Group #28311459 — subgroups

Subgroups — #28311459

Order 13568.

374 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 112 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	#20291682	C₂	1	1	yes	maximal, characteristic
2	#20291052	C₂	1	2	yes	maximal, complemented
3	#20290945	C₂	1	1	yes	maximal, characteristic
4	#20289766	C₂	1	2	yes	maximal, complemented
5	#20274518	C₂	1	1	yes	maximal, characteristic
6	#16811681	—	2	4	no	—
7	C₂×#13133291	—	2	2	no	—
8	C₂×#13133291	C₂²	1	1	yes	characteristic
9	C₂×#13133291	C₂²	1	2	yes	—
10	C₂²×#11719910	C₂²	1	1	yes	characteristic
11	#16811544	—	2	4	no	—
12	C₂²⋊#11719910	—	2	2	no	—
13	C₂×#13133224	C₂²	1	1	yes	characteristic
14	#16810597	C₄	1	1	yes	characteristic
15	#16810581	C₂²	1	2	yes	—
16	#16810395	C₄	1	2	yes	—
17	C₅₃×#413	C₄	1	1	yes	characteristic
18	#13133291	—	2	4	no	—
19	#13133291	—	2	2	no	—
20	#13133291	—	2	2	no	—
21	#13133291	—	2	2	no	—
22	C₂×#11719910	—	2	4	no	—
23	C₂×#11719910	—	2	2	no	—
24	C₂×#11719910	—	2	1	no	—
25	C₂×#11719910	—	2	1	no	—
26	C₂×#11719910	D₄	1	4	yes	—
27	#13133224	—	2	2	no	—
28	#13133224	—	2	2	no	—
29	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719917	Q₈	1	2	yes	—
31	C₂²×#88322	D₄	1	2	yes	—
32	C₂²×#88322	C₂³	1	1	yes	characteristic
33	#13133176	C₂×C₄	1	2	yes	—
34	#13133132	C₈	1	4	yes	—
35	#13133127	—	2	2	no	—
36	C₈×Dic₅₃	—	2	2	no	—
37	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	#101×C₅₃	—	2	1	no	—
40	#100×C₅₃	—	2	1	no	—
41	#41×C₅₃	—	2	2	no	—
42	#41×C₅₃	—	2	2	no	—
43	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
44	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
45	C₂×D₄×C₅₃	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₅₃	—	2	2	no	—
49	M₁₆×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
52	C₄⋊Dic₅₃	—	2	2	no	—
53	C₄×Dic₅₃	—	2	2	no	—
54	#11719917	—	2	2	no	—
55	#11719917	—	2	4	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	C₄⋊C₄	1	2	yes	—
59	C₂×#88322	C₂×D₄	1	2	yes	—
60	C₂×#88322	C₂×D₄	1	1	yes	characteristic
61	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
62	C₂×#88322	#41	1	2	yes	—
63	#11719910	—	2	4	no	—
64	#11719910	—	4	1	no	—
65	#11719910	—	4	1	no	—
66	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
67	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₅₃	—	2	2	no	—
69	D₄×C₅₃	—	2	4	no	—
70	D₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#131	1	2	yes	—
78	C₂×Dic₅₃	—	2	4	no	—
79	C₈×C₅₃	—	4	2	no	—
80	#88322	—	2	2	no	—
81	#88322	#120	1	1	yes	characteristic
82	#88322	C₂²⋊Q₈	1	2	yes	—
83	#88322	#123	1	1	yes	characteristic
84	#12915	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	4	1	no	—
87	C₂²×C₅₃	—	4	2	no	—
88	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₅₃	C₈×D₄	1	2	yes	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	#385	1	1	yes	characteristic
92	C₄×C₅₃	#424	1	1	yes	characteristic
93	C₄×C₅₃	#435	1	2	yes	—
94	Dic₅₃	—	4	4	no	—
95	#3407	—	53	1	no	—
96	#2151	—	53	2	no	—
97	#2113	—	53	1	no	—
98	#1757	—	53	1	no	—
99	#1378	—	53	2	no	—
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	—	2	1	no	—
102	C₂×C₅₃	—	4	2	no	—
103	C₂×C₅₃	#1850	1	1	yes	characteristic
104	C₂×#131	—	53	1	no	—
105	#504	—	106	4	no	—
106	C₂×M₃₂	—	53	2	no	—
107	C₂×M₃₂	—	53	1	no	—
108	C₂×M₃₂	—	106	2	no	—
109	C₂²×C₁₆	—	53	1	no	—
110	#413	#6157	1	1	yes	characteristic
111	#410	—	53	1	no	—
112	#405	—	53	2	no	—
113	C₄⋊C₁₆	—	106	4	no	—
114	C₂²⋊C₁₆	—	106	2	no	—
115	#344	—	53	2	no	—
116	C₅₃	#12915	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	C₂×#6157	1	1	yes	characteristic
118	#131	—	106	2	no	—
119	#131	—	106	2	no	—
120	C₂×M₁₆	—	53	2	no	—
121	C₂×M₁₆	—	53	1	no	—
122	C₂×M₁₆	C₂×#6157	1	2	yes	—
123	C₂²×C₈	—	53	1	no	—
124	C₂²×C₈	—	53	2	no	—
125	#117	—	53	2	no	—
126	M₃₂	—	106	2	no	—
127	M₃₂	—	106	2	no	—
128	M₃₂	—	106	4	no	—
129	M₃₂	—	106	2	no	—
130	C₂×C₁₆	—	53	4	no	—
131	C₂×C₁₆	—	106	4	no	—
132	C₂×C₁₆	—	106	1	no	—
133	C₂×C₁₆	—	106	1	no	—
134	C₂×C₁₆	—	106	2	no	—
135	#101	—	2	1	no	—
136	#100	—	2	1	no	—
137	#99	—	53	4	no	—
138	#97	—	106	2	no	—
139	C₄×C₈	—	106	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	#11719940	1	1	yes	characteristic
143	C₂×D₄	D₅₃⋊C₈	1	2	yes	—
144	C₂×D₄	#11719940	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊#6157	1	2	yes	—
146	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	2	no	—
149	M₁₆	—	106	4	no	—
150	M₁₆	—	106	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	53	1	no	—
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	53	2	no	—
156	C₂×C₈	—	53	2	no	—
157	C₂×C₈	—	106	2	no	—
158	C₂×C₈	—	106	2	no	—
159	C₄⋊C₄	—	106	2	no	—
160	C₂²⋊C₄	—	53	4	no	—
161	C₄²	—	106	2	no	—
162	C₁₆	—	106	4	no	—
163	C₁₆	—	212	1	no	—
164	C₁₆	—	212	1	no	—
165	C₂³	#13133284	1	2	yes	—
166	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	106	4	no	—
173	C₂×C₄	#13133284	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
175	C₂×C₄	D₄×#6157	1	2	yes	—
176	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
177	C₂×C₄	#13133304	1	2	yes	—
178	C₈	—	4	2	no	—
179	C₈	—	53	1	no	—
180	C₈	—	53	2	no	—
181	C₈	—	53	1	no	—
182	C₈	—	106	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×#13133284	1	1	yes	characteristic
187	C₂²	#16811650	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	212	4	no	—
190	C₄	#16811570	1	1	yes	characteristic
191	C₄	#16811629	1	1	yes	characteristic
192	C₄	#16811654	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#20290659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291682

C₂

yes

maximal, characteristic

#20291052

C₂

yes

maximal, complemented

#20290945

C₂

yes

maximal, characteristic

#20289766

C₂

yes

maximal, complemented

#20274518

C₂

yes

maximal, characteristic

#16811681

—

C₂×#13133291

—

C₂×#13133291

C₂²

yes

characteristic

C₂×#13133291

C₂²

yes

—

C₂²×#11719910

C₂²

yes

—

—

C₂²

yes

characteristic

#16810597

C₄

yes

characteristic

#16810581

C₂²

yes

—

#16810395

C₄

yes

—

C₅₃×#413

C₄

yes

—

—

—

—

—

—

—

—

D₄

yes

—

#13133224

—

#13133224

—

C₂×#11719917

C₂×C₄

yes

characteristic

C₂×#11719917

Q₈

yes

—

C₂²×#88322

D₄

yes

—

C₂²×#88322

C₂³

yes

characteristic

#13133176

C₂×C₄

yes

—

#13133132

C₈

yes

—

#13133127

—

C₈×Dic₅₃

—

C₂×#41×C₅₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

#101×C₅₃

—

#100×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

M₁₆

yes

characteristic

C₂×D₄×C₅₃

C₂×C₈

yes

—

C₂×D₄×C₅₃

M₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

C₂×C₈

yes

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#88322

C₂×D₄

yes

—

C₂×#88322

C₂×D₄

yes

characteristic

C₂×#88322

C₂×Q₈

yes

characteristic

C₂×#88322

#41

yes

—

#11719910

—

#11719910

—

#11719910

—

C₂³×C₅₃

C₄⋊C₈

yes

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

C₈×C₅₃

—

—

yes

C₂²⋊Q₈

yes

—

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₃

C₈×D₄

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

—

—

—

—

—

100

C₂×C₅₃

—

101

C₂×C₅₃

—

102

C₂×C₅₃

—

103

C₂×C₅₃

#1850

yes

characteristic

104

C₂×#131

—

105

#504

—

106

—

106

C₂×M₃₂

—

107

C₂×M₃₂

—

108

C₂×M₃₂

—

106

—

109

C₂²×C₁₆

—

110

yes

111

—

112

#405

—

113

C₄⋊C₁₆

—

106

—

114

C₂²⋊C₁₆

—

106

—

115

#344

—

116

C₅₃

#12915

yes

characteristic, complemented, Sylow

117

C₂×#41

C₂×#6157

yes

characteristic

118

#131

—

106

—

119

#131

—

106

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

C₂×#6157

yes

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

#117

—

126

M₃₂

—

106

—

127

M₃₂

—

106

—

128

M₃₂

—

106

—

129

M₃₂

—

106

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

106

—

132

C₂×C₁₆

—

106

—

133

C₂×C₁₆

—

106

—

134

C₂×C₁₆

—

106

—

135

#101

—

136

#100

—

137

#99

—

138

#97

—

106

—

139

C₄×C₈

—

106

—

140

#41

—

141

#41

—

142

C₂×Q₈

#11719940

yes

characteristic

143

C₂×D₄

D₅₃⋊C₈

yes

—

144

C₂×D₄

#11719940

yes

characteristic

145

C₂²×C₄

C₄⋊#6157

yes

—

146

C₂²×C₄

C₂²×#6157

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

106

—

150

M₁₆

—

106

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

106

—

158

C₂×C₈

—

106

—

159

C₄⋊C₄

—

106

—

160

C₂²⋊C₄

—

161

C₄²

—

106

—

162

C₁₆

—

106

—

163

C₁₆

—

212

—

164

C₁₆

—

212

—

165

C₂³

#13133284

yes

—

166

C₂³

C₂×D₅₃⋊C₈

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

106

—

173

C₂×C₄

#13133284

yes

—

174

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

175

C₂×C₄

D₄×#6157

yes

—

176

C₂×C₄

C₂×#11719940

yes

characteristic

177

C₂×C₄

#13133304

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

106

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×#13133284

yes

characteristic

187

C₂²

#16811650

yes

—

188

C₄

—

189

C₄

—

212

—

190

C₄

#16811570

yes

characteristic

191

C₄

#16811629

yes

characteristic

192

C₄

#16811654

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#20290659

yes

characteristic

	subgroup	G/H
center	C₄	#16811570
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#413	C₄
socle	C₂×C₅₃	#1850
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂²×#88322	C₂³
hypercenter	#413	#6157
hypocenter	C₅₃	#12915
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#16811570

commutator

C₂³×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₅₃×#413

C₄

socle

C₂×C₅₃

#1850

prime radical

C₂×#41×C₅₃

C₂×C₄

cosocle

C₂²×#88322

C₂³

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. #20291052	C₂	semidirect
4. #20289766	C₂	semidirect
116. C₅₃	#12915	semidirect

H ≅ G/N

type

2. #20291052

C₂

semidirect

4. #20289766

C₂

semidirect

116. C₅₃

#12915

semidirect

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

subgroup	index
#20291682	2
#20291052	2
#20290945	2
#20289766	2
#20274518	2
#12915	53

subgroup

index

Subgroups — #28311459

Conjugacy classes of subgroups

Complements

Maximal subgroups