Group #27277854 — subgroups

Subgroups — #27277854

Order 13568.

382 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 163 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	#20274832	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810375	C₂	1	2	yes	maximal
4	C₂×#16810373	C₂	1	2	yes	maximal
5	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810378	—	2	2	no	—
10	#16810378	C₄	1	2	yes	—
11	#16810378	C₄	1	2	yes	—
12	#16810375	—	2	4	no	—
13	#16810373	—	2	4	no	—
14	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	—	2	1	no	—
18	C₂²×#88322	—	2	1	no	—
19	C₂²×#88322	C₂×C₄	1	4	yes	—
20	C₄⋊#88322	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄×#88322	D₄	1	4	yes	—
23	C₄×#88322	Q₈	1	4	yes	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	2	no	—
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	C₄⋊C₈×C₅₃	—	4	4	no	—
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	4	no	—
40	C₄²×C₅₃	—	2	4	no	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	C₂×D₄	1	2	yes	—
46	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
47	C₂×#88322	—	2	4	no	—
48	C₂×#88322	—	2	4	no	—
49	C₂×#88322	—	4	1	no	—
50	C₂×#88322	—	4	1	no	—
51	C₂×#88322	—	4	2	no	—
52	C₂×#88322	—	4	2	no	—
53	C₂×#88322	C₄⋊C₄	1	8	yes	—
54	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	#108	1	4	yes	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₅₃	#131	1	2	yes	—
67	C₂×C₄×C₅₃	#131	1	2	yes	—
68	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₄×C₅₃	#137	1	2	yes	—
70	C₈×C₅₃	—	8	2	no	—
71	#88322	—	2	8	no	—
72	#88322	—	8	1	no	—
73	#88322	—	8	1	no	—
74	#8812	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	#384	1	1	yes	characteristic
76	C₂²×C₅₃	#424	1	1	yes	characteristic
77	C₂²×C₅₃	#424	1	1	yes	characteristic
78	C₂²×C₅₃	#428	1	2	yes	—
79	C₂²×C₅₃	C₂×#108	1	2	yes	—
80	C₄×C₅₃	—	2	4	no	—
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	—	4	2	no	—
83	C₄×C₅₃	#424	1	4	yes	—
84	C₄×C₅₃	#496	1	1	yes	characteristic
85	C₄×C₅₃	#497	1	1	yes	characteristic
86	C₄×C₅₃	#501	1	2	yes	—
87	#1842	D₅₃	1	1	yes	characteristic
88	#1831	—	53	1	no	—
89	#1744	—	53	1	no	—
90	C₂×#335	—	53	2	no	—
91	C₂×#334	—	53	2	no	—
92	C₂×C₅₃	#1573	1	2	yes	—
93	C₂×C₅₃	#1574	1	2	yes	—
94	C₂×C₅₃	#1843	1	1	yes	characteristic
95	C₂×C₅₃	#1847	1	1	yes	characteristic
96	C₂×C₅₃	#1848	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	2	2	no	—
98	C₂×C₄⋊C₈	—	53	1	no	—
99	C₂×C₄⋊C₈	—	53	1	no	—
100	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
101	C₂×C₄×C₈	—	53	2	no	—
102	C₄³	C₂×D₅₃	1	1	yes	characteristic
103	#336	—	53	2	no	—
104	#336	—	53	2	no	—
105	#336	—	106	2	no	—
106	#335	—	106	4	no	—
107	#334	—	106	4	no	—
108	C₅₃	#8812	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	53	4	no	—
111	C₂²×C₈	—	106	1	no	—
112	C₂²×C₈	—	106	1	no	—
113	C₂×C₄²	Dic₁₀₆	1	2	yes	—
114	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
117	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	4	4	no	—
121	C₄⋊C₈	—	106	2	no	—
122	C₄⋊C₈	—	106	2	no	—
123	C₄×C₈	—	53	4	no	—
124	C₄×C₈	—	53	4	no	—
125	C₂²×C₄	#11719919	1	2	yes	—
126	C₂²×C₄	#11719919	1	2	yes	—
127	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	—	53	8	no	—
133	C₂×C₈	—	106	4	no	—
134	C₂×C₈	—	106	4	no	—
135	C₂×C₈	—	212	1	no	—
136	C₂×C₈	—	212	1	no	—
137	C₂×C₈	—	212	2	no	—
138	C₂×C₈	—	212	2	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	D₄×D₅₃	1	2	yes	—
145	C₄²	Q₈×D₅₃	1	2	yes	—
146	C₂³	C₂×#11719919	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#13133152	1	4	yes	—
154	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
155	C₂×C₄	#13133213	1	1	yes	characteristic
156	C₂×C₄	#13133215	1	1	yes	characteristic
157	C₂×C₄	#13133216	1	2	yes	—
158	C₂×C₄	#13133223	1	2	yes	—
159	C₂×C₄	#13133225	1	2	yes	—
160	C₂×C₄	#13133243	1	2	yes	—
161	C₂×C₄	#13133248	1	2	yes	—
162	C₈	—	8	2	no	—
163	C₈	—	106	8	no	—
164	C₈	—	424	1	no	—
165	C₈	—	424	1	no	—
166	C₂²	#16810492	1	1	yes	characteristic
167	C₂²	#16810649	1	1	yes	characteristic
168	C₂²	#16810651	1	1	yes	characteristic
169	C₂²	#16810663	1	2	yes	—
170	C₂²	C₂×#13133152	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#16810648	1	4	yes	—
175	C₄	#16811010	1	1	yes	characteristic
176	C₄	#16811015	1	1	yes	characteristic
177	C₄	#16811029	1	2	yes	—
178	C₂	#20273498	1	2	yes	—
179	C₂	#20273506	1	2	yes	—
180	C₂	#20274868	1	1	yes	characteristic
181	C₂	#20274881	1	1	yes	characteristic
182	C₂	#20274886	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274832

C₂

yes

maximal, characteristic

#20274461

C₂

yes

maximal, characteristic

C₂×#16810375

C₂

yes

maximal

C₂×#16810373

C₂

yes

maximal

C₅₃×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

characteristic

C₂×C₄⋊#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

—

#16810378

—

#16810378

C₄

yes

—

#16810378

C₄

yes

—

#16810375

—

#16810373

—

C₂×C₄⋊C₈×C₅₃

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₂²×#88322

—

C₂²×#88322

C₂×C₄

yes

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄×#88322

D₄

yes

—

C₄×#88322

Q₈

yes

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#108

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#108

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#424

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1842

D₅₃

yes

—

—

—

—

C₂×C₅₃

#1573

yes

—

C₂×C₅₃

#1574

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

100

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

101

C₂×C₄×C₈

—

102

C₄³

C₂×D₅₃

yes

characteristic

103

#336

—

104

#336

—

105

#336

—

106

—

106

#335

—

106

—

107

#334

—

106

—

108

C₅₃

#8812

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

—

106

—

112

C₂²×C₈

—

106

—

113

C₂×C₄²

Dic₁₀₆

yes

—

114

C₂×C₄²

C₄×D₅₃

yes

characteristic

115

C₂×C₄²

C₄×D₅₃

yes

characteristic

116

C₂×C₄²

C₂²⋊D₅₃

yes

—

117

C₂×C₄²

C₂²×D₅₃

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

106

—

122

C₄⋊C₈

—

106

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₂²×C₄

#11719919

yes

—

126

C₂²×C₄

#11719919

yes

—

127

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

128

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

129

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

106

—

134

C₂×C₈

—

106

—

135

C₂×C₈

—

212

—

136

C₂×C₈

—

212

—

137

C₂×C₈

—

212

—

138

C₂×C₈

—

212

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

D₄×D₅₃

yes

—

145

C₄²

Q₈×D₅₃

yes

—

146

C₂³

C₂×#11719919

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#13133152

yes

—

154

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

155

C₂×C₄

#13133213

yes

characteristic

156

C₂×C₄

#13133215

yes

characteristic

157

C₂×C₄

#13133216

yes

—

158

C₂×C₄

#13133223

yes

—

159

C₂×C₄

#13133225

yes

—

160

C₂×C₄

#13133243

yes

—

161

C₂×C₄

#13133248

yes

—

162

C₈

—

163

C₈

—

106

—

164

C₈

—

424

—

165

C₈

—

424

—

166

C₂²

#16810492

yes

characteristic

167

C₂²

#16810649

yes

characteristic

168

C₂²

#16810651

yes

characteristic

169

C₂²

#16810663

yes

—

170

C₂²

C₂×#13133152

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#16810648

yes

—

175

C₄

#16811010

yes

characteristic

176

C₄

#16811015

yes

characteristic

177

C₄

#16811029

yes

—

178

C₂

#20273498

yes

—

179

C₂

#20273506

yes

—

180

C₂

#20274868

yes

characteristic

181

C₂

#20274881

yes

characteristic

182

C₂

#20274886

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#8812
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1842

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
108. C₅₃	#8812	semidirect

H ≅ G/N

type

108. C₅₃

#8812

semidirect

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

subgroup	index
#20274832	2
#20274461	2
C₂×#16810375	2
C₂×#16810373	2
C₅₃×#1842	2
#8812	53

subgroup

index

C₅₃×#1842

Subgroups — #27277854

Conjugacy classes of subgroups

Complements

Maximal subgroups