Group #27282711 — subgroups

Subgroups — #27282711

Order 13568.

294 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
2	#20273465	C₂	1	2	yes	maximal
3	#20273007	C₂	1	1	yes	maximal, characteristic
4	#20272565	C₂	1	2	yes	maximal
5	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	1	yes	characteristic
7	#16810676	C₂²	1	1	yes	characteristic
8	C₂×#13133153	C₄	1	2	yes	—
9	C₄×#11719917	C₂²	1	2	yes	—
10	#16810388	C₄	1	1	yes	characteristic
11	#16810388	C₄	1	1	yes	characteristic
12	#16810375	—	2	2	no	—
13	#16810373	—	2	2	no	—
14	#16810345	—	2	2	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#334	—	2	2	no	—
18	C₂×#11719917	C₂×C₄	1	2	yes	—
19	C₂×#11719917	C₂×C₄	1	2	yes	—
20	#13133153	—	2	4	no	—
21	C₄⋊#88322	—	2	1	no	—
22	C₄⋊#88322	D₄	1	2	yes	—
23	C₂²⋊#88322	—	2	1	no	—
24	C₂²⋊#88322	Q₈	1	2	yes	—
25	#13133128	—	2	2	no	—
26	C₄×#88322	—	2	2	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	8	2	no	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₃	#41	1	1	yes	characteristic
50	#11719917	—	2	2	no	—
51	#11719917	—	2	2	no	—
52	#11719917	—	2	2	no	—
53	#11719917	M₁₆	1	4	yes	—
54	C₂×#88322	—	2	2	no	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	4	1	no	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₅₃	#131	1	2	yes	—
69	C₈×C₅₃	—	2	2	no	—
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	—	8	2	no	—
72	#88322	—	2	4	no	—
73	#88322	—	4	2	no	—
74	#88322	—	4	2	no	—
75	#88322	—	4	2	no	—
76	#9609	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	1	no	—
78	C₂²×C₅₃	—	2	1	no	—
79	C₂²×C₅₃	#389	1	1	yes	characteristic
80	C₂²×C₅₃	#446	1	2	yes	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	#423	1	1	yes	characteristic
85	C₄×C₅₃	#424	1	1	yes	characteristic
86	C₄×C₅₃	#435	1	2	yes	—
87	C₄×C₅₃	#471	1	2	yes	—
88	C₄×#108	—	53	1	no	—
89	#1565	—	53	2	no	—
90	C₂×#334	D₅₃	1	1	yes	characteristic
91	#1452	—	53	1	no	—
92	#1283	—	53	2	no	—
93	C₂×C₅₃	—	2	2	no	—
94	C₂×C₅₃	#1940	1	1	yes	characteristic
95	C₂×C₅₃	#1951	1	1	yes	characteristic
96	C₂×C₅₃	#1953	1	1	yes	characteristic
97	#431	—	53	1	no	—
98	#431	—	53	1	no	—
99	C₂×#108	—	53	2	no	—
100	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
101	C₄×M₁₆	—	53	2	no	—
102	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
103	#341	—	53	1	no	—
104	#341	—	53	1	no	—
105	#335	—	106	2	no	—
106	#334	—	2	2	no	—
107	#334	—	106	2	no	—
108	#322	—	106	2	no	—
109	C₅₃	#9609	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	—	53	2	no	—
111	C₂×M₁₆	—	53	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
116	#108	—	106	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	53	2	no	—
120	C₄⋊C₈	—	106	1	no	—
121	C₂²⋊C₈	—	53	2	no	—
122	C₂²⋊C₈	—	106	1	no	—
123	#97	—	106	2	no	—
124	C₄×C₈	—	106	2	no	—
125	C₄×C₈	—	106	1	no	—
126	C₄×C₈	—	106	1	no	—
127	C₄×C₈	Dic₁₀₆	1	2	yes	—
128	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
129	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂²×C₄	#11719947	1	1	yes	characteristic
131	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
132	M₁₆	—	53	4	no	—
133	M₁₆	—	106	2	no	—
134	M₁₆	—	106	2	no	—
135	M₁₆	—	106	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	—	106	2	no	—
141	C₂×C₈	—	106	2	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	—	106	2	no	—
144	C₂×C₈	#11719919	1	2	yes	—
145	C₂×C₈	#11719919	1	2	yes	—
146	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
147	C₄²	D₄×D₅₃	1	1	yes	characteristic
148	C₄²	#11719949	1	1	yes	characteristic
149	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂³	#13133182	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
154	C₂×C₄	#13133179	1	1	yes	characteristic
155	C₂×C₄	#13133188	1	1	yes	characteristic
156	C₂×C₄	#13133193	1	1	yes	characteristic
157	C₂×C₄	#13133208	1	1	yes	characteristic
158	C₂×C₄	#13133216	1	1	yes	characteristic
159	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
160	C₂×C₄	#13133223	1	2	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	8	2	no	—
164	C₈	—	106	4	no	—
165	C₈	—	212	2	no	—
166	C₈	—	212	2	no	—
167	C₈	—	212	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16810519	1	1	yes	characteristic
171	C₂²	#16810752	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#16810646	1	1	yes	characteristic
176	C₄	#16810649	1	1	yes	characteristic
177	C₄	#16810693	1	2	yes	—
178	C₄	#16810883	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#20275436	1	1	yes	characteristic
181	C₂	#20275493	1	1	yes	characteristic
182	C₂	#20275503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133153

C₂

yes

maximal, characteristic

#20273465

C₂

yes

maximal

#20273007

C₂

yes

maximal, characteristic

#20272565

C₂

yes

maximal

C₂×C₅₃×#334

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

characteristic

#16810676

C₂²

yes

characteristic

C₂×#13133153

C₄

yes

—

C₄×#11719917

C₂²

yes

—

#16810388

C₄

yes

characteristic

#16810388

C₄

yes

—

—

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

C₂×#11719917

C₂×C₄

yes

—

C₂×#11719917

C₂×C₄

yes

—

#13133153

—

C₄⋊#88322

—

C₄⋊#88322

D₄

yes

—

C₂²⋊#88322

—

C₂²⋊#88322

Q₈

yes

—

—

—

—

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

—

—

—

M₁₆

yes

—

—

—

—

—

C₂³×C₅₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#125

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#389

yes

characteristic

C₂²×C₅₃

#446

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#471

yes

—

C₄×#108

—

#1565

—

C₂×#334

D₅₃

yes

characteristic

#1452

—

#1283

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

100

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

101

C₄×M₁₆

—

102

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

103

#341

—

104

#341

—

105

#335

—

106

—

106

#334

—

107

#334

—

106

—

108

#322

—

106

—

109

C₅₃

#9609

yes

characteristic, complemented, Sylow

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

C₄×D₅₃

yes

characteristic

114

C₂²×C₈

C₄×D₅₃

yes

characteristic

115

C₂×C₄²

C₂²×D₅₃

yes

characteristic

116

#108

—

106

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

106

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

—

106

—

123

#97

—

106

—

124

C₄×C₈

—

106

—

125

C₄×C₈

—

106

—

126

C₄×C₈

—

106

—

127

C₄×C₈

Dic₁₀₆

yes

—

128

C₄×C₈

C₂²⋊D₅₃

yes

—

129

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

130

C₂²×C₄

#11719947

yes

characteristic

131

C₂²×C₄

Q₈×D₅₃

yes

characteristic

132

M₁₆

—

133

M₁₆

—

106

—

134

M₁₆

—

106

—

135

M₁₆

—

106

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

106

—

141

C₂×C₈

—

106

—

142

C₂×C₈

—

106

—

143

C₂×C₈

—

106

—

144

C₂×C₈

#11719919

yes

—

145

C₂×C₈

#11719919

yes

—

146

C₄²

C₂×Dic₁₀₆

yes

characteristic

147

C₄²

D₄×D₅₃

yes

characteristic

148

C₄²

#11719949

yes

characteristic

149

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

150

C₂³

#13133182

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×#11719919

yes

characteristic

154

C₂×C₄

#13133179

yes

characteristic

155

C₂×C₄

#13133188

yes

characteristic

156

C₂×C₄

#13133193

yes

characteristic

157

C₂×C₄

#13133208

yes

characteristic

158

C₂×C₄

#13133216

yes

characteristic

159

C₂×C₄

M₁₆×D₅₃

yes

—

160

C₂×C₄

#13133223

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

106

—

165

C₈

—

212

—

166

C₈

—

212

—

167

C₈

—

212

—

168

C₂²

—

169

C₂²

—

170

C₂²

#16810519

yes

characteristic

171

C₂²

#16810752

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#16810646

yes

characteristic

176

C₄

#16810649

yes

characteristic

177

C₄

#16810693

yes

—

178

C₄

#16810883

yes

—

179

C₂

—

180

C₂

#20275436

yes

characteristic

181

C₂

#20275493

yes

characteristic

182

C₂

#20275503

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#9609
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

subgroup

G/H

center

C₂×C₄

#13133188

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₅₃×#334

C₂

socle

C₂²×C₅₃

#389

prime radical

C₂³×C₅₃

C₄×Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

N	H ≅ G/N	type
109. C₅₃	#9609	semidirect

H ≅ G/N

type

109. C₅₃

#9609

semidirect

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

subgroup	index
C₄×#13133153	2
#20273465	2
#20273007	2
#20272565	2
C₂×C₅₃×#334	2
#9609	53

subgroup

index

C₂×C₅₃×#334

Subgroups — #27282711

Conjugacy classes of subgroups

Complements

Maximal subgroups