Group #27496229 — subgroups

Subgroups — #27496229

Order 13568.

446 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20276184	C₂	1	3	yes	maximal, complemented
2	#20276177	C₂	1	3	yes	maximal
3	#20276117	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276114	C₂	1	1	yes	maximal, characteristic
5	#20275618	C₂	1	3	yes	maximal
6	#20275505	C₂	1	3	yes	maximal, complemented
7	C₅₃×#1747	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719913	C₂²	1	1	yes	characteristic
9	#16810671	C₂²	1	3	yes	—
10	#16810671	C₂²	1	3	yes	—
11	C₂×#13133153	C₂²	1	3	yes	—
12	#16810631	C₂²	1	3	yes	—
13	#16810627	C₂²	1	3	yes	—
14	C₂×#13133135	C₂²	1	3	yes	—
15	C₂×#13133133	C₂²	1	3	yes	—
16	#16810393	C₂²	1	6	yes	—
17	C₅₃×#405	C₂²	1	3	yes	—
18	C₅₃×#338	C₂²	1	1	yes	characteristic
19	C₅₃×H(3,ℤ/4)	C₂²	1	3	yes	—
20	C₂²×Dic₁₀₆	C₂³	1	1	yes	characteristic
21	C₂²×D₂₁₂	C₂³	1	1	yes	characteristic
22	C₂×#11719913	—	2	3	no	—
23	C₂×#11719913	—	2	3	no	—
24	C₂×#11719917	C₂³	1	3	yes	—
25	C₂×#11719917	C₂³	1	3	yes	—
26	#13133153	—	2	3	no	—
27	#13133153	—	2	3	no	—
28	#13133152	—	4	3	no	—
29	#13133152	—	4	3	no	—
30	#13133140	—	4	3	no	—
31	#13133137	—	4	3	no	—
32	#13133135	—	4	3	no	—
33	#13133133	—	4	3	no	—
34	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	#117×C₅₃	C₂³	1	3	yes	—
37	#97×C₅₃	—	2	3	no	—
38	C₄×C₈×C₅₃	—	2	3	no	—
39	C₂³×D₅₃	—	2	1	no	—
40	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	1	no	—
42	C₂×D₂₁₂	—	2	3	no	—
43	C₂×D₂₁₂	—	2	3	no	—
44	C₂×Dic₁₀₆	—	2	3	no	—
45	C₂×Dic₁₀₆	—	2	3	no	—
46	M₁₆×C₅₃	—	2	3	no	—
47	M₁₆×C₅₃	—	2	3	no	—
48	C₂×C₈×C₅₃	—	2	3	no	—
49	C₂×C₈×C₅₃	#41	1	3	yes	—
50	C₂×C₈×C₅₃	#41	1	3	yes	—
51	C₄⋊C₄×C₅₃	—	2	3	no	—
52	C₂²⋊C₄×C₅₃	—	2	3	no	—
53	C₄²×C₅₃	—	2	3	no	—
54	#11719917	—	2	3	no	—
55	#11719917	—	2	3	no	—
56	#11719917	—	2	3	no	—
57	#11719917	—	2	3	no	—
58	C₂×#88322	—	2	3	no	—
59	C₂×#88322	—	2	3	no	—
60	#11719913	—	4	1	no	—
61	#11719913	—	4	3	no	—
62	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
63	C₂²×D₅₃	—	4	3	no	—
64	C₂²×D₅₃	—	16	1	no	—
65	C₂×C₄×C₅₃	—	4	3	no	—
66	C₂×C₄×C₅₃	C₂×#41	1	3	yes	—
67	C₂×C₄×C₅₃	H(5,𝔽₂)	1	3	yes	—
68	C₂×Dic₅₃	—	4	3	no	—
69	D₂₁₂	—	4	1	no	—
70	D₂₁₂	—	4	3	no	—
71	Dic₁₀₆	—	4	1	no	—
72	Dic₁₀₆	—	4	3	no	—
73	C₈×C₅₃	—	4	3	no	—
74	C₈×C₅₃	#126	1	1	yes	characteristic
75	C₈×C₅₃	#126	1	3	yes	—
76	#88322	—	4	3	no	—
77	#88322	—	4	3	no	—
78	#30236	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	4	1	no	—
80	C₂²×C₅₃	#542	1	3	yes	—
81	C₂×D₅₃	—	8	1	no	—
82	C₂×D₅₃	—	16	3	no	—
83	C₄×C₅₃	—	8	3	no	—
84	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
85	C₄×C₅₃	#540	1	3	yes	—
86	Dic₅₃	—	8	1	no	—
87	#2082	—	53	3	no	—
88	#2081	—	53	3	no	—
89	#2070	—	53	1	no	—
90	#2069	—	53	1	no	—
91	#1977	—	53	3	no	—
92	#1953	—	53	3	no	—
93	#1747	D₅₃	1	1	yes	characteristic, complemented
94	C₂×C₅₃	—	2	3	no	—
95	C₂×C₅₃	#2644	1	1	yes	characteristic
96	D₅₃	—	16	1	no	—
97	C₂²×SD₁₆	—	53	1	no	—
98	#430	—	53	3	no	—
99	#430	—	53	3	no	—
100	C₂×#108	—	53	3	no	—
101	#419	—	53	3	no	—
102	#418	—	53	3	no	—
103	C₂×#101	—	53	3	no	—
104	C₂×#100	—	53	3	no	—
105	#405	C₂×D₅₃	1	3	yes	—
106	#344	—	53	6	no	—
107	#338	C₂×D₅₃	1	1	yes	characteristic
108	H(3,ℤ/4)	C₂×D₅₃	1	3	yes	—
109	C₅₃	#30236	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	53	1	no	—
111	C₂²×D₄	—	53	1	no	—
112	C₂×SD₁₆	—	106	3	no	—
113	C₂×SD₁₆	—	106	3	no	—
114	C₂×M₁₆	—	53	3	no	—
115	C₂×M₁₆	—	53	3	no	—
116	C₂×M₁₆	C₂²×D₅₃	1	3	yes	—
117	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	#117	C₂²×D₅₃	1	3	yes	—
119	#108	—	106	3	no	—
120	#108	—	106	3	no	—
121	#108	—	212	3	no	—
122	#108	—	212	3	no	—
123	#103	—	212	3	no	—
124	#102	—	212	3	no	—
125	#101	—	212	3	no	—
126	#100	—	212	3	no	—
127	#97	—	2	3	no	—
128	C₄×C₈	—	2	3	no	—
129	C₂⁴	—	106	1	no	—
130	C₂×Q₈	—	106	3	no	—
131	C₂×Q₈	—	106	3	no	—
132	C₂×D₄	—	106	3	no	—
133	C₂×D₄	—	106	3	no	—
134	C₂²×C₄	—	106	1	no	—
135	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
136	SD₁₆	—	212	1	no	—
137	SD₁₆	—	212	3	no	—
138	M₁₆	—	2	3	no	—
139	M₁₆	—	2	3	no	—
140	M₁₆	—	106	3	no	—
141	M₁₆	—	106	3	no	—
142	M₁₆	—	106	3	no	—
143	M₁₆	—	106	3	no	—
144	C₂×C₈	—	2	3	no	—
145	C₂×C₈	—	106	3	no	—
146	C₂×C₈	—	106	3	no	—
147	C₂×C₈	#11719945	1	3	yes	—
148	C₂×C₈	#11719945	1	3	yes	—
149	C₄⋊C₄	—	2	3	no	—
150	C₂²⋊C₄	—	2	3	no	—
151	C₄²	—	2	3	no	—
152	C₂³	—	848	1	no	—
153	C₂³	—	212	3	no	—
154	C₂³	#13133264	1	1	yes	characteristic
155	Q₈	—	212	1	no	—
156	Q₈	—	212	3	no	—
157	D₄	—	212	1	no	—
158	D₄	—	212	3	no	—
159	C₂×C₄	—	4	3	no	—
160	C₂×C₄	—	212	3	no	—
161	C₂×C₄	C₂×#11719945	1	3	yes	—
162	C₂×C₄	#13133263	1	3	yes	—
163	C₈	—	4	3	no	—
164	C₈	—	212	3	no	—
165	C₈	—	212	3	no	—
166	C₈	#13133211	1	1	yes	characteristic
167	C₈	#13133211	1	3	yes	—
168	C₂²	—	4	1	no	—
169	C₂²	—	848	3	no	—
170	C₂²	—	424	1	no	—
171	C₂²	#16811244	1	3	yes	—
172	C₄	—	8	3	no	—
173	C₄	—	424	1	no	—
174	C₄	C₂×#13133211	1	1	yes	characteristic
175	C₄	#16811228	1	3	yes	—
176	C₂	—	2	3	no	—
177	C₂	—	848	1	no	—
178	C₂	#20280675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276184

C₂

yes

maximal, complemented

#20276177

C₂

yes

maximal

#20276117

C₂

yes

maximal, characteristic, complemented

#20276114

C₂

yes

maximal, characteristic

#20275618

C₂

yes

maximal

#20275505

C₂

yes

maximal, complemented

C₅₃×#1747

C₂

yes

maximal, characteristic, complemented

C₂²×#11719913

C₂²

yes

characteristic

#16810671

C₂²

yes

—

#16810671

C₂²

yes

—

C₂×#13133153

C₂²

yes

—

#16810631

C₂²

yes

—

#16810627

C₂²

yes

—

C₂×#13133135

C₂²

yes

—

C₂×#13133133

C₂²

yes

—

#16810393

C₂²

yes

—

C₅₃×#405

C₂²

yes

—

C₅₃×#338

C₂²

yes

characteristic

C₅₃×H(3,ℤ/4)

C₂²

yes

—

C₂²×Dic₁₀₆

C₂³

yes

characteristic

C₂²×D₂₁₂

C₂³

yes

—

—

C₂³

yes

—

C₂×#11719917

C₂³

yes

—

—

—

—

—

—

—

—

—

C₂×M₁₆×C₅₃

C₂³

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

C₂³

yes

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

—

—

—

—

—

—

—

—

C₂³×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×#41

yes

—

C₂×C₄×C₅₃

H(5,𝔽₂)

yes

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#126

yes

characteristic

C₈×C₅₃

#126

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#542

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#126

yes

characteristic

C₄×C₅₃

#540

yes

—

Dic₅₃

—

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

#2644

yes

characteristic

D₅₃

—

C₂²×SD₁₆

—

#430

—

#430

—

100

C₂×#108

—

101

#419

—

102

#418

—

103

C₂×#101

—

104

C₂×#100

—

105

#405

C₂×D₅₃

yes

—

106

#344

—

107

#338

C₂×D₅₃

yes

characteristic

108

H(3,ℤ/4)

C₂×D₅₃

yes

—

109

C₅₃

#30236

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂²×D₄

—

112

C₂×SD₁₆

—

106

—

113

C₂×SD₁₆

—

106

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

C₂²×D₅₃

yes

—

117

C₂²×C₈

C₂²×D₅₃

yes

characteristic

118

#117

C₂²×D₅₃

yes

—

119

#108

—

106

—

120

#108

—

106

—

121

#108

—

212

—

122

#108

—

212

—

123

#103

—

212

—

124

#102

—

212

—

125

#101

—

212

—

126

#100

—

212

—

127

#97

—

128

C₄×C₈

—

129

C₂⁴

—

106

—

130

C₂×Q₈

—

106

—

131

C₂×Q₈

—

106

—

132

C₂×D₄

—

106

—

133

C₂×D₄

—

106

—

134

C₂²×C₄

—

106

—

135

C₂²×C₄

C₂³×D₅₃

yes

characteristic

136

SD₁₆

—

212

—

137

SD₁₆

—

212

—

138

M₁₆

—

139

M₁₆

—

140

M₁₆

—

106

—

141

M₁₆

—

106

—

142

M₁₆

—

106

—

143

M₁₆

—

106

—

144

C₂×C₈

—

145

C₂×C₈

—

106

—

146

C₂×C₈

—

106

—

147

C₂×C₈

#11719945

yes

—

148

C₂×C₈

#11719945

yes

—

149

C₄⋊C₄

—

150

C₂²⋊C₄

—

151

C₄²

—

152

C₂³

—

848

—

153

C₂³

—

212

—

154

C₂³

#13133264

yes

characteristic

155

Q₈

—

212

—

156

Q₈

—

212

—

157

D₄

—

212

—

158

D₄

—

212

—

159

C₂×C₄

—

160

C₂×C₄

—

212

—

161

C₂×C₄

C₂×#11719945

yes

—

162

C₂×C₄

#13133263

yes

—

163

C₈

—

164

C₈

—

212

—

165

C₈

—

212

—

166

C₈

#13133211

yes

characteristic

167

C₈

#13133211

yes

—

168

C₂²

—

169

C₂²

—

848

—

170

C₂²

—

424

—

171

C₂²

#16811244

yes

—

172

C₄

—

173

C₄

—

424

—

174

C₄

C₂×#13133211

yes

characteristic

175

C₄

#16811228

yes

—

176

C₂

—

177

C₂

—

848

—

178

C₂

#20280675

yes

characteristic

	subgroup	G/H
center	C₂	#20280675
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1747	C₂
socle	C₂×C₅₃	#2644
prime radical	C₂²×D₂₁₂	C₂³
hypercenter	#1747	D₅₃
hypocenter	C₅₃	#30236
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747

subgroup

G/H

center

C₂

#20280675

commutator, cosocle

C₂²×C₄×C₅₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting

C₅₃×#1747

C₂

socle

C₂×C₅₃

#2644

prime radical

C₂²×D₂₁₂

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747

N	H ≅ G/N	type
1. #20276184	C₂	semidirect
3. #20276117	C₂	semidirect
6. #20275505	C₂	semidirect
7. C₅₃×#1747	C₂	semidirect
93. #1747	D₅₃	semidirect
109. C₅₃	#30236	semidirect

H ≅ G/N

type

1. #20276184

C₂

semidirect

3. #20276117

C₂

semidirect

6. #20275505

C₂

semidirect

7. C₅₃×#1747

C₂

semidirect

93. #1747

D₅₃

semidirect

109. C₅₃

#30236

semidirect

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

subgroup	index
#20276184	2
#20276177	2
#20276117	2
#20276114	2
#20275618	2
#20275505	2
C₅₃×#1747	2
#30236	53

subgroup

index

C₅₃×#1747

Subgroups — #27496229

Conjugacy classes of subgroups

Complements

Maximal subgroups