Group #27277121 — subgroups

Subgroups — #27277121

Order 13568.

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

normal subgroups only

This group has 336 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	#20283645	C₂	1	1	yes	maximal, characteristic
2	#20273473	C₂	1	2	yes	maximal
3	#20273452	C₂	1	2	yes	maximal
4	#20272628	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1309	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	1	yes	characteristic
7	#16810750	—	2	2	no	—
8	#16810747	—	2	2	no	—
9	#16810651	C₄	1	2	yes	—
10	#16810647	C₂²	1	2	yes	—
11	#16810581	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810375	—	2	2	no	—
14	#16810375	—	2	2	no	—
15	#16810370	—	2	2	no	—
16	#16810368	—	2	2	no	—
17	C₅₃×#423	C₂²	1	2	yes	—
18	C₅₃×#423	C₂²	1	1	yes	characteristic
19	C₂×#11719917	—	2	2	no	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	#13133217	D₄	1	2	yes	—
23	#13133209	Q₈	1	2	yes	—
24	#13133196	—	2	2	no	—
25	C₄×Dic₁₀₆	—	2	2	no	—
26	#13133176	C₂×C₄	1	2	yes	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	#13133146	—	2	4	no	—
30	#13133127	—	2	2	no	—
31	C₄×#88322	D₄	1	2	yes	—
32	C₄×#88322	D₄	1	2	yes	—
33	C₈×Dic₅₃	—	2	2	no	—
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	#41	1	2	yes	—
42	C₂×Dic₁₀₆	—	2	2	no	—
43	M₁₆×C₅₃	—	4	2	no	—
44	M₁₆×C₅₃	—	8	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₄²×C₅₃	C₂×D₄	1	2	yes	—
48	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²⋊Dic₅₃	—	2	2	no	—
51	C₄⋊Dic₅₃	—	2	2	no	—
52	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
53	#11719919	—	2	4	no	—
54	C₄×Dic₅₃	—	2	2	no	—
55	#11719917	—	8	2	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	4	1	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×#88322	C₂²⋊C₄	1	2	yes	—
61	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×C₅₃	#117	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#121	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×Dic₅₃	—	2	4	no	—
72	Dic₁₀₆	—	4	2	no	—
73	Dic₁₀₆	—	4	2	no	—
74	C₈×C₅₃	—	4	2	no	—
75	C₈×C₅₃	—	8	2	no	—
76	#88322	—	2	1	no	—
77	#88322	—	2	2	no	—
78	#88322	—	8	2	no	—
79	#88322	#101	1	2	yes	—
80	#8639	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	—	2	2	no	—
83	C₂²×C₅₃	#380	1	1	yes	characteristic
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
87	C₄×C₅₃	#413	1	1	yes	characteristic
88	C₄×C₅₃	#472	1	2	yes	—
89	C₄×C₅₃	#473	1	2	yes	—
90	Dic₅₃	—	4	4	no	—
91	#2960	—	53	1	no	—
92	#1567	—	53	2	no	—
93	#1561	—	53	2	no	—
94	#1309	—	53	1	no	—
95	#1309	D₅₃	1	1	yes	characteristic
96	C₂×C₅₃	—	4	1	no	—
97	C₂×C₅₃	#1801	1	1	yes	characteristic
98	C₂×C₅₃	#1816	1	2	yes	—
99	#533	—	53	1	no	—
100	#446	—	106	2	no	—
101	C₈×Q₈	—	106	2	no	—
102	#424	—	53	2	no	—
103	#423	—	53	2	no	—
104	#423	C₂×D₅₃	1	2	yes	—
105	#423	C₂×D₅₃	1	1	yes	characteristic
106	#405	—	53	2	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	#335	—	106	2	no	—
109	#335	—	106	2	no	—
110	#332	—	106	2	no	—
111	#332	—	106	2	no	—
112	C₅₃	#8639	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	106	2	no	—
115	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
116	C₂²×C₈	—	53	1	no	—
117	C₂²×C₈	—	53	1	no	—
118	C₄⋊Q₈	—	53	2	no	—
119	#125	—	53	2	no	—
120	C₂²⋊Q₈	—	106	2	no	—
121	C₄×Q₈	—	106	2	no	—
122	#117	—	53	2	no	—
123	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	106	4	no	—
128	C₄⋊C₈	—	106	2	no	—
129	C₄⋊C₈	—	106	2	no	—
130	#97	—	106	2	no	—
131	C₄×C₈	—	53	2	no	—
132	C₄×C₈	—	53	2	no	—
133	C₄×C₈	—	106	2	no	—
134	C₂×Q₈	—	106	2	no	—
135	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₄	#11719945	1	2	yes	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	8	2	no	—
139	M₁₆	—	424	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	1	no	—
144	C₂×C₈	—	53	2	no	—
145	C₂×C₈	—	106	2	no	—
146	C₂×C₈	—	212	1	no	—
147	C₂×C₈	—	212	2	no	—
148	C₄⋊C₄	—	53	4	no	—
149	C₄⋊C₄	—	106	2	no	—
150	C₄⋊C₄	—	106	4	no	—
151	C₂²⋊C₄	—	106	2	no	—
152	C₄²	—	106	2	no	—
153	C₄²	D₄×D₅₃	1	2	yes	—
154	C₄²	D₄×D₅₃	1	1	yes	characteristic
155	C₄²	Q₈×D₅₃	1	1	yes	characteristic
156	C₂³	#13133174	1	1	yes	characteristic
157	Q₈	—	212	2	no	—
158	Q₈	—	212	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	106	4	no	—
164	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
165	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
166	C₂×C₄	#13133190	1	2	yes	—
167	C₂×C₄	#13133195	1	2	yes	—
168	C₈	—	4	2	no	—
169	C₈	—	8	2	no	—
170	C₈	—	53	2	no	—
171	C₈	—	106	1	no	—
172	C₈	—	106	2	no	—
173	C₈	—	424	2	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	2	no	—
176	C₂²	#16810482	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	2	no	—
179	C₄	—	212	4	no	—
180	C₄	#101×D₅₃	1	1	yes	characteristic
181	C₄	#16810604	1	1	yes	characteristic
182	C₄	#16810892	1	2	yes	—
183	C₄	#16810897	1	2	yes	—
184	C₂	—	4	1	no	—
185	C₂	#20274710	1	1	yes	characteristic
186	C₂	#20274778	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283645

C₂

yes

maximal, characteristic

#20273473

C₂

yes

maximal

#20273452

C₂

yes

maximal

#20272628

C₂

yes

maximal, characteristic

C₅₃×#1309

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

—

—

C₄

yes

—

#16810647

C₂²

yes

—

#16810581

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

—

—

—

—

C₅₃×#423

C₂²

yes

—

C₅₃×#423

C₂²

yes

characteristic

C₂×#11719917

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

#13133217

D₄

yes

—

#13133209

Q₈

yes

—

#13133196

—

C₄×Dic₁₀₆

—

#13133176

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#88322

D₄

yes

—

C₈×Dic₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₄⋊C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂²⋊C₄

yes

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

yes

—

#8639

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#380

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#473

yes

—

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1801

yes

characteristic

C₂×C₅₃

#1816

yes

—

#533

—

100

#446

—

106

—

101

C₈×Q₈

—

106

—

102

#424

—

103

#423

—

104

#423

C₂×D₅₃

yes

—

105

#423

C₂×D₅₃

yes

characteristic

106

#405

—

107

C₂×C₄×C₈

—

108

#335

—

106

—

109

#335

—

106

—

110

#332

—

106

—

111

#332

—

106

—

112

C₅₃

#8639

yes

characteristic, complemented, Sylow

113

C₂×M₁₆

—

114

C₂×M₁₆

—

106

—

115

C₂×M₁₆

C₄×D₅₃

yes

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₄⋊Q₈

—

119

#125

—

120

C₂²⋊Q₈

—

106

—

121

C₄×Q₈

—

106

—

122

#117

—

123

C₂×C₄²

C₂²×D₅₃

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

106

—

128

C₄⋊C₈

—

106

—

129

C₄⋊C₈

—

106

—

130

#97

—

106

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

106

—

134

C₂×Q₈

—

106

—

135

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

136

C₂²×C₄

#11719945

yes

—

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

424

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

106

—

146

C₂×C₈

—

212

—

147

C₂×C₈

—

212

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

106

—

150

C₄⋊C₄

—

106

—

151

C₂²⋊C₄

—

106

—

152

C₄²

—

106

—

153

C₄²

D₄×D₅₃

yes

—

154

C₄²

D₄×D₅₃

yes

characteristic

155

C₄²

Q₈×D₅₃

yes

characteristic

156

C₂³

#13133174

yes

characteristic

157

Q₈

—

212

—

158

Q₈

—

212

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

106

—

164

C₂×C₄

C₂²⋊C₄×D₅₃

yes

characteristic

165

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

166

C₂×C₄

#13133190

yes

—

167

C₂×C₄

#13133195

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

106

—

172

C₈

—

106

—

173

C₈

—

424

—

174

C₂²

—

175

C₂²

—

176

C₂²

#16810482

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

212

—

180

C₄

#101×D₅₃

yes

characteristic

181

C₄

#16810604

yes

characteristic

182

C₄

#16810892

yes

—

183

C₄

#16810897

yes

—

184

C₂

—

185

C₂

#20274710

yes

characteristic

186

C₂

#20274778

yes

—

	subgroup	G/H
center	C₂²	#16810482
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1309	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1309	D₅₃
hypocenter	C₅₃	#8639
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#16810482

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1309

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
112. C₅₃	#8639	semidirect

H ≅ G/N

type

112. C₅₃

#8639

semidirect

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

subgroup	index
#20283645	2
#20273473	2
#20273452	2
#20272628	2
C₅₃×#1309	2
#8639	53

subgroup

index

C₅₃×#1309

Subgroups — #27277121

Conjugacy classes of subgroups

Complements

Maximal subgroups