Group #27496234 — subgroups

Subgroups — #27496234

Order 13568.

446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 125 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	#20276182	C₂	1	3	yes	maximal, complemented
2	#20276173	C₂	1	3	yes	maximal
3	#20276114	C₂	1	1	yes	maximal, characteristic
4	#20275619	C₂	1	3	yes	maximal
5	#20275502	C₂	1	3	yes	maximal, complemented
6	#20274471	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2070	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719924	C₂²	1	1	yes	characteristic
9	#16810671	C₂²	1	3	yes	—
10	C₂×#13133152	C₂²	1	3	yes	—
11	#16810632	C₂²	1	3	yes	—
12	#16810629	C₂²	1	3	yes	—
13	C₂×#13133133	C₂²	1	3	yes	—
14	#16810581	C₂²	1	3	yes	—
15	#16810395	C₂²	1	3	yes	—
16	#16810394	C₂²	1	3	yes	—
17	#16810380	C₂²	1	3	yes	—
18	C₅₃×#430	C₂²	1	3	yes	—
19	C₂×#101×C₅₃	C₂²	1	3	yes	—
20	C₅₃×#338	C₂²	1	1	yes	characteristic
21	C₂²×D₂₁₂	C₂³	1	1	yes	characteristic
22	C₂×#11719924	—	2	3	no	—
23	C₂×#11719924	—	2	3	no	—
24	C₂×#11719917	C₂³	1	3	yes	—
25	C₂²×#88322	C₂³	1	1	yes	characteristic
26	#13133176	C₂³	1	3	yes	—
27	#13133152	—	2	3	no	—
28	#13133152	—	2	3	no	—
29	#13133152	—	4	3	no	—
30	#13133141	—	4	3	no	—
31	#13133139	—	4	3	no	—
32	#13133133	—	4	3	no	—
33	#13133127	—	2	3	no	—
34	C₈×Dic₅₃	—	2	3	no	—
35	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
37	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
38	#108×C₅₃	—	4	3	no	—
39	#101×C₅₃	—	4	3	no	—
40	C₂³×D₅₃	—	2	1	no	—
41	C₂×Q₈×C₅₃	—	2	3	no	—
42	C₂×Q₈×C₅₃	—	2	3	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₂₁₂	—	2	3	no	—
46	C₂×D₂₁₂	—	2	3	no	—
47	M₁₆×C₅₃	—	2	3	no	—
48	M₁₆×C₅₃	—	2	3	no	—
49	M₁₆×C₅₃	—	2	3	no	—
50	M₁₆×C₅₃	—	2	3	no	—
51	C₂×C₈×C₅₃	—	2	3	no	—
52	C₂×C₈×C₅₃	—	2	3	no	—
53	C₂²⋊Dic₅₃	—	2	3	no	—
54	#11719924	—	4	1	no	—
55	#11719924	—	4	3	no	—
56	C₄⋊Dic₅₃	—	2	3	no	—
57	C₄×Dic₅₃	—	2	3	no	—
58	#11719917	—	2	3	no	—
59	#11719917	—	2	3	no	—
60	C₂×#88322	—	2	3	no	—
61	C₂×#88322	#41	1	3	yes	—
62	C₂×#88322	#41	1	3	yes	—
63	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
64	C₂²×D₅₃	—	4	3	no	—
65	C₂²×D₅₃	—	16	1	no	—
66	Q₈×C₅₃	—	4	1	no	—
67	Q₈×C₅₃	—	4	3	no	—
68	C₂×C₄×C₅₃	—	4	3	no	—
69	C₂×C₄×C₅₃	C₂×#41	1	3	yes	—
70	C₂×C₄×C₅₃	H(5,𝔽₂)	1	3	yes	—
71	C₂×Dic₅₃	—	4	3	no	—
72	D₂₁₂	—	4	1	no	—
73	D₂₁₂	—	4	3	no	—
74	C₈×C₅₃	—	4	3	no	—
75	C₈×C₅₃	—	4	3	no	—
76	#88322	—	4	3	no	—
77	#88322	#126	1	1	yes	characteristic
78	#88322	#126	1	3	yes	—
79	#30236	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	4	1	no	—
81	C₂²×C₅₃	#542	1	3	yes	—
82	C₂×D₅₃	—	8	1	no	—
83	C₂×D₅₃	—	16	3	no	—
84	C₄×C₅₃	—	8	1	no	—
85	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
86	C₄×C₅₃	#540	1	3	yes	—
87	Dic₅₃	—	8	3	no	—
88	#2082	—	53	3	no	—
89	#2081	—	53	3	no	—
90	#2070	D₅₃	1	1	yes	characteristic, complemented
91	#2069	—	53	1	no	—
92	#1977	—	53	3	no	—
93	#1953	—	53	3	no	—
94	#1747	—	53	1	no	—
95	C₂×C₅₃	—	2	3	no	—
96	C₂×C₅₃	#2644	1	1	yes	characteristic
97	D₅₃	—	16	1	no	—
98	C₂²×SD₁₆	—	53	1	no	—
99	#430	—	53	3	no	—
100	#430	C₂×D₅₃	1	3	yes	—
101	C₂×#108	—	53	3	no	—
102	#419	—	53	3	no	—
103	#418	—	53	3	no	—
104	C₂×#101	C₂×D₅₃	1	3	yes	—
105	C₂×#100	—	53	3	no	—
106	#405	—	53	3	no	—
107	#344	—	53	3	no	—
108	#344	—	53	3	no	—
109	#338	C₂×D₅₃	1	1	yes	characteristic
110	H(3,ℤ/4)	—	53	3	no	—
111	C₅₃	#30236	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
113	C₂²×D₄	—	53	1	no	—
114	C₂×SD₁₆	—	106	3	no	—
115	C₂×SD₁₆	—	106	3	no	—
116	C₂×M₁₆	—	53	3	no	—
117	C₂×M₁₆	C₂²×D₅₃	1	3	yes	—
118	C₂×M₁₆	C₂²×D₅₃	1	3	yes	—
119	C₂²×C₈	—	53	1	no	—
120	#117	—	53	3	no	—
121	#108	—	4	3	no	—
122	#108	—	106	3	no	—
123	#108	—	106	3	no	—
124	#108	—	212	3	no	—
125	#103	—	212	3	no	—
126	#102	—	212	3	no	—
127	#101	—	4	3	no	—
128	#100	—	212	3	no	—
129	#97	—	106	3	no	—
130	C₄×C₈	—	106	3	no	—
131	C₂⁴	—	106	1	no	—
132	C₂×Q₈	—	2	3	no	—
133	C₂×Q₈	—	2	3	no	—
134	C₂×D₄	—	106	3	no	—
135	C₂×D₄	—	106	3	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
138	SD₁₆	—	212	1	no	—
139	SD₁₆	—	212	3	no	—
140	M₁₆	—	2	3	no	—
141	M₁₆	—	2	3	no	—
142	M₁₆	—	2	3	no	—
143	M₁₆	—	2	3	no	—
144	M₁₆	—	106	3	no	—
145	M₁₆	—	106	3	no	—
146	C₂×C₈	—	2	3	no	—
147	C₂×C₈	—	2	3	no	—
148	C₂×C₈	—	53	3	no	—
149	C₂×C₈	—	53	3	no	—
150	C₂×C₈	—	106	3	no	—
151	C₄⋊C₄	—	106	3	no	—
152	C₂²⋊C₄	—	106	3	no	—
153	C₄²	—	106	3	no	—
154	C₂³	—	848	1	no	—
155	C₂³	—	212	3	no	—
156	C₂³	#13133265	1	1	yes	characteristic
157	Q₈	—	4	1	no	—
158	Q₈	—	4	3	no	—
159	D₄	—	212	1	no	—
160	D₄	—	212	3	no	—
161	C₂×C₄	—	4	3	no	—
162	C₂×C₄	—	212	3	no	—
163	C₂×C₄	#41×D₅₃	1	3	yes	—
164	C₂×C₄	#13133262	1	3	yes	—
165	C₈	—	4	3	no	—
166	C₈	—	4	3	no	—
167	C₈	—	53	1	no	—
168	C₈	—	53	3	no	—
169	C₈	—	212	3	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	848	3	no	—
172	C₂²	—	424	1	no	—
173	C₂²	#16811243	1	3	yes	—
174	C₄	—	8	1	no	—
175	C₄	—	424	3	no	—
176	C₄	#126×D₅₃	1	1	yes	characteristic
177	C₄	#16811223	1	3	yes	—
178	C₂	—	2	3	no	—
179	C₂	—	848	1	no	—
180	C₂	#20280677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276182

C₂

yes

maximal, complemented

#20276173

C₂

yes

maximal

#20276114

C₂

yes

maximal, characteristic

#20275619

C₂

yes

maximal

#20275502

C₂

yes

maximal, complemented

#20274471

C₂

yes

maximal, characteristic, complemented

C₅₃×#2070

C₂

yes

maximal, characteristic, complemented

C₂²×#11719924

C₂²

yes

characteristic

#16810671

C₂²

yes

—

C₂×#13133152

C₂²

yes

—

#16810632

C₂²

yes

—

#16810629

C₂²

yes

—

C₂×#13133133

C₂²

yes

—

#16810581

C₂²

yes

—

#16810395

C₂²

yes

—

#16810394

C₂²

yes

—

#16810380

C₂²

yes

—

C₅₃×#430

C₂²

yes

—

C₂×#101×C₅₃

C₂²

yes

—

C₅₃×#338

C₂²

yes

characteristic

C₂²×D₂₁₂

C₂³

yes

—

—

C₂³

yes

—

C₂²×#88322

C₂³

yes

characteristic

#13133176

C₂³

yes

—

—

—

—

—

—

—

—

C₈×Dic₅₃

—

C₂²×Q₈×C₅₃

C₂³

yes

characteristic

C₂×M₁₆×C₅₃

C₂³

yes

—

C₂×M₁₆×C₅₃

C₂³

yes

—

#108×C₅₃

—

#101×C₅₃

—

C₂³×D₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

—

#11719924

—

#11719924

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

yes

—

C₂×#88322

#41

yes

—

C₂³×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×#41

yes

—

C₂×C₄×C₅₃

H(5,𝔽₂)

yes

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

—

—

yes

yes

—

#30236

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#542

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#126

yes

characteristic

C₄×C₅₃

#540

yes

—

Dic₅₃

—

#2082

—

#2081

—

#2070

D₅₃

yes

characteristic, complemented

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#2644

yes

characteristic

D₅₃

—

C₂²×SD₁₆

—

#430

—

100

#430

C₂×D₅₃

yes

—

101

C₂×#108

—

102

#419

—

103

#418

—

104

C₂×#101

C₂×D₅₃

yes

—

105

C₂×#100

—

106

#405

—

107

#344

—

108

#344

—

109

#338

C₂×D₅₃

yes

characteristic

110

H(3,ℤ/4)

—

111

C₅₃

#30236

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

C₂²×D₅₃

yes

characteristic

113

C₂²×D₄

—

114

C₂×SD₁₆

—

106

—

115

C₂×SD₁₆

—

106

—

116

C₂×M₁₆

—

117

C₂×M₁₆

C₂²×D₅₃

yes

—

118

C₂×M₁₆

C₂²×D₅₃

yes

—

119

C₂²×C₈

—

120

#117

—

121

#108

—

122

#108

—

106

—

123

#108

—

106

—

124

#108

—

212

—

125

#103

—

212

—

126

#102

—

212

—

127

#101

—

128

#100

—

212

—

129

#97

—

106

—

130

C₄×C₈

—

106

—

131

C₂⁴

—

106

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×D₄

—

106

—

135

C₂×D₄

—

106

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂³×D₅₃

yes

characteristic

138

SD₁₆

—

212

—

139

SD₁₆

—

212

—

140

M₁₆

—

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

106

—

145

M₁₆

—

106

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

106

—

151

C₄⋊C₄

—

106

—

152

C₂²⋊C₄

—

106

—

153

C₄²

—

106

—

154

C₂³

—

848

—

155

C₂³

—

212

—

156

C₂³

#13133265

yes

characteristic

157

Q₈

—

158

Q₈

—

159

D₄

—

212

—

160

D₄

—

212

—

161

C₂×C₄

—

162

C₂×C₄

—

212

—

163

C₂×C₄

#41×D₅₃

yes

—

164

C₂×C₄

#13133262

yes

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₈

—

212

—

170

C₂²

—

171

C₂²

—

848

—

172

C₂²

—

424

—

173

C₂²

#16811243

yes

—

174

C₄

—

175

C₄

—

424

—

176

C₄

#126×D₅₃

yes

characteristic

177

C₄

#16811223

yes

—

178

C₂

—

179

C₂

—

848

—

180

C₂

#20280677

yes

characteristic

	subgroup	G/H
center	C₂	#20280677
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2070	C₂
socle	C₂×C₅₃	#2644
prime radical	C₂²×D₂₁₂	C₂³
hypercenter	#2070	D₅₃
hypocenter	C₅₃	#30236
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2070

subgroup

G/H

center

C₂

#20280677

commutator, cosocle

C₂²×C₄×C₅₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting

C₅₃×#2070

C₂

socle

C₂×C₅₃

#2644

prime radical

C₂²×D₂₁₂

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2070

N	H ≅ G/N	type
1. #20276182	C₂	semidirect
5. #20275502	C₂	semidirect
6. #20274471	C₂	semidirect
7. C₅₃×#2070	C₂	semidirect
90. #2070	D₅₃	semidirect
111. C₅₃	#30236	semidirect

H ≅ G/N

type

1. #20276182

C₂

semidirect

5. #20275502

C₂

semidirect

6. #20274471

C₂

semidirect

7. C₅₃×#2070

C₂

semidirect

90. #2070

D₅₃

semidirect

111. C₅₃

#30236

semidirect

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

subgroup	index
#20276182	2
#20276173	2
#20276114	2
#20275619	2
#20275502	2
#20274471	2
C₅₃×#2070	2
#30236	53

subgroup

index

C₅₃×#2070

Subgroups — #27496234

Conjugacy classes of subgroups

Complements

Maximal subgroups