	C₂×C₄⋊C₈	—	53	2	no	—
103	C₂×C₄⋊C₈	—	106	2	no	—
104	C₂×#104	—	106	2	no	—
105	#376	C₂×D₅₃	1	1	yes	characteristic
106	#335	—	106	4	no	—
107	D₄⋊C₈	—	106	4	no	—
108	C₅₃	#10667	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	106	1	no	—
110	C₂×M₁₆	—	106	1	no	—
111	C₂²×C₈	—	106	2	no	—
112	C₄⋊Q₈	—	53	1	no	—
113	#127	—	53	1	no	—
114	#124	—	106	1	no	—
115	#121	—	106	2	no	—
116	C₄×D₄	—	106	2	no	—
117	C₂×C₄²	D₂₁₂	1	2	yes	—
118	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	106	2	no	—
121	C₄⋊C₈	—	106	2	no	—
122	C₄⋊C₈	—	106	2	no	—
123	C₄⋊C₈	—	212	4	no	—
124	#104	—	212	4	no	—
125	C₂²⋊C₈	—	2	2	no	—
126	C₄×C₈	D₂₁₂	1	4	yes	—
127	#95	—	2	2	no	—
128	#41	—	212	2	no	—
129	#41	—	212	2	no	—
130	C₂×Q₈	—	106	1	no	—
131	C₂×D₄	—	106	2	no	—
132	C₂×D₄	—	106	1	no	—
133	C₂×D₄	—	212	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	212	1	no	—
136	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
137	C₂²×C₄	#11719947	1	2	yes	—
138	M₁₆	—	212	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	212	2	no	—
141	C₂×C₈	—	212	1	no	—
142	C₂×C₈	—	212	4	no	—
143	C₄⋊C₄	—	106	2	no	—
144	C₂²⋊C₄	—	212	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	D₄₂₄	1	4	yes	—
147	C₄²	C₂×D₂₁₂	1	2	yes	—
148	C₄²	#11719947	1	1	yes	characteristic
149	C₄²	Q₈×D₅₃	1	1	yes	characteristic
150	C₂³	—	212	1	no	—
151	C₂³	#13133201	1	1	yes	characteristic
152	Q₈	—	212	2	no	—
153	D₄	—	424	2	no	—
154	D₄	—	212	4	no	—
155	D₄	—	212	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	424	1	no	—
160	C₂×C₄	—	424	1	no	—
161	C₂×C₄	—	212	1	no	—
162	C₂×C₄	#13133197	1	2	yes	—
163	C₂×C₄	#13133197	1	1	yes	characteristic
164	C₂×C₄	#13133201	1	2	yes	—
165	C₂×C₄	#13133214	1	1	yes	characteristic
166	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
167	C₂×C₄	#13133250	1	2	yes	—
168	C₈	—	4	4	no	—
169	C₈	—	424	2	no	—
170	C₈	—	424	1	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	424	1	no	—
173	C₂²	—	424	2	no	—
174	C₂²	#16810560	1	1	yes	characteristic
175	C₂²	#16810937	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	424	1	no	—
180	C₄	#16810904	1	2	yes	—
181	C₄	#16811000	1	1	yes	characteristic
182	C₄	#16811017	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	848	1	no	—
185	C₂	#20276242	1	1	yes	characteristic
186	C₂	#20276269	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275547

C₂

yes

maximal, characteristic

#20274869

C₂

yes

maximal, characteristic, complemented

#20273490

C₂

yes

maximal, complemented

#20273115

C₂

yes

maximal

C₅₃×#1274

C₂

yes

maximal, characteristic, complemented

#16811173

C₂²

yes

characteristic

#16810647

C₂²

yes

characteristic

C₂×C₄⋊#88322

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×#13133145

—

#16810376

—

#16810352

—

C₅₃×#431

C₂²

yes

—

C₅₃×#376

C₂²

yes

characteristic

C₂×#11719945

—

C₂×#11719917

—

C₂²×#88322

—

#13133217

Q₈

yes

characteristic

#13133214

Q₈

yes

characteristic

#13133205

—

#13133191

—

C₄×D₂₁₂

—

—

—

—

—

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₂²⋊D₅₃

—

#11719945

—

#11719945

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

—

C₄⋊Dic₅₃

—

—

—

—

—

C₂³×C₅₃

#123

yes

characteristic

C₂²×D₅₃

—

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

#123

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#399

yes

characteristic

C₂²×C₅₃

#480

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#474

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

#2093

yes

characteristic

C₂×C₅₃

#2098

yes

—

D₅₃

—

#532

—

100

#431

C₂×D₅₃

yes

—

101

#423

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄⋊C₈

—

106

—

104

C₂×#104

—

106

—

105

#376

C₂×D₅₃

yes

characteristic

106

#335

—

106

—

107

D₄⋊C₈

—

106

—

108

C₅₃

#10667

yes

characteristic, complemented, Sylow

109

C₂×#41

—

106

—

110

C₂×M₁₆

—

106

—

111

C₂²×C₈

—

106

—

112

C₄⋊Q₈

—

113

#127

—

114

#124

—

106

—

115

#121

—

106

—

116

C₄×D₄

—

106

—

117

C₂×C₄²

D₂₁₂

yes

—

118

C₂×C₄²

C₂²×D₅₃

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

106

—

121

C₄⋊C₈

—

106

—

122

C₄⋊C₈

—

106

—

123

C₄⋊C₈

—

212

—

124

#104

—

212

—

125

C₂²⋊C₈

—

126

C₄×C₈

D₂₁₂

yes

—

127

#95

—

128

#41

—

212

—

129

#41

—

212

—

130

C₂×Q₈

—

106

—

131

C₂×D₄

—

106

—

132

C₂×D₄

—

106

—

133

C₂×D₄

—

212

—

134

C₂²×C₄

—

135

C₂²×C₄

—

212

—

136

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

137

C₂²×C₄

#11719947

yes

—

138

M₁₆

—

212

—

139

C₂×C₈

—

140

C₂×C₈

—

212

—

141

C₂×C₈

—

212

—

142

C₂×C₈

—

212

—

143

C₄⋊C₄

—

106

—

144

C₂²⋊C₄

—

212

—

145

C₄²

—

146

C₄²

D₄₂₄

yes

—

147

C₄²

C₂×D₂₁₂

yes

—

148

C₄²

#11719947

yes

characteristic

149

C₄²

Q₈×D₅₃

yes

characteristic

150

C₂³

—

212

—

151

C₂³

#13133201

yes

characteristic

152

Q₈

—

212

—

153

D₄

—

424

—

154

D₄

—

212

—

155

D₄

—

212

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

424

—

160

C₂×C₄

—

424

—

161

C₂×C₄

—

212

—

162

C₂×C₄

#13133197

yes

—

163

C₂×C₄

#13133197

yes

characteristic

164

C₂×C₄

#13133201

yes

—

165

C₂×C₄

#13133214

yes

characteristic

166

C₂×C₄

C₂×D₄₂₄

yes

—

167

C₂×C₄

#13133250

yes

—

168

C₈

—

169

C₈

—

424

—

170

C₈

—

424

—

171

C₂²

—

172

C₂²

—

424

—

173

C₂²

—

424

—

174

C₂²

#16810560

yes

characteristic

175

C₂²

#16810937

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

424

—

180

C₄

#16810904

yes

—

181

C₄

#16811000

yes

characteristic

182

C₄

#16811017

yes

characteristic

183

C₂

—

184

C₂

—

848

—

185

C₂

#20276242

yes

characteristic

186

C₂

#20276269

yes

—

	subgroup	G/H
center	C₂×C₄	#13133214
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1274	C₂
socle	C₂²×C₅₃	#399
prime radical	#16811173	C₂²
hypercenter	#1274	D₅₃
hypocenter	C₅₃	#10667
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#13133214

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1274

C₂

C₂²×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
2. #20274869	C₂	semidirect
3. #20273490	C₂	semidirect
5. C₅₃×#1274	C₂	semidirect
94. #1274	D₅₃	semidirect
108. C₅₃	#10667	semidirect

H ≅ G/N

type

2. #20274869

C₂

semidirect

3. #20273490

C₂

semidirect

5. C₅₃×#1274

C₂

semidirect

94. #1274

D₅₃

semidirect

108. C₅₃

#10667

semidirect

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

subgroup	index
#20275547	2
#20274869	2
#20273490	2
#20273115	2
C₅₃×#1274	2
#10667	53

subgroup

index

C₅₃×#1274

Subgroups — #27289108

Conjugacy classes of subgroups

Complements

Maximal subgroups