Group #27289081 — subgroups

Subgroups — #27289081

Order 13568.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20275655	C₂	1	1	yes	maximal, characteristic
2	#20273491	C₂	1	2	yes	maximal
3	#20273143	C₂	1	2	yes	maximal
4	#20272546	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	#16810676	C₂²	1	2	yes	—
8	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
9	#16810377	—	2	2	no	—
10	#16810375	—	2	4	no	—
11	#16810353	—	2	4	no	—
12	C₂×#108×C₅₃	—	2	2	no	—
13	C₅₃×#423	C₂²	1	2	yes	—
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₆	Q₈	1	1	yes	characteristic
16	C₂²×#88322	—	2	1	no	—
17	C₄×Dic₁₀₆	—	2	2	no	—
18	C₄×Dic₁₀₆	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
20	C₂×#11719919	—	2	1	no	—
21	C₂×C₄×Dic₅₃	—	2	1	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₂²⋊#88322	—	2	2	no	—
25	C₄×#88322	D₄	1	4	yes	—
26	C₂×M₁₆×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	#108×C₅₃	—	4	4	no	—
30	C₄⋊C₈×C₅₃	—	2	2	no	—
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	#97×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	2	1	no	—
37	C₂²×Dic₅₃	—	2	1	no	—
38	C₂×Dic₁₀₆	—	2	2	no	—
39	C₂×Dic₁₀₆	—	4	2	no	—
40	M₁₆×C₅₃	—	4	4	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
45	C₄²×C₅₃	C₂×D₄	1	2	yes	—
46	C₄²×C₅₃	#41	1	2	yes	—
47	C₄⋊Dic₅₃	—	2	2	no	—
48	#11719919	—	4	2	no	—
49	C₄×Dic₅₃	—	4	1	no	—
50	C₄×Dic₅₃	—	4	1	no	—
51	C₂×#88322	—	2	4	no	—
52	C₂×#88322	—	4	1	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	4	1	no	—
56	C₂×C₄×C₅₃	—	4	1	no	—
57	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#123	1	2	yes	—
60	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
62	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
63	C₂×Dic₅₃	—	2	2	no	—
64	C₂×Dic₅₃	—	4	1	no	—
65	C₂×Dic₅₃	—	4	1	no	—
66	C₂×Dic₅₃	—	4	2	no	—
67	C₂×Dic₅₃	—	4	1	no	—
68	Dic₁₀₆	—	2	4	no	—
69	Dic₁₀₆	—	4	2	no	—
70	C₈×C₅₃	—	2	4	no	—
71	C₈×C₅₃	—	8	2	no	—
72	#88322	—	4	4	no	—
73	#88322	—	8	1	no	—
74	#10662	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	—	2	1	no	—
77	C₂²×C₅₃	#399	1	1	yes	characteristic
78	C₂²×C₅₃	#477	1	2	yes	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	#484	1	2	yes	—
82	C₄×C₅₃	#494	1	1	yes	characteristic
83	C₄×C₅₃	#496	1	1	yes	characteristic
84	Dic₅₃	—	4	2	no	—
85	Dic₅₃	—	8	1	no	—
86	#1985	—	53	1	no	—
87	#1849	D₅₃	1	1	yes	characteristic
88	#1571	—	53	2	no	—
89	#1487	—	53	2	no	—
90	#1275	—	53	1	no	—
91	C₂×C₅₃	—	2	2	no	—
92	C₂×C₅₃	#2091	1	1	yes	characteristic
93	C₂×C₅₃	#2095	1	1	yes	characteristic
94	C₂×C₅₃	#2098	1	1	yes	characteristic
95	C₂×C₄×Q₈	—	53	1	no	—
96	#431	—	53	2	no	—
97	C₂×#108	—	2	2	no	—
98	#423	C₂×D₅₃	1	2	yes	—
99	C₂×C₄⋊C₈	—	53	1	no	—
100	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
101	#336	—	106	2	no	—
102	#335	—	106	4	no	—
103	Q₈⋊C₈	—	106	4	no	—
104	C₅₃	#10662	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	53	1	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂²×C₈	—	106	1	no	—
108	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
109	C₄×Q₈	—	106	2	no	—
110	C₄×Q₈	—	106	2	no	—
111	C₂×C₄⋊C₄	—	53	1	no	—
112	C₂×C₄⋊C₄	—	106	1	no	—
113	C₂×C₄²	—	106	1	no	—
114	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
115	#108	—	4	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	106	2	no	—
119	C₄⋊C₈	—	106	2	no	—
120	C₂²⋊C₈	—	106	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	53	4	no	—
123	C₂×Q₈	—	106	2	no	—
124	C₂×Q₈	—	212	2	no	—
125	C₂²×C₄	—	106	1	no	—
126	C₂²×C₄	—	106	1	no	—
127	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
128	C₂²×C₄	#11719949	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	M₁₆	—	4	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	106	4	no	—
135	C₂×C₈	—	212	1	no	—
136	C₂×C₈	—	212	2	no	—
137	C₂×C₈	#11719925	1	4	yes	—
138	C₄⋊C₄	—	106	2	no	—
139	C₄⋊C₄	—	212	2	no	—
140	C₄²	—	212	1	no	—
141	C₄²	—	212	1	no	—
142	C₄²	#11719945	1	2	yes	—
143	C₄²	D₄×D₅₃	1	2	yes	—
144	C₂³	#13133193	1	1	yes	characteristic
145	Q₈	—	106	4	no	—
146	Q₈	—	212	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	106	2	no	—
150	C₂×C₄	—	212	2	no	—
151	C₂×C₄	—	212	1	no	—
152	C₂×C₄	—	212	1	no	—
153	C₂×C₄	—	212	1	no	—
154	C₂×C₄	#13133195	1	2	yes	—
155	C₂×C₄	#13133200	1	1	yes	characteristic
156	C₂×C₄	#13133202	1	2	yes	—
157	C₂×C₄	#13133213	1	1	yes	characteristic
158	C₂×C₄	C₂×#11719925	1	2	yes	—
159	C₂×C₄	#13133243	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	8	2	no	—
162	C₈	—	424	1	no	—
163	C₈	—	212	4	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#16810558	1	1	yes	characteristic
167	C₂²	#16810915	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	424	1	no	—
171	C₄	—	212	2	no	—
172	C₄	#16810958	1	2	yes	—
173	C₄	#16811002	1	1	yes	characteristic
174	C₄	#16811010	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#20276230	1	1	yes	characteristic
177	C₂	#20276254	1	1	yes	characteristic
178	C₂	#20276267	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275655

C₂

yes

maximal, characteristic

#20273491

C₂

yes

maximal

#20273143

C₂

yes

maximal

#20272546

C₂

yes

maximal, characteristic

C₅₃×#1849

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₀₆

C₂²

yes

characteristic

#16810676

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

—

—

—

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

Q₈

yes

characteristic

C₂²×#88322

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₂×C₄⋊Dic₅₃

Q₈

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

#97×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#399

yes

characteristic

C₂²×C₅₃

#477

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#484

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#1985

—

#1849

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

—

#431

—

C₂×#108

—

#423

C₂×D₅₃

yes

—

C₂×C₄⋊C₈

—

100

C₂×#97

C₂×D₅₃

yes

characteristic

101

#336

—

106

—

102

#335

—

106

—

103

Q₈⋊C₈

—

106

—

104

C₅₃

#10662

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

106

—

108

C₂²×C₈

C₂²⋊D₅₃

yes

—

109

C₄×Q₈

—

106

—

110

C₄×Q₈

—

106

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

106

—

113

C₂×C₄²

—

106

—

114

C₂×C₄²

C₂²×D₅₃

yes

characteristic

115

#108

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

106

—

119

C₄⋊C₈

—

106

—

120

C₂²⋊C₈

—

106

—

121

#97

—

122

C₄×C₈

—

123

C₂×Q₈

—

106

—

124

C₂×Q₈

—

212

—

125

C₂²×C₄

—

106

—

126

C₂²×C₄

—

106

—

127

C₂²×C₄

Q₈×D₅₃

yes

characteristic

128

C₂²×C₄

#11719949

yes

characteristic

129

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

106

—

135

C₂×C₈

—

212

—

136

C₂×C₈

—

212

—

137

C₂×C₈

#11719925

yes

—

138

C₄⋊C₄

—

106

—

139

C₄⋊C₄

—

212

—

140

C₄²

—

212

—

141

C₄²

—

212

—

142

C₄²

#11719945

yes

—

143

C₄²

D₄×D₅₃

yes

—

144

C₂³

#13133193

yes

characteristic

145

Q₈

—

106

—

146

Q₈

—

212

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

212

—

151

C₂×C₄

—

212

—

152

C₂×C₄

—

212

—

153

C₂×C₄

—

212

—

154

C₂×C₄

#13133195

yes

—

155

C₂×C₄

#13133200

yes

characteristic

156

C₂×C₄

#13133202

yes

—

157

C₂×C₄

#13133213

yes

characteristic

158

C₂×C₄

C₂×#11719925

yes

—

159

C₂×C₄

#13133243

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

424

—

163

C₈

—

212

—

164

C₂²

—

165

C₂²

—

166

C₂²

#16810558

yes

characteristic

167

C₂²

#16810915

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

424

—

171

C₄

—

212

—

172

C₄

#16810958

yes

—

173

C₄

#16811002

yes

characteristic

174

C₄

#16811010

yes

characteristic

175

C₂

—

176

C₂

#20276230

yes

characteristic

177

C₂

#20276254

yes

characteristic

178

C₂

#20276267

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1849	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂³×C₅₃	C₂²⋊Q₈
hypercenter	#1849	D₅₃
hypocenter	C₅₃	#10662
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂×C₄

#13133213

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1849

C₂

socle

C₂²×C₅₃

#399

prime radical

C₂³×C₅₃

C₂²⋊Q₈

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
104. C₅₃	#10662	semidirect

H ≅ G/N

type

104. C₅₃

#10662

semidirect

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

subgroup	index
#20275655	2
#20273491	2
#20273143	2
#20272546	2
C₅₃×#1849	2
#10662	53

subgroup

index

C₅₃×#1849

Subgroups — #27289081

Conjugacy classes of subgroups

Complements

Maximal subgroups