Group #28303536 — subgroups

Subgroups — #28303536

Order 13568.

378 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 144 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	#20290647	C₂	1	1	yes	maximal, characteristic
2	#20290630	C₂	1	1	yes	maximal, characteristic
3	#20289545	C₂	1	4	yes	maximal
4	#97×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88324	—	2	2	no	—
6	C₂×C₄⋊#88324	C₂²	1	2	yes	—
7	C₂×C₄⋊#88324	C₂²	1	2	yes	—
8	#16811489	—	2	2	no	—
9	C₂×#13133128	C₄	1	1	yes	characteristic
10	C₂×#13133127	C₄	1	2	yes	—
11	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
12	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
14	C₂²×#88324	—	2	2	no	—
15	C₂²×#88324	—	2	2	no	—
16	C₄⋊#88324	—	2	4	no	—
17	C₄⋊#88324	—	2	4	no	—
18	C₄⋊#88324	—	4	4	no	—
19	C₂²×#88322	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
22	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
23	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
24	#13133128	C₈	1	4	yes	complemented
25	#13133127	—	2	4	no	—
26	C₈×Dic₅₃	—	2	2	no	—
27	C₈×Dic₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	#97×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
34	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×#88324	—	4	2	no	—
37	C₂×#88324	—	4	2	no	—
38	C₂×#88324	—	4	4	no	—
39	C₂×#88324	—	4	2	no	—
40	C₂×#88324	—	4	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
44	C₄²×C₅₃	C₂×C₈	1	2	yes	—
45	C₄²×C₅₃	M₁₆	1	2	yes	—
46	C₄×Dic₅₃	—	2	2	no	—
47	C₄×Dic₅₃	—	2	2	no	—
48	C₄×Dic₅₃	—	2	2	no	—
49	C₄×Dic₅₃	—	4	2	no	—
50	C₄×Dic₅₃	#41	1	2	yes	—
51	C₄×Dic₅₃	#41	1	2	yes	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	2	no	—
54	C₂×#88322	C₂×C₈	1	4	yes	—
55	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₅₃	#117	1	2	yes	—
60	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
63	C₂×Dic₅₃	—	2	2	no	—
64	C₂×Dic₅₃	—	2	2	no	—
65	C₂×Dic₅₃	—	2	2	no	—
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	#124	1	1	yes	characteristic
68	C₂×Dic₅₃	#124	1	1	yes	characteristic
69	C₂×Dic₅₃	#125	1	2	yes	—
70	C₂×Dic₅₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₅₃	#137	1	2	yes	—
72	#88324	—	8	2	no	—
73	#88324	—	8	2	no	—
74	C₈×C₅₃	—	2	4	no	—
75	#88322	—	2	4	no	—
76	#8786	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	#382	1	1	yes	characteristic
78	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₅₃	#423	1	1	yes	characteristic
80	C₂²×C₅₃	#428	1	2	yes	—
81	C₂²×C₅₃	#430	1	2	yes	—
82	C₄×C₅₃	—	2	4	no	—
83	C₄×C₅₃	#431	1	2	yes	—
84	C₄×C₅₃	#432	1	2	yes	—
85	Dic₅₃	—	2	2	no	—
86	Dic₅₃	—	4	2	no	—
87	Dic₅₃	#489	1	2	yes	—
88	Dic₅₃	#490	1	1	yes	characteristic
89	Dic₅₃	#491	1	1	yes	characteristic
90	#1842	—	53	1	no	—
91	#1831	—	53	1	no	—
92	C₄×#97	—	53	1	no	—
93	#1277	—	53	4	no	—
94	C₂×C₅₃	#1565	1	2	yes	—
95	C₂×C₅₃	#1569	1	2	yes	—
96	C₂×C₅₃	#1831	1	1	yes	characteristic
97	C₂×C₅₃	#1836	1	1	yes	characteristic
98	C₂×C₅₃	#1838	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	53	2	no	—
100	C₂×C₄⋊C₈	—	53	2	no	—
101	C₂×C₄⋊C₈	—	106	2	no	—
102	C₂×#97	—	53	1	no	—
103	C₂×#97	—	53	2	no	—
104	C₂×#97	#6157	1	1	yes	characteristic
105	C₂×C₄×C₈	—	53	2	no	—
106	C₄³	—	53	1	no	—
107	#336	—	106	2	no	—
108	C₅₃	#8786	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	53	2	no	—
110	C₂²×C₈	—	106	2	no	—
111	C₂²×C₈	—	106	2	no	—
112	C₂²×C₈	C₂×#6157	1	2	yes	—
113	C₂×C₄²	—	53	1	no	—
114	C₂×C₄²	—	53	2	no	—
115	C₂×C₄²	—	53	1	no	—
116	C₂×C₄²	—	53	2	no	—
117	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
118	C₄⋊C₈	—	106	4	no	—
119	C₄⋊C₈	—	106	4	no	—
120	C₄⋊C₈	—	212	4	no	—
121	#97	—	2	2	no	—
122	#97	—	53	4	no	—
123	#97	—	106	4	no	—
124	C₄×C₈	—	106	2	no	—
125	C₄×C₈	—	106	2	no	—
126	C₂²×C₄	—	53	1	no	—
127	C₂²×C₄	—	53	1	no	—
128	C₂²×C₄	—	53	2	no	—
129	C₂²×C₄	C₄⋊#6157	1	2	yes	—
130	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	53	4	no	—
134	C₂×C₈	—	106	2	no	—
135	C₂×C₈	—	106	2	no	—
136	C₂×C₈	—	212	2	no	—
137	C₂×C₈	—	212	2	no	—
138	C₂×C₈	—	212	4	no	—
139	C₂×C₈	—	212	2	no	—
140	C₂×C₈	—	212	2	no	—
141	C₂×C₈	#11719936	1	4	yes	—
142	C₄²	—	53	2	no	—
143	C₄²	—	53	2	no	—
144	C₄²	—	106	2	no	—
145	C₄²	—	106	2	no	—
146	C₄²	—	106	2	no	—
147	C₄²	—	212	2	no	—
148	C₄²	D₅₃⋊C₈	1	2	yes	—
149	C₄²	#11719940	1	2	yes	—
150	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	53	1	no	—
154	C₂×C₄	—	53	1	no	—
155	C₂×C₄	—	53	2	no	—
156	C₂×C₄	—	53	2	no	—
157	C₂×C₄	—	53	2	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	—	106	2	no	—
160	C₂×C₄	—	106	2	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	#13133284	1	4	yes	—
163	C₂×C₄	#13133296	1	2	yes	—
164	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
165	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
166	C₂×C₄	C₂×#11719936	1	2	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	106	4	no	—
169	C₈	—	424	2	no	—
170	C₈	—	424	2	no	—
171	C₂²	#16811567	1	1	yes	characteristic
172	C₂²	C₂×#13133284	1	1	yes	characteristic
173	C₂²	#16811627	1	1	yes	characteristic
174	C₂²	#16811634	1	2	yes	—
175	C₂²	#16811638	1	2	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	53	1	no	—
178	C₄	—	53	2	no	—
179	C₄	—	53	1	no	—
180	C₄	—	106	2	no	—
181	C₄	—	212	2	no	—
182	C₄	#16811642	1	2	yes	—
183	C₄	#16811644	1	2	yes	—
184	C₂	#20290268	1	2	yes	—
185	C₂	#20290280	1	2	yes	—
186	C₂	#20290632	1	1	yes	characteristic
187	C₂	#20290638	1	1	yes	characteristic
188	C₂	#20290641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290647

C₂

yes

maximal, characteristic

#20290630

C₂

yes

maximal, characteristic

#20289545

C₂

yes

maximal

#97×Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#88324

—

C₂×C₄⋊#88324

C₂²

yes

—

C₂×C₄⋊#88324

C₂²

yes

—

#16811489

—

C₂×#13133128

C₄

yes

characteristic

C₂×#13133127

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

—

C₄²×Dic₅₃

C₂²

yes

characteristic

C₂×#97×C₅₃

C₄

yes

characteristic

C₂²×#88324

—

C₂²×#88324

—

C₄⋊#88324

—

C₄⋊#88324

—

C₄⋊#88324

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

—

C₂×C₄×Dic₅₃

Q₈

yes

—

C₂×C₄×Dic₅₃

C₂³

yes

characteristic

#13133128

C₈

yes

complemented

#13133127

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#97×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₄⋊C₄

yes

—

C₂²×Dic₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

C₂×Q₈

yes

—

—

—

—

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

#41

yes

—

C₄×Dic₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₈

yes

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

—

C₂×Dic₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

#137

yes

—

#88324

—

#88324

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₄⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#430

yes

—

C₄×C₅₃

—

C₄×C₅₃

#431

yes

—

C₄×C₅₃

#432

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#489

yes

—

Dic₅₃

yes

Dic₅₃

yes

—

—

C₄×#97

—

#1277

—

C₂×C₅₃

#1565

yes

—

C₂×C₅₃

#1569

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

—

106

—

102

C₂×#97

—

103

C₂×#97

—

104

C₂×#97

#6157

yes

characteristic

105

C₂×C₄×C₈

—

106

C₄³

—

107

#336

—

106

—

108

C₅₃

#8786

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

110

C₂²×C₈

—

106

—

111

C₂²×C₈

—

106

—

112

C₂²×C₈

C₂×#6157

yes

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×#6157

yes

characteristic

118

C₄⋊C₈

—

106

—

119

C₄⋊C₈

—

106

—

120

C₄⋊C₈

—

212

—

121

#97

—

122

#97

—

123

#97

—

106

—

124

C₄×C₈

—

106

—

125

C₄×C₈

—

106

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₄⋊#6157

yes

—

130

C₂²×C₄

C₂²×#6157

yes

characteristic

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

106

—

135

C₂×C₈

—

106

—

136

C₂×C₈

—

212

—

137

C₂×C₈

—

212

—

138

C₂×C₈

—

212

—

139

C₂×C₈

—

212

—

140

C₂×C₈

—

212

—

141

C₂×C₈

#11719936

yes

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

106

—

145

C₄²

—

106

—

146

C₄²

—

106

—

147

C₄²

—

212

—

148

C₄²

D₅₃⋊C₈

yes

—

149

C₄²

#11719940

yes

—

150

C₂³

C₂×C₄⋊#6157

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

106

—

159

C₂×C₄

—

106

—

160

C₂×C₄

—

106

—

161

C₂×C₄

—

106

—

162

C₂×C₄

#13133284

yes

—

163

C₂×C₄

#13133296

yes

—

164

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

165

C₂×C₄

C₂×#11719940

yes

characteristic

166

C₂×C₄

C₂×#11719936

yes

—

167

C₈

—

168

C₈

—

106

—

169

C₈

—

424

—

170

C₈

—

424

—

171

C₂²

#16811567

yes

characteristic

172

C₂²

C₂×#13133284

yes

characteristic

173

C₂²

#16811627

yes

characteristic

174

C₂²

#16811634

yes

—

175

C₂²

#16811638

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

106

—

181

C₄

—

212

—

182

C₄

#16811642

yes

—

183

C₄

#16811644

yes

—

184

C₂

#20290268

yes

—

185

C₂

#20290280

yes

—

186

C₂

#20290632

yes

characteristic

187

C₂

#20290638

yes

characteristic

188

C₂

#20290641

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#97×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	#8786
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₂×#97×C₅₃

C₄

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₅₃

C₂³

C₂×#97

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
24. #13133128	C₈	semidirect
108. C₅₃	#8786	semidirect

H ≅ G/N

type

24. #13133128

C₈

semidirect

108. C₅₃

#8786

semidirect

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

subgroup	index
#20290647	2
#20290630	2
#20289545	2
#97×Dic₅₃	2
#8786	53

subgroup

index

#97×Dic₅₃

Subgroups — #28303536

Conjugacy classes of subgroups

Complements

Maximal subgroups