Group #27288870 — subgroups

Subgroups — #27288870

Order 13568.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20274835	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273364	C₂	1	2	yes	maximal
3	#20272610	C₂	1	2	yes	maximal, complemented
4	#20272544	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1963	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	#16810378	—	2	2	no	—
9	#16810369	—	2	2	no	—
10	#16810367	—	2	2	no	—
11	C₅₃×#532	C₂²	1	1	yes	characteristic
12	C₅₃×#423	C₂²	1	1	yes	characteristic
13	C₂×#104×C₅₃	—	2	2	no	—
14	C₅₃×#376	C₂²	1	1	yes	characteristic
15	C₂²×#88322	—	2	2	no	—
16	C₄⋊#88322	—	2	2	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₄⋊#88322	—	2	2	no	—
19	C₂²⋊#88322	—	2	2	no	—
20	C₄×#88322	—	2	2	no	—
21	C₂×#41×C₅₃	—	2	1	no	—
22	C₂×M₁₆×C₅₃	—	2	1	no	—
23	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
24	#127×C₅₃	D₄	1	1	yes	characteristic
25	#124×C₅₃	—	2	1	no	—
26	#121×C₅₃	—	2	2	no	—
27	C₄×D₄×C₅₃	—	2	2	no	—
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	2	no	—
31	#104×C₅₃	—	4	4	no	—
32	#95×C₅₃	—	2	2	no	—
33	#41×C₅₃	—	4	2	no	—
34	#41×C₅₃	—	4	2	no	—
35	C₂×Q₈×C₅₃	—	2	1	no	—
36	C₂×D₄×C₅₃	—	2	1	no	—
37	C₂×D₄×C₅₃	—	2	2	no	—
38	C₂×D₄×C₅₃	—	4	1	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	4	1	no	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	2	yes	—
43	M₁₆×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	—	4	1	no	—
45	C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂²⋊C₄×C₅₃	—	4	2	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₅₃	#41	1	1	yes	characteristic
51	C₄²×C₅₃	#41	1	2	yes	—
52	C₂×#88322	—	4	2	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	—	4	2	no	—
55	C₂×#88322	—	4	2	no	—
56	C₂³×C₅₃	—	4	1	no	—
57	C₂³×C₅₃	#123	1	1	yes	characteristic
58	Q₈×C₅₃	—	4	2	no	—
59	D₄×C₅₃	—	4	2	no	—
60	D₄×C₅₃	—	8	2	no	—
61	D₄×C₅₃	—	8	2	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	4	1	no	—
65	C₂×C₄×C₅₃	—	4	2	no	—
66	C₂×C₄×C₅₃	—	8	1	no	—
67	C₂×C₄×C₅₃	—	8	1	no	—
68	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#123	1	2	yes	—
70	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#126	1	2	yes	—
72	C₂×C₄×C₅₃	#135	1	2	yes	—
73	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₈×C₅₃	—	8	1	no	—
75	#88322	—	8	2	no	—
76	#88322	—	8	2	no	—
77	#10629	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	—	2	2	no	—
79	C₂²×C₅₃	—	8	1	no	—
80	C₂²×C₅₃	—	8	2	no	—
81	C₂²×C₅₃	#397	1	1	yes	characteristic
82	C₂²×C₅₃	#482	1	2	yes	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	8	2	no	—
86	C₄×C₅₃	—	8	1	no	—
87	C₄×C₅₃	#463	1	2	yes	—
88	C₄×C₅₃	#488	1	1	yes	characteristic
89	C₄×C₅₃	#490	1	1	yes	characteristic
90	#1963	D₅₃	1	1	yes	characteristic
91	#1832	—	53	1	no	—
92	#1540	—	53	2	no	—
93	#1301	—	53	2	no	—
94	#1274	—	53	1	no	—
95	C₂×C₅₃	—	2	2	no	—
96	C₂×C₅₃	—	16	1	no	—
97	C₂×C₅₃	#2073	1	1	yes	characteristic
98	C₂×C₅₃	#2080	1	2	yes	—
99	#532	C₂×D₅₃	1	1	yes	characteristic
100	#431	—	53	2	no	—
101	#423	C₂×D₅₃	1	1	yes	characteristic
102	C₂×C₄⋊C₈	—	53	2	no	—
103	C₂×#104	—	2	2	no	—
104	#376	C₂×D₅₃	1	1	yes	characteristic
105	#336	—	106	2	no	—
106	#332	—	106	2	no	—
107	#331	—	106	2	no	—
108	C₅₃	#10629	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	2	1	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂²×C₈	—	106	2	no	—
112	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
113	#127	D₂₁₂	1	1	yes	characteristic
114	#124	—	2	1	no	—
115	#121	—	2	2	no	—
116	C₄×D₄	—	2	2	no	—
117	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
118	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	106	2	no	—
121	C₄⋊C₈	—	106	2	no	—
122	C₄⋊C₈	—	106	2	no	—
123	#104	—	4	4	no	—
124	C₂²⋊C₈	—	106	2	no	—
125	C₄×C₈	—	106	2	no	—
126	#95	—	2	2	no	—
127	#41	—	4	2	no	—
128	#41	—	4	2	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	4	1	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	#11719947	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
137	M₁₆	—	4	2	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	212	2	no	—
140	C₂×C₈	—	212	2	no	—
141	C₂×C₈	—	212	2	no	—
142	C₂×C₈	—	212	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
148	C₄²	#11719945	1	1	yes	characteristic
149	C₄²	#11719947	1	2	yes	—
150	C₂³	—	4	1	no	—
151	C₂³	#13133198	1	1	yes	characteristic
152	Q₈	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	8	2	no	—
155	D₄	—	8	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	#13133191	1	1	yes	characteristic
163	C₂×C₄	#13133201	1	2	yes	—
164	C₂×C₄	#13133204	1	1	yes	characteristic
165	C₂×C₄	#13133210	1	2	yes	—
166	C₂×C₄	#13133237	1	2	yes	—
167	C₂×C₄	#13133243	1	2	yes	—
168	C₈	—	8	1	no	—
169	C₈	—	424	2	no	—
170	C₈	—	424	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	#16810548	1	1	yes	characteristic
175	C₂²	#16810944	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	8	2	no	—
179	C₄	—	8	1	no	—
180	C₄	#16810838	1	2	yes	—
181	C₄	#16810977	1	1	yes	characteristic
182	C₄	#16810987	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	16	1	no	—
185	C₂	#20276128	1	1	yes	characteristic
186	C₂	#20276165	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274835

C₂

yes

maximal, characteristic, complemented

#20273364

C₂

yes

maximal

#20272610

C₂

yes

maximal, complemented

#20272544

C₂

yes

maximal, characteristic, complemented

C₅₃×#1963

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

—

#16810378

—

#16810369

—

#16810367

—

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#423

C₂²

yes

characteristic

C₂×#104×C₅₃

—

C₅₃×#376

C₂²

yes

characteristic

C₂²×#88322

—

—

—

—

—

—

C₂×#41×C₅₃

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

characteristic

#127×C₅₃

D₄

yes

characteristic

#124×C₅₃

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

#104×C₅₃

—

#95×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

#123

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#126

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#397

yes

characteristic

C₂²×C₅₃

#482

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#463

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#2073

yes

characteristic

C₂×C₅₃

#2080

yes

—

#532

C₂×D₅₃

yes

characteristic

100

#431

—

101

#423

C₂×D₅₃

yes

characteristic

102

C₂×C₄⋊C₈

—

103

C₂×#104

—

104

#376

C₂×D₅₃

yes

characteristic

105

#336

—

106

—

106

#332

—

106

—

107

#331

—

106

—

108

C₅₃

#10629

yes

characteristic, complemented, Sylow

109

C₂×#41

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

106

—

112

C₄⋊Q₈

D₂₁₂

yes

characteristic

113

#127

D₂₁₂

yes

characteristic

114

#124

—

115

#121

—

116

C₄×D₄

—

117

C₂×C₄²

C₂²⋊D₅₃

yes

—

118

C₂×C₄²

C₂²×D₅₃

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

106

—

121

C₄⋊C₈

—

106

—

122

C₄⋊C₈

—

106

—

123

#104

—

124

C₂²⋊C₈

—

106

—

125

C₄×C₈

—

106

—

126

#95

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

#11719947

yes

—

136

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

212

—

140

C₂×C₈

—

212

—

141

C₂×C₈

—

212

—

142

C₂×C₈

—

212

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

C₂×D₂₁₂

yes

characteristic

148

C₄²

#11719945

yes

characteristic

149

C₄²

#11719947

yes

—

150

C₂³

—

151

C₂³

#13133198

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#13133191

yes

characteristic

163

C₂×C₄

#13133201

yes

—

164

C₂×C₄

#13133204

yes

characteristic

165

C₂×C₄

#13133210

yes

—

166

C₂×C₄

#13133237

yes

—

167

C₂×C₄

#13133243

yes

—

168

C₈

—

169

C₈

—

424

—

170

C₈

—

424

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#16810548

yes

characteristic

175

C₂²

#16810944

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#16810838

yes

—

181

C₄

#16810977

yes

characteristic

182

C₄

#16810987

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

#20276128

yes

characteristic

186

C₂

#20276165

yes

—

	subgroup	G/H
center	C₂×C₄	#13133204
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1963	C₂
socle	C₂²×C₅₃	#397
prime radical	C₅₃×#532	C₂²
hypercenter	#1963	D₅₃
hypocenter	C₅₃	#10629
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

subgroup

G/H

center

C₂×C₄

#13133204

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1963

C₂

socle

C₂²×C₅₃

#397

prime radical

C₅₃×#532

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

N	H ≅ G/N	type
1. #20274835	C₂	semidirect
3. #20272610	C₂	semidirect
4. #20272544	C₂	semidirect
108. C₅₃	#10629	semidirect

H ≅ G/N

type

1. #20274835

C₂

semidirect

3. #20272610

C₂

semidirect

4. #20272544

C₂

semidirect

108. C₅₃

#10629

semidirect

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

subgroup	index
#20274835	2
#20273364	2
#20272610	2
#20272544	2
C₅₃×#1963	2
#10629	53

subgroup

index

C₅₃×#1963

Subgroups — #27288870

Conjugacy classes of subgroups

Complements

Maximal subgroups