Group #27300284 — subgroups

Subgroups — #27300284

Order 13568.

354 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂×#16811041	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810452	C₂	1	2	yes	maximal, complemented
3	#20274762	C₂	1	2	yes	maximal
4	#20272819	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133223	C₂²	1	1	yes	characteristic
7	#16811041	—	2	4	no	—
8	C₂×#13133156	C₄	1	2	yes	—
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
11	C₂×#13133153	C₂²	1	2	yes	—
12	#16810605	—	2	2	no	—
13	#16810452	—	2	4	no	—
14	#16810381	—	2	2	no	—
15	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#108×C₅₃	C₂²	1	2	yes	—
17	C₂×#11719945	D₄	1	1	yes	characteristic
18	#13133223	—	2	2	no	—
19	#13133223	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	D₄	1	1	yes	characteristic
22	C₂×#11719912	Q₈	1	2	yes	—
23	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
24	#13133156	—	2	4	no	—
25	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
26	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
27	C₂×#11719908	D₄	1	4	yes	—
28	C₁₆×D₅₃	—	2	4	no	—
29	#13133153	—	2	2	no	—
30	#13133153	—	2	2	no	—
31	#13133135	—	4	2	no	—
32	#13133133	—	4	2	no	—
33	C₂×M₁₆×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	M₃₂×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
37	#108×C₅₃	—	2	2	no	—
38	#108×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₂²⋊D₅₃	Q₁₆	1	2	yes	—
41	#11719945	—	2	2	no	—
42	#11719945	—	2	2	no	—
43	C₂×D₂₁₂	D₈	1	1	yes	characteristic
44	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
45	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
46	M₁₆×C₅₃	—	4	2	no	—
47	M₁₆×C₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₅₃	#41	1	2	yes	—
53	#11719917	—	2	2	no	—
54	#11719917	—	4	2	no	—
55	#11719917	—	4	2	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂×#88322	C₄⋊C₄	1	2	yes	—
58	#11719912	—	2	4	no	—
59	C₈×D₅₃	—	2	2	no	—
60	C₈×D₅₃	—	2	2	no	—
61	C₁₆×C₅₃	—	2	2	no	—
62	#11719908	C₂²⋊C₄	1	2	yes	—
63	#11719908	C₂²⋊C₄	1	2	yes	—
64	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×D₅₃	#107	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#121	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#135	1	2	yes	—
70	C₂²⋊D₅₃	—	2	4	no	—
71	C₂×Dic₅₃	#107	1	2	yes	—
72	D₂₁₂	—	2	2	no	—
73	C₄×D₅₃	—	2	2	no	—
74	C₄×D₅₃	—	2	2	no	—
75	Dic₁₀₆	—	2	2	no	—
76	C₈×C₅₃	—	8	2	no	—
77	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₅₃	#117	1	1	yes	characteristic
79	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
80	#88322	—	2	2	no	—
81	#88322	—	8	2	no	—
82	#12840	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	4	1	no	—
84	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
85	C₂²×C₅₃	#467	1	2	yes	—
86	C₂×D₅₃	—	2	2	no	—
87	C₂×D₅₃	—	4	2	no	—
88	C₄×C₅₃	#380	1	1	yes	characteristic
89	C₄×C₅₃	#427	1	1	yes	characteristic
90	C₄×C₅₃	#466	1	2	yes	—
91	Dic₅₃	—	2	2	no	—
92	C₂×#504	—	53	1	no	—
93	C₂×#368	—	53	2	no	—
94	#1813	—	53	2	no	—
95	#1385	—	53	1	no	—
96	#1385	D₅₃	1	1	yes	characteristic, complemented
97	C₂×C₅₃	—	2	1	no	—
98	C₂×C₅₃	—	2	2	no	—
99	C₂×C₅₃	#1808	1	1	yes	characteristic
100	D₅₃	—	4	2	no	—
101	C₂×#131	—	53	1	no	—
102	#504	—	106	4	no	—
103	C₂×M₃₂	—	53	2	no	—
104	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
105	C₂²×C₁₆	—	53	1	no	—
106	C₂²×C₁₆	—	53	2	no	—
107	C₂×#108	—	53	2	no	—
108	C₂×#108	C₂×D₅₃	1	2	yes	—
109	#413	—	106	2	no	—
110	#368	—	106	4	no	—
111	#338	—	106	2	no	—
112	C₅₃	#12840	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	—	53	1	no	—
114	#131	—	106	2	no	—
115	#131	—	106	2	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	53	1	no	—
118	C₂×M₁₆	—	53	2	no	—
119	C₂×M₁₆	—	106	2	no	—
120	C₂²×C₈	—	53	2	no	—
121	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	—	106	4	no	—
124	C₂×C₁₆	—	53	1	no	—
125	C₂×C₁₆	—	53	1	no	—
126	C₂×C₁₆	—	53	4	no	—
127	C₂×C₁₆	—	106	4	no	—
128	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
129	#108	—	2	2	no	—
130	#108	—	2	2	no	—
131	#108	—	106	2	no	—
132	#108	—	106	2	no	—
133	#101	—	212	2	no	—
134	#100	—	212	2	no	—
135	#41	—	106	2	no	—
136	#41	—	106	2	no	—
137	C₂×Q₈	—	53	1	no	—
138	C₂×D₄	—	53	2	no	—
139	C₂×D₄	—	53	1	no	—
140	C₂²×C₄	—	53	2	no	—
141	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	106	2	no	—
145	M₁₆	—	106	4	no	—
146	M₁₆	—	212	2	no	—
147	M₁₆	—	212	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	53	2	no	—
150	C₂×C₈	—	106	2	no	—
151	C₂×C₈	—	106	2	no	—
152	C₂×C₈	—	212	2	no	—
153	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₈	#11719945	1	2	yes	—
155	C₂×C₈	D₄×D₅₃	1	2	yes	—
156	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
157	C₁₆	—	2	2	no	—
158	C₁₆	—	53	2	no	—
159	C₁₆	—	53	2	no	—
160	C₂³	—	53	2	no	—
161	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
162	Q₈	—	106	2	no	—
163	D₄	—	106	4	no	—
164	D₄	—	106	2	no	—
165	C₂×C₄	—	53	2	no	—
166	C₂×C₄	—	106	2	no	—
167	C₂×C₄	—	106	2	no	—
168	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
169	C₂×C₄	#13133190	1	2	yes	—
170	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
171	C₂×C₄	#13133238	1	2	yes	—
172	C₈	—	8	2	no	—
173	C₈	—	106	2	no	—
174	C₈	—	424	2	no	—
175	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
176	C₈	#13133174	1	1	yes	characteristic
177	C₈	#13133195	1	2	yes	—
178	C₂²	—	4	1	no	—
179	C₂²	—	106	2	no	—
180	C₂²	—	212	2	no	—
181	C₂²	#107×D₅₃	1	1	yes	characteristic
182	C₂²	#16810863	1	2	yes	—
183	C₄	—	106	2	no	—
184	C₄	#16810482	1	1	yes	characteristic
185	C₄	#16810658	1	1	yes	characteristic
186	C₄	#16810858	1	2	yes	—
187	C₂	—	2	1	no	—
188	C₂	—	2	2	no	—
189	C₂	—	212	2	no	—
190	C₂	#20274743	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811041

C₂

yes

maximal, characteristic

C₂×#16810452

C₂

yes

maximal, complemented

#20274762

C₂

yes

maximal

#20272819

C₂

yes

maximal, characteristic, complemented

C₅₃×#1385

C₂

yes

maximal, characteristic, complemented

C₂×#13133223

C₂²

yes

characteristic

#16811041

—

C₂×#13133156

C₄

yes

—

C₂²×#11719908

C₂²

yes

characteristic

C₂×C₁₆×D₅₃

C₄

yes

—

C₂×#13133153

C₂²

yes

—

#16810605

—

#16810452

—

#16810381

—

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₂×#11719945

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11719912

Q₈

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

—

#13133156

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

D₄

yes

—

C₁₆×D₅₃

—

—

—

—

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

#108×C₅₃

—

#108×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₅₃

Q₁₆

yes

—

#11719945

—

#11719945

—

C₂×D₂₁₂

D₈

yes

characteristic

C₂×C₄×D₅₃

C₄⋊C₄

yes

—

C₂×Dic₁₀₆

D₈

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

—

—

—

—

C₄⋊C₄

yes

—

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

C₂²⋊C₄

yes

—

#11719908

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂²×D₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

#107

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₂²⋊Q₈

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

#467

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1808

yes

characteristic

100

D₅₃

—

101

C₂×#131

—

102

#504

—

106

—

103

C₂×M₃₂

—

104

C₂×M₃₂

C₂×D₅₃

yes

characteristic

105

C₂²×C₁₆

—

106

C₂²×C₁₆

—

107

C₂×#108

—

108

C₂×#108

C₂×D₅₃

yes

—

109

#413

—

106

—

110

#368

—

106

—

111

#338

—

106

—

112

C₅₃

#12840

yes

characteristic, complemented, Sylow

113

C₂×#41

—

114

#131

—

106

—

115

#131

—

106

—

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

106

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂²×D₅₃

yes

characteristic

122

M₃₂

—

123

M₃₂

—

106

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

106

—

128

C₂×C₁₆

C₄×D₅₃

yes

—

129

#108

—

130

#108

—

131

#108

—

106

—

132

#108

—

106

—

133

#101

—

212

—

134

#100

—

212

—

135

#41

—

106

—

136

#41

—

106

—

137

C₂×Q₈

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₅₃

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

106

—

145

M₁₆

—

106

—

146

M₁₆

—

212

—

147

M₁₆

—

212

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

106

—

151

C₂×C₈

—

106

—

152

C₂×C₈

—

212

—

153

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

154

C₂×C₈

#11719945

yes

—

155

C₂×C₈

D₄×D₅₃

yes

—

156

C₂×C₈

Q₈×D₅₃

yes

characteristic

157

C₁₆

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

—

161

C₂³

Q₁₆×D₅₃

yes

characteristic

162

Q₈

—

106

—

163

D₄

—

106

—

164

D₄

—

106

—

165

C₂×C₄

—

166

C₂×C₄

—

106

—

167

C₂×C₄

—

106

—

168

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

169

C₂×C₄

#13133190

yes

—

170

C₂×C₄

D₈×D₅₃

yes

characteristic

171

C₂×C₄

#13133238

yes

—

172

C₈

—

173

C₈

—

106

—

174

C₈

—

424

—

175

C₈

C₂²⋊C₄×D₅₃

yes

characteristic

176

C₈

#13133174

yes

characteristic

177

C₈

#13133195

yes

—

178

C₂²

—

179

C₂²

—

106

—

180

C₂²

—

212

—

181

C₂²

#107×D₅₃

yes

characteristic

182

C₂²

#16810863

yes

—

183

C₄

—

106

—

184

C₄

#16810482

yes

characteristic

185

C₄

#16810658

yes

characteristic

186

C₄

#16810858

yes

—

187

C₂

—

188

C₂

—

189

C₂

—

212

—

190

C₂

#20274743

yes

characteristic

	subgroup	G/H
center	C₄	#16810658
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1385	C₂
socle	C₂×C₅₃	#1808
prime radical	C₂×#11719945	D₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1385	D₅₃
hypocenter	C₅₃	#12840
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#16810658

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1385

C₂

C₂×C₅₃

D₄

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
2. C₂×#16810452	C₂	semidirect
4. #20272819	C₂	semidirect
5. C₅₃×#1385	C₂	semidirect
96. #1385	D₅₃	semidirect
112. C₅₃	#12840	semidirect

H ≅ G/N

type

2. C₂×#16810452

C₂

semidirect

4. #20272819

C₂

semidirect

5. C₅₃×#1385

C₂

semidirect

96. #1385

D₅₃

semidirect

112. C₅₃

#12840

semidirect

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

subgroup	index
C₂×#16811041	2
C₂×#16810452	2
#20274762	2
#20272819	2
C₅₃×#1385	2
#12840	53

subgroup

index

C₅₃×#1385

Subgroups — #27300284

Conjugacy classes of subgroups

Complements

Maximal subgroups