Group #27303641 — subgroups

Subgroups — #27303641

Order 13568.

378 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20276473	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810439	C₂	1	2	yes	maximal
3	#20275631	C₂	1	2	yes	maximal
4	#20272809	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	1	yes	characteristic
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810632	—	2	2	no	—
9	C₂×#13133136	—	2	2	no	—
10	#16810578	C₂²	1	1	yes	characteristic
11	#16810441	—	2	2	no	—
12	#16810439	—	2	4	no	—
13	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₅₃	—	2	4	no	—
15	C₂×Q₃₂×C₅₃	—	2	2	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719917	—	2	2	no	—
18	C₂×#11719917	D₄	1	1	yes	characteristic
19	#13133176	D₄	1	1	yes	characteristic
20	#13133157	—	2	2	no	—
21	C₂×#11719908	—	2	1	no	—
22	#13133153	—	2	2	no	—
23	#13133153	—	2	2	no	—
24	#13133141	—	4	4	no	—
25	#13133136	—	4	4	no	—
26	#13133127	—	2	1	no	—
27	C₈×Dic₅₃	—	2	1	no	—
28	C₂²×Q₈×C₅₃	—	2	2	no	—
29	C₂×Q₁₆×C₅₃	—	2	2	no	—
30	C₂×Q₁₆×C₅₃	—	2	2	no	—
31	C₂×Q₁₆×C₅₃	—	2	2	no	—
32	C₂×Q₁₆×C₅₃	—	2	2	no	—
33	C₂×Q₁₆×C₅₃	—	2	2	no	—
34	C₂×Q₁₆×C₅₃	D₄	1	4	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	Q₃₂×C₅₃	—	2	8	no	—
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
39	C₂×Q₈×C₅₃	—	2	4	no	—
40	C₂×Q₈×C₅₃	—	4	2	no	—
41	C₂×Q₈×C₅₃	—	4	2	no	—
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂×Q₈×C₅₃	—	4	2	no	—
44	C₂×Q₈×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₅₃	—	2	4	no	—
48	Q₁₆×C₅₃	—	2	4	no	—
49	Q₁₆×C₅₃	—	4	2	no	—
50	Q₁₆×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₅₃	#41	1	2	yes	—
55	C₂²⋊Dic₅₃	—	2	1	no	—
56	C₄⋊Dic₅₃	D₈	1	2	yes	—
57	C₄×Dic₅₃	—	2	1	no	—
58	#11719917	—	2	2	no	—
59	#11719917	—	4	2	no	—
60	#11719917	—	4	2	no	—
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	—	4	2	no	—
63	C₁₆×C₅₃	—	2	2	no	—
64	#11719908	—	4	1	no	—
65	C₂³×C₅₃	#135	1	1	yes	characteristic
66	Q₈×C₅₃	—	4	4	no	—
67	Q₈×C₅₃	—	4	4	no	—
68	Q₈×C₅₃	—	8	2	no	—
69	Q₈×C₅₃	—	8	2	no	—
70	C₂×C₄×C₅₃	—	4	4	no	—
71	C₂×C₄×C₅₃	—	8	2	no	—
72	C₂×C₄×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#121	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×Dic₅₃	—	2	2	no	—
78	C₈×C₅₃	#120	1	1	yes	characteristic
79	C₈×C₅₃	#123	1	2	yes	—
80	C₈×C₅₃	#124	1	1	yes	characteristic
81	#88322	—	4	1	no	—
82	#88322	—	8	2	no	—
83	#13441	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	—	2	2	no	—
85	C₂²×C₅₃	#463	1	2	yes	—
86	C₂²×C₅₃	#466	1	1	yes	characteristic
87	C₄×C₅₃	—	8	4	no	—
88	C₄×C₅₃	#394	1	1	yes	characteristic
89	C₄×C₅₃	#459	1	2	yes	—
90	C₄×C₅₃	#469	1	1	yes	characteristic
91	Dic₅₃	—	4	2	no	—
92	C₂²×Q₃₂	D₅₃	1	1	yes	characteristic
93	#2151	—	53	1	no	—
94	C₂×#362	—	53	2	no	—
95	#1979	—	53	2	no	—
96	#1381	—	53	1	no	—
97	C₂×C₅₃	—	2	2	no	—
98	C₂×C₅₃	#2054	1	1	yes	characteristic
99	C₂×C₅₃	#2222	1	2	yes	—
100	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
101	C₂×Q₃₂	—	2	4	no	—
102	C₂×Q₃₂	—	2	2	no	—
103	C₂×M₃₂	—	53	1	no	—
104	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
105	C₂×#108	—	53	2	no	—
106	#419	—	106	2	no	—
107	C₂×#101	—	106	2	no	—
108	#405	—	53	1	no	—
109	C₄⋊C₁₆	—	106	2	no	—
110	#362	—	106	4	no	—
111	C₅₃	#13441	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	2	2	no	—
113	C₂×Q₁₆	—	2	2	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	C₂²⋊D₅₃	1	4	yes	—
119	C₂×M₁₆	—	53	1	no	—
120	C₂×M₁₆	—	106	2	no	—
121	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
122	#117	—	53	1	no	—
123	Q₃₂	—	2	8	no	—
124	M₃₂	—	106	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	106	1	no	—
127	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
128	#108	—	106	2	no	—
129	#108	—	106	2	no	—
130	#103	—	212	4	no	—
131	#101	—	212	4	no	—
132	#97	—	106	1	no	—
133	C₄×C₈	—	106	1	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
142	Q₁₆	—	2	4	no	—
143	Q₁₆	—	2	4	no	—
144	Q₁₆	—	4	2	no	—
145	Q₁₆	—	4	2	no	—
146	M₁₆	—	106	2	no	—
147	M₁₆	—	212	2	no	—
148	M₁₆	—	212	2	no	—
149	C₂×C₈	—	106	1	no	—
150	C₂×C₈	—	212	2	no	—
151	C₂×C₈	#11719947	1	1	yes	characteristic
152	C₂×C₈	#11719947	1	2	yes	—
153	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
154	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	C₄⋊C₄	—	53	2	no	—
156	C₂²⋊C₄	—	106	1	no	—
157	C₄²	—	106	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	212	1	no	—
160	C₂³	#13133236	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	Q₈	—	4	4	no	—
163	Q₈	—	8	2	no	—
164	Q₈	—	8	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	106	2	no	—
169	C₂×C₄	#13133188	1	1	yes	characteristic
170	C₂×C₄	#13133189	1	2	yes	—
171	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
172	C₂×C₄	#13133241	1	2	yes	—
173	C₈	—	424	2	no	—
174	C₈	—	212	1	no	—
175	C₈	#13133186	1	1	yes	characteristic
176	C₈	#13133198	1	2	yes	—
177	C₈	#13133204	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#16810836	1	2	yes	—
180	C₂²	#16810856	1	1	yes	characteristic
181	C₄	—	8	4	no	—
182	C₄	—	212	2	no	—
183	C₄	#16810537	1	1	yes	characteristic
184	C₄	#16810816	1	2	yes	—
185	C₄	#16810873	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	#20276023	1	1	yes	characteristic
188	C₂	#20276837	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276473

C₂

yes

maximal, characteristic

C₂×#16810439

C₂

yes

maximal

#20275631

C₂

yes

maximal

#20272809

C₂

yes

maximal, characteristic

C₂²×Q₃₂×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

characteristic

C₂×#13133153

C₂²

yes

—

#16810632

—

C₂×#13133136

—

#16810578

C₂²

yes

characteristic

#16810441

—

#16810439

—

C₂²×Q₁₆×C₅₃

C₂²

yes

—

C₂×Q₃₂×C₅₃

—

C₂×Q₃₂×C₅₃

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#11719917

—

C₂×#11719917

D₄

yes

characteristic

#13133176

D₄

yes

—

—

—

—

—

—

—

C₈×Dic₅₃

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

Q₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

D₈

yes

—

C₄×Dic₅₃

—

—

—

—

—

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

#135

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₈×C₅₃

#120

yes

characteristic

C₈×C₅₃

#123

yes

—

C₈×C₅₃

yes

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#463

yes

—

C₂²×C₅₃

#466

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#394

yes

characteristic

C₄×C₅₃

#459

yes

—

C₄×C₅₃

#469

yes

characteristic

Dic₅₃

—

C₂²×Q₃₂

D₅₃

yes

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#2054

yes

characteristic

C₂×C₅₃

#2222

yes

—

100

C₂²×Q₁₆

C₂×D₅₃

yes

—

101

C₂×Q₃₂

—

102

C₂×Q₃₂

—

103

C₂×M₃₂

—

104

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

105

C₂×#108

—

106

#419

—

106

—

107

C₂×#101

—

106

—

108

#405

—

109

C₄⋊C₁₆

—

106

—

110

#362

—

106

—

111

C₅₃

#13441

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

C₂²⋊D₅₃

yes

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

106

—

121

C₂²×C₈

C₂²×D₅₃

yes

characteristic

122

#117

—

123

Q₃₂

—

124

M₃₂

—

106

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

106

—

127

C₂×C₁₆

C₂²⋊D₅₃

yes

—

128

#108

—

106

—

129

#108

—

106

—

130

#103

—

212

—

131

#101

—

212

—

132

#97

—

106

—

133

C₄×C₈

—

106

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₅₃

yes

characteristic

142

Q₁₆

—

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

146

M₁₆

—

106

—

147

M₁₆

—

212

—

148

M₁₆

—

212

—

149

C₂×C₈

—

106

—

150

C₂×C₈

—

212

—

151

C₂×C₈

#11719947

yes

characteristic

152

C₂×C₈

#11719947

yes

—

153

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

154

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

106

—

157

C₄²

—

106

—

158

C₁₆

—

159

C₁₆

—

212

—

160

C₂³

#13133236

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

106

—

169

C₂×C₄

#13133188

yes

characteristic

170

C₂×C₄

#13133189

yes

—

171

C₂×C₄

D₈×D₅₃

yes

characteristic

172

C₂×C₄

#13133241

yes

—

173

C₈

—

424

—

174

C₈

—

212

—

175

C₈

#13133186

yes

characteristic

176

C₈

#13133198

yes

—

177

C₈

#13133204

yes

characteristic

178

C₂²

—

179

C₂²

#16810836

yes

—

180

C₂²

#16810856

yes

characteristic

181

C₄

—

182

C₄

—

212

—

183

C₄

#16810537

yes

characteristic

184

C₄

#16810816

yes

—

185

C₄

#16810873

yes

characteristic

186

C₂

—

187

C₂

#20276023

yes

characteristic

188

C₂

#20276837

yes

—

	subgroup	G/H
center	C₂²	#16810856
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	#466
prime radical	C₂³×C₅₃	#135
hypercenter	C₂²×Q₃₂	D₅₃
hypocenter	C₅₃	#13441
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂²

#16810856

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂²×Q₃₂×C₅₃

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂²×Q₃₂

D₅₃

hypocenter

C₅₃

#13441

LCS

G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
111. C₅₃	#13441	semidirect

H ≅ G/N

type

111. C₅₃

#13441

semidirect

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

subgroup	index
#20276473	2
C₂×#16810439	2
#20275631	2
#20272809	2
C₂²×Q₃₂×C₅₃	2
#13441	53

subgroup

index

C₂²×Q₃₂×C₅₃

Subgroups — #27303641

Conjugacy classes of subgroups

Complements

Maximal subgroups