Group #27303737 — subgroups

Subgroups — #27303737

Order 13568.

356 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16810445	C₂	1	2	yes	maximal
2	#20275463	C₂	1	1	yes	maximal, characteristic
3	#20272822	C₂	1	2	yes	maximal
4	#20272797	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133153	C₂²	1	1	yes	characteristic
8	#16810661	C₂²	1	1	yes	characteristic
9	#16810649	—	2	1	no	—
10	#16810445	—	2	4	no	—
11	#16810383	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
13	C₅₃×#498	—	2	4	no	—
14	C₂×#106×C₅₃	—	2	1	no	—
15	C₂×#106×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719917	—	2	1	no	—
18	#13133176	—	2	1	no	—
19	#13133157	—	2	2	no	—
20	#13133157	Q₈	1	4	yes	complemented
21	C₂×#11719908	—	2	2	no	—
22	#13133153	—	2	2	no	—
23	#13133151	Q₈	1	2	yes	complemented
24	#13133149	—	2	1	no	—
25	#13133146	—	4	2	no	—
26	C₂²×Q₈×C₅₃	—	4	1	no	—
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₅₃	—	2	4	no	—
30	C₄⋊Q₈×C₅₃	—	4	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	#106×C₅₃	—	2	4	no	—
35	#106×C₅₃	—	2	2	no	—
36	#106×C₅₃	D₄	1	4	yes	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₂×Q₈×C₅₃	—	4	4	no	—
39	C₂×Q₈×C₅₃	—	8	4	no	—
40	C₂²×C₄×C₅₃	—	2	1	no	—
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₅₃	—	2	4	no	—
51	C₄⋊C₄×C₅₃	—	4	4	no	—
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₄²×C₅₃	—	2	1	no	—
54	C₄²×C₅₃	SD₁₆	1	2	yes	—
55	C₂²⋊Dic₅₃	—	4	1	no	—
56	C₄⋊Dic₅₃	—	2	2	no	—
57	C₄×Dic₅₃	—	4	1	no	—
58	#11719917	—	4	1	no	—
59	#11719917	—	4	1	no	—
60	C₂×#88322	—	4	1	no	—
61	#11719908	—	2	4	no	—
62	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	Q₈×C₅₃	—	8	8	no	—
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	4	4	no	—
67	C₂×C₄×C₅₃	—	8	4	no	—
68	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
74	C₂×Dic₅₃	—	4	2	no	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
78	#88322	—	8	1	no	—
79	#13457	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	2	no	—
81	C₂²×C₅₃	#450	1	1	yes	characteristic
82	C₂²×C₅₃	#476	1	1	yes	characteristic
83	C₂²×C₅₃	#501	1	1	yes	characteristic
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	—	8	4	no	—
86	C₄×C₅₃	#395	1	1	yes	characteristic
87	C₄×C₅₃	#448	1	1	yes	characteristic
88	C₄×C₅₃	#475	1	1	yes	characteristic
89	C₄×C₅₃	#498	1	1	yes	characteristic
90	Dic₅₃	—	8	2	no	—
91	C₂×#498	D₅₃	1	1	yes	characteristic
92	C₂×#365	—	53	2	no	—
93	#1945	—	53	1	no	—
94	#1386	—	53	2	no	—
95	#1376	—	53	1	no	—
96	C₂×C₅₃	—	2	1	no	—
97	C₂×C₅₃	—	2	1	no	—
98	C₂×C₅₃	#2058	1	1	yes	characteristic
99	C₂×C₅₃	#2230	1	2	yes	—
100	C₂×C₄⋊Q₈	—	2	1	no	—
101	C₂×M₃₂	—	53	2	no	—
102	#498	—	2	4	no	—
103	C₂×#108	—	53	1	no	—
104	#427	—	53	1	no	—
105	C₂×#106	—	2	1	no	—
106	C₂×#106	C₂×D₅₃	1	2	yes	—
107	#424	—	106	1	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	#365	—	106	4	no	—
110	#339	—	106	1	no	—
111	C₅₃	#13457	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	4	1	no	—
113	C₂×M₁₆	—	106	1	no	—
114	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	—	4	2	no	—
118	#117	—	106	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
122	M₃₂	—	53	4	no	—
123	M₃₂	—	106	2	no	—
124	C₂×C₁₆	—	106	2	no	—
125	#108	—	106	2	no	—
126	#107	—	53	2	no	—
127	#106	—	2	4	no	—
128	#106	—	2	2	no	—
129	#106	—	106	1	no	—
130	#106	C₂²⋊D₅₃	1	4	yes	—
131	C₄⋊C₈	—	212	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×Q₈	—	8	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
138	M₁₆	—	212	1	no	—
139	M₁₆	—	212	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	212	1	no	—
143	C₂×C₈	#11719924	1	2	yes	—
144	C₂×C₈	Q₈×D₅₃	1	2	yes	—
145	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
146	C₂×C₈	#11719949	1	1	yes	characteristic
147	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	106	2	no	—
152	C₂²⋊C₄	—	212	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	212	1	no	—
155	C₄²	#11719923	1	2	yes	—
156	C₁₆	—	106	4	no	—
157	C₂³	#13133243	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	212	2	no	—
164	C₂×C₄	#13133186	1	1	yes	characteristic
165	C₂×C₄	#13133193	1	1	yes	characteristic
166	C₂×C₄	#13133215	1	1	yes	characteristic
167	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
168	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
169	C₂×C₄	#13133248	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	424	1	no	—
172	C₈	#13133193	1	2	yes	—
173	C₈	#13133215	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#16810774	1	1	yes	characteristic
176	C₂²	#16810910	1	1	yes	characteristic
177	C₂²	#16811029	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	4	no	—
180	C₄	—	424	2	no	—
181	C₄	#16810542	1	1	yes	characteristic
182	C₄	#16810762	1	1	yes	characteristic
183	C₄	#16810905	1	1	yes	characteristic
184	C₄	#16811019	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#20276045	1	1	yes	characteristic
188	C₂	#20276883	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810445

C₂

yes

maximal

#20275463

C₂

yes

maximal, characteristic

#20272822

C₂

yes

maximal

#20272797

C₂

yes

maximal, characteristic

C₂×C₅₃×#498

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×#13133153

C₂²

yes

characteristic

#16810661

C₂²

yes

—

—

—

C₂×C₄⋊Q₈×C₅₃

—

C₅₃×#498

—

C₂×#106×C₅₃

—

C₂×#106×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

Q₈

yes

—

—

Q₈

yes

complemented

#13133149

—

#13133146

—

C₂²×Q₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

#106×C₅₃

—

#106×C₅₃

—

#106×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

SD₁₆

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

C₂³×C₅₃

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

#137

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

C₄⋊Q₈

yes

—

#88322

—

#13457

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

C₂×#498

D₅₃

yes

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#2058

yes

characteristic

C₂×C₅₃

#2230

yes

—

100

C₂×C₄⋊Q₈

—

101

C₂×M₃₂

—

102

#498

—

103

C₂×#108

—

104

#427

—

105

C₂×#106

—

106

C₂×#106

C₂×D₅₃

yes

—

107

#424

—

106

—

108

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

109

#365

—

106

—

110

#339

—

106

—

111

C₅₃

#13457

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂×M₁₆

—

106

—

114

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

115

C₂²×C₈

C₂²×D₅₃

yes

characteristic

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

#117

—

106

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

122

M₃₂

—

123

M₃₂

—

106

—

124

C₂×C₁₆

—

106

—

125

#108

—

106

—

126

#107

—

127

#106

—

128

#106

—

129

#106

—

106

—

130

#106

C₂²⋊D₅₃

yes

—

131

C₄⋊C₈

—

212

—

132

C₄×C₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

138

M₁₆

—

212

—

139

M₁₆

—

212

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

212

—

143

C₂×C₈

#11719924

yes

—

144

C₂×C₈

Q₈×D₅₃

yes

—

145

C₂×C₈

Q₈×D₅₃

yes

characteristic

146

C₂×C₈

#11719949

yes

characteristic

147

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

106

—

152

C₂²⋊C₄

—

212

—

153

C₄²

—

154

C₄²

—

212

—

155

C₄²

#11719923

yes

—

156

C₁₆

—

106

—

157

C₂³

#13133243

yes

characteristic

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

212

—

164

C₂×C₄

#13133186

yes

characteristic

165

C₂×C₄

#13133193

yes

characteristic

166

C₂×C₄

#13133215

yes

characteristic

167

C₂×C₄

C₂×#11719924

yes

characteristic

168

C₂×C₄

C₂×#11719923

yes

characteristic

169

C₂×C₄

#13133248

yes

characteristic

170

C₈

—

171

C₈

—

424

—

172

C₈

#13133193

yes

—

173

C₈

#13133215

yes

—

174

C₂²

—

175

C₂²

#16810774

yes

characteristic

176

C₂²

#16810910

yes

characteristic

177

C₂²

#16811029

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

424

—

181

C₄

#16810542

yes

characteristic

182

C₄

#16810762

yes

characteristic

183

C₄

#16810905

yes

characteristic

184

C₄

#16811019

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#20276045

yes

characteristic

188

C₂

#20276883

yes

—

	subgroup	G/H
center	C₂²	#16810910
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#498	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
hypercenter	C₂×#498	D₅₃
hypocenter	C₅₃	#13457
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

subgroup

G/H

center

C₂²

#16810910

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#498

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

N	H ≅ G/N	type
20. #13133157	Q₈	semidirect
23. #13133151	Q₈	semidirect
111. C₅₃	#13457	semidirect

H ≅ G/N

type

20. #13133157

Q₈

semidirect

23. #13133151

Q₈

semidirect

111. C₅₃

#13457

semidirect

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

subgroup	index
C₂×#16810445	2
#20275463	2
#20272822	2
#20272797	2
C₂×C₅₃×#498	2
#13457	53

subgroup

index

C₂×C₅₃×#498

Subgroups — #27303737

Conjugacy classes of subgroups

Complements

Maximal subgroups