Group #27282856 — subgroups

Subgroups — #27282856

Order 13568.

260 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20273504	C₂	1	2	yes	maximal
2	#20273471	C₂	1	1	yes	maximal, characteristic
3	#20273004	C₂	1	1	yes	maximal, characteristic
4	#20272606	C₂	1	2	yes	maximal
5	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	1	yes	characteristic
7	#16810647	C₂²	1	2	yes	—
8	C₄×#11719917	C₂²	1	1	yes	characteristic
9	#16810376	—	2	2	no	—
10	#16810374	—	2	2	no	—
11	#16810374	—	2	1	no	—
12	#16810374	—	2	1	no	—
13	#16810345	—	2	1	no	—
14	#16810344	—	2	1	no	—
15	C₅₃×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	C₄	1	2	yes	—
18	C₅₃×#334	C₄	1	2	yes	—
19	C₂×#11719917	—	2	1	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	D₄	1	2	yes	—
24	C₂²⋊#88322	—	2	1	no	—
25	#13133128	Q₈	1	2	yes	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²⋊C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	4	1	no	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	#11719917	—	4	2	no	—
53	#11719917	—	8	2	no	—
54	C₂×#88322	—	2	2	no	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂³×C₅₃	#125	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	2	1	no	—
71	C₈×C₅₃	—	4	4	no	—
72	C₈×C₅₃	#108	1	2	yes	—
73	#88322	—	4	2	no	—
74	#88322	—	4	2	no	—
75	#88322	—	8	2	no	—
76	#9635	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	1	no	—
78	C₂²×C₅₃	—	2	1	no	—
79	C₂²×C₅₃	—	2	1	no	—
80	C₂²×C₅₃	#389	1	1	yes	characteristic
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
85	C₄×C₅₃	#430	1	1	yes	characteristic
86	C₄×C₅₃	#444	1	2	yes	—
87	C₄×C₅₃	#471	1	2	yes	—
88	#1574	—	53	2	no	—
89	#1566	—	53	1	no	—
90	#1564	D₅₃	1	1	yes	characteristic
91	#1451	—	53	1	no	—
92	#1299	—	53	2	no	—
93	C₂×C₅₃	—	4	1	no	—
94	C₂×C₅₃	#1946	1	1	yes	characteristic
95	C₂×C₅₃	#1950	1	1	yes	characteristic
96	C₂×C₅₃	#1951	1	1	yes	characteristic
97	#431	—	53	1	no	—
98	#431	C₂×D₅₃	1	2	yes	—
99	#423	—	53	2	no	—
100	C₄×M₁₆	—	53	1	no	—
101	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
102	#335	—	106	2	no	—
103	#335	Dic₅₃	1	2	yes	—
104	#334	—	106	2	no	—
105	#334	—	106	1	no	—
106	#334	—	106	1	no	—
107	#334	Dic₅₃	1	2	yes	—
108	#322	—	106	1	no	—
109	#322	—	106	1	no	—
110	C₅₃	#9635	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₁₆	—	106	1	no	—
112	C₂×M₁₆	—	106	2	no	—
113	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
114	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
116	C₄⋊C₈	—	53	2	no	—
117	C₄⋊C₈	—	106	2	no	—
118	C₄⋊C₈	—	106	2	no	—
119	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	106	1	no	—
122	#97	—	53	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	106	1	no	—
125	C₄×C₈	—	106	1	no	—
126	C₄×C₈	D₂₁₂	1	1	yes	characteristic
127	C₄×C₈	D₂₁₂	1	1	yes	characteristic
128	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
129	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
131	C₂²×C₄	#11719947	1	1	yes	characteristic
132	C₂²×C₄	#11719949	1	1	yes	characteristic
133	M₁₆	—	424	2	no	—
134	M₁₆	—	212	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	106	2	no	—
139	C₂×C₈	—	106	2	no	—
140	C₂×C₈	—	212	2	no	—
141	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
145	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
146	C₄²	D₄×D₅₃	1	1	yes	characteristic
147	C₄²	Q₈×D₅₃	1	1	yes	characteristic
148	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
149	C₂³	#13133208	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
155	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
156	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
157	C₂×C₄	#13133192	1	1	yes	characteristic
158	C₂×C₄	#13133197	1	1	yes	characteristic
159	C₂×C₄	#13133216	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	4	4	no	—
164	C₈	—	424	2	no	—
165	C₈	—	212	2	no	—
166	C₈	—	212	2	no	—
167	C₈	#13133153	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#16810522	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	C₂×#13133153	1	1	yes	characteristic
176	C₄	#16810672	1	1	yes	characteristic
177	C₄	#16810745	1	2	yes	—
178	C₄	#16810887	1	2	yes	—
179	C₂	—	4	1	no	—
180	C₂	#20275469	1	1	yes	characteristic
181	C₂	#20275491	1	1	yes	characteristic
182	C₂	#20275496	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273504

C₂

yes

maximal

#20273471

C₂

yes

maximal, characteristic

#20273004

C₂

yes

maximal, characteristic

#20272606

C₂

yes

maximal

C₅₃×#1564

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

characteristic

#16810647

C₂²

yes

—

C₄×#11719917

C₂²

yes

—

—

—

—

—

—

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

C₄

yes

—

C₅₃×#334

C₄

yes

—

—

—

—

—

D₄

yes

—

C₂²⋊#88322

—

#13133128

Q₈

yes

—

C₄×#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

C₂²×C₄×C₅₃

yes

C₂²×C₄×C₅₃

yes

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

—

—

—

—

C₂³×C₅₃

#125

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#108

yes

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#389

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#471

yes

—

#1574

—

#1566

—

#1564

D₅₃

yes

characteristic

#1451

—

#1299

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×D₅₃

yes

—

#423

—

100

C₄×M₁₆

—

101

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

102

#335

—

106

—

103

#335

Dic₅₃

yes

—

104

#334

—

106

—

105

#334

—

106

—

106

#334

—

106

—

107

#334

Dic₅₃

yes

—

108

#322

—

106

—

109

#322

—

106

—

110

C₅₃

#9635

yes

characteristic, complemented, Sylow

111

C₂×M₁₆

—

106

—

112

C₂×M₁₆

—

106

—

113

C₂²×C₈

Dic₁₀₆

yes

characteristic

114

C₂²×C₈

Dic₁₀₆

yes

characteristic

115

C₂×C₄²

C₂²×D₅₃

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

106

—

118

C₄⋊C₈

—

106

—

119

C₄⋊C₈

C₂×Dic₅₃

yes

—

120

C₂²⋊C₈

—

121

C₂²⋊C₈

—

106

—

122

#97

—

123

C₄×C₈

—

124

C₄×C₈

—

106

—

125

C₄×C₈

—

106

—

126

C₄×C₈

D₂₁₂

yes

characteristic

127

C₄×C₈

D₂₁₂

yes

characteristic

128

C₄×C₈

C₂×Dic₅₃

yes

characteristic

129

C₄×C₈

C₂×Dic₅₃

yes

characteristic

130

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

131

C₂²×C₄

#11719947

yes

characteristic

132

C₂²×C₄

#11719949

yes

characteristic

133

M₁₆

—

424

—

134

M₁₆

—

212

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

106

—

139

C₂×C₈

—

106

—

140

C₂×C₈

—

212

—

141

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

142

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

143

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

144

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

145

C₄²

C₂×D₂₁₂

yes

characteristic

146

C₄²

D₄×D₅₃

yes

characteristic

147

C₄²

Q₈×D₅₃

yes

characteristic

148

C₄²

C₂²×Dic₅₃

yes

characteristic

149

C₂³

#13133208

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

155

C₂×C₄

D₄×Dic₅₃

yes

characteristic

156

C₂×C₄

Q₈×Dic₅₃

yes

characteristic

157

C₂×C₄

#13133192

yes

characteristic

158

C₂×C₄

#13133197

yes

characteristic

159

C₂×C₄

#13133216

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

424

—

165

C₈

—

212

—

166

C₈

—

212

—

167

C₈

#13133153

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#16810522

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

C₂×#13133153

yes

characteristic

176

C₄

#16810672

yes

characteristic

177

C₄

#16810745

yes

—

178

C₄

#16810887

yes

—

179

C₂

—

180

C₂

#20275469

yes

characteristic

181

C₂

#20275491

yes

characteristic

182

C₂

#20275496

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	#125
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#9635
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

subgroup

G/H

center

C₂×C₄

#13133192

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1564

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

N	H ≅ G/N	type
110. C₅₃	#9635	semidirect

H ≅ G/N

type

110. C₅₃

#9635

semidirect

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

subgroup	index
#20273504	2
#20273471	2
#20273004	2
#20272606	2
C₅₃×#1564	2
#9635	53

subgroup

index

C₅₃×#1564

Subgroups — #27282856

Conjugacy classes of subgroups

Complements

Maximal subgroups