Group #27271019 — subgroups

Subgroups — #27271019

Order 13568.

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

normal subgroups only

This group has 262 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×#13133149	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810676	—	2	1	no	—
5	C₂×#13133149	C₄	1	2	yes	—
6	C₄×C₄⋊Dic₅₃	—	2	1	no	—
7	#16810386	—	2	1	no	—
8	#16810376	C₄	1	2	yes	complemented
9	#16810374	—	2	1	no	—
10	C₂×M₃₂×C₅₃	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#346	C₄	1	2	yes	complemented
13	C₄×C₁₆×C₅₃	—	2	1	no	—
14	C₂×C₄⋊Dic₅₃	—	2	2	no	—
15	C₂×C₄×Dic₅₃	—	4	1	no	—
16	#13133149	—	2	2	no	—
17	#13133149	C₈	1	4	yes	complemented
18	C₄⋊#88322	—	2	2	no	—
19	C₂²⋊#88322	—	4	1	no	—
20	C₄×#88322	—	4	1	no	—
21	#13133124	—	4	1	no	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	M₃₂×C₅₃	—	2	4	no	—
26	C₂×C₁₆×C₅₃	—	2	2	no	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
34	C₂²×Dic₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
42	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
46	C₄⋊Dic₅₃	—	2	4	no	—
47	C₄⋊Dic₅₃	—	4	2	no	—
48	C₄×Dic₅₃	—	8	2	no	—
49	C₂×#88322	—	4	2	no	—
50	C₁₆×C₅₃	—	2	4	no	—
51	C₂³×C₅₃	#102	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#104	1	2	yes	—
60	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
62	C₂×Dic₅₃	—	4	4	no	—
63	C₂×Dic₅₃	—	8	2	no	—
64	C₂×Dic₅₃	—	8	2	no	—
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
68	C₈×C₅₃	#97	1	1	yes	characteristic
69	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
70	#88322	—	8	2	no	—
71	#7175	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
75	C₂²×C₅₃	#340	1	1	yes	characteristic
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	—	2	1	no	—
79	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
80	C₄×C₅₃	#336	1	1	yes	characteristic
81	C₄×C₅₃	#339	1	1	yes	characteristic
82	C₄×C₅₃	#360	1	2	yes	—
83	Dic₅₃	—	8	4	no	—
84	C₄×M₃₂	D₅₃	1	1	yes	characteristic
85	C₄×#106	—	53	1	no	—
86	#1564	—	53	1	no	—
87	C₂×C₅₃	—	2	2	no	—
88	C₂×C₅₃	#1279	1	1	yes	characteristic
89	C₂×C₅₃	#1386	1	1	yes	characteristic
90	C₂×C₅₃	#1389	1	1	yes	characteristic
91	C₂×M₃₂	—	2	1	no	—
92	#431	—	106	1	no	—
93	C₂×#106	—	53	2	no	—
94	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
95	C₄×C₄⋊C₄	—	106	1	no	—
96	#346	Dic₅₃	1	2	yes	complemented
97	C₄×C₁₆	—	2	1	no	—
98	#340	—	106	1	no	—
99	#335	—	53	2	no	—
100	#334	—	106	1	no	—
101	C₅₃	#7175	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	106	2	no	—
105	C₂×C₄²	—	212	1	no	—
106	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
107	M₃₂	—	2	4	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#106	—	53	4	no	—
110	#106	—	106	2	no	—
111	C₄⋊C₈	—	106	2	no	—
112	C₂²⋊C₈	—	212	1	no	—
113	C₄×C₈	—	212	1	no	—
114	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
115	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
117	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
118	#95	—	212	1	no	—
119	C₂²×C₄	—	212	2	no	—
120	C₂²×C₄	#11719913	1	1	yes	characteristic
121	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
122	C₂²×C₄	#11719921	1	1	yes	characteristic
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	212	2	no	—
125	C₂×C₈	C₈×D₅₃	1	2	yes	—
126	C₂×C₈	#11719912	1	2	yes	—
127	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
128	C₂×C₈	#11719919	1	1	yes	characteristic
129	C₂×C₈	#11719919	1	1	yes	characteristic
130	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
131	C₄⋊C₄	—	106	4	no	—
132	C₄⋊C₄	—	212	2	no	—
133	C₄²	—	424	2	no	—
134	C₄²	#11719913	1	1	yes	characteristic
135	C₄²	Dic₂₁₂	1	1	yes	characteristic
136	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
137	C₄²	#11719921	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₂³	#13133139	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	424	2	no	—
143	C₂×C₄	—	424	2	no	—
144	C₂×C₄	—	212	4	no	—
145	C₂×C₄	#13133124	1	1	yes	characteristic
146	C₂×C₄	#13133129	1	2	yes	—
147	C₂×C₄	#13133139	1	1	yes	characteristic
148	C₂×C₄	#13133142	1	1	yes	characteristic
149	C₂×C₄	#13133142	1	1	yes	characteristic
150	C₂×C₄	#13133145	1	2	yes	—
151	C₂×C₄	#13133149	1	1	yes	characteristic
152	C₂×C₄	#13133151	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	424	2	no	—
156	C₈	C₈×Dic₅₃	1	1	yes	characteristic
157	C₈	#13133127	1	1	yes	characteristic
158	C₈	#13133146	1	2	yes	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	#16810352	1	2	yes	—
162	C₂²	#16810386	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	424	4	no	—
167	C₄	#16810355	1	2	yes	—
168	C₄	#16810377	1	1	yes	characteristic
169	C₄	#16810383	1	1	yes	characteristic
170	C₄	#16810434	1	2	yes	—
171	C₂	—	2	2	no	—
172	C₂	#20272555	1	1	yes	characteristic
173	C₂	#20272821	1	1	yes	characteristic
174	C₂	#20272830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133149

C₂

yes

maximal, characteristic

#20273463

C₂

yes

maximal, characteristic

C₄×M₃₂×C₅₃

C₂

yes

maximal, characteristic

#16810676

—

C₂×#13133149

C₄

yes

—

C₄×C₄⋊Dic₅₃

—

#16810386

—

#16810376

C₄

yes

complemented

#16810374

—

C₂×M₃₂×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

complemented

C₄×C₁₆×C₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄×Dic₅₃

—

#13133149

—

#13133149

C₈

yes

—

—

—

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

SD₁₆

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

SD₁₆

yes

characteristic

C₄²×C₅₃

Q₁₆

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₁₆×C₅₃

—

C₂³×C₅₃

#102

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

characteristic

C₈×C₅₃

#97

yes

characteristic

C₈×C₅₃

C₄⋊C₈

yes

—

#88322

—

#7175

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

D₄⋊C₈

yes

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

Q₈⋊C₈

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

C₄×M₃₂

D₅₃

yes

characteristic

C₄×#106

—

#1564

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×M₃₂

—

#431

—

106

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₄⋊C₄

—

106

—

#346

Dic₅₃

yes

complemented

C₄×C₁₆

—

#340

—

106

—

#335

—

100

#334

—

106

—

101

C₅₃

#7175

yes

characteristic, complemented, Sylow

102

C₂²×C₈

C₄×D₅₃

yes

characteristic

103

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

104

C₂×C₄⋊C₄

—

106

—

105

C₂×C₄²

—

212

—

106

C₂×C₄²

D₂₁₂

yes

characteristic

107

M₃₂

—

108

C₂×C₁₆

—

109

#106

—

110

#106

—

106

—

111

C₄⋊C₈

—

106

—

112

C₂²⋊C₈

—

212

—

113

C₄×C₈

—

212

—

114

C₄×C₈

Dic₁₀₆

yes

characteristic

115

C₄×C₈

C₄×D₅₃

yes

characteristic

116

C₄×C₈

C₂×Dic₅₃

yes

characteristic

117

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

118

#95

—

212

—

119

C₂²×C₄

—

212

—

120

C₂²×C₄

#11719913

yes

characteristic

121

C₂²×C₄

D₄₂₄

yes

characteristic

122

C₂²×C₄

#11719921

yes

characteristic

123

C₂×C₈

—

124

C₂×C₈

—

212

—

125

C₂×C₈

C₈×D₅₃

yes

—

126

C₂×C₈

#11719912

yes

—

127

C₂×C₈

C₄×Dic₅₃

yes

characteristic

128

C₂×C₈

#11719919

yes

characteristic

129

C₂×C₈

#11719919

yes

characteristic

130

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

131

C₄⋊C₄

—

106

—

132

C₄⋊C₄

—

212

—

133

C₄²

—

424

—

134

C₄²

#11719913

yes

characteristic

135

C₄²

Dic₂₁₂

yes

characteristic

136

C₄²

C₄⋊Dic₅₃

yes

characteristic

137

C₄²

#11719921

yes

characteristic

138

C₁₆

—

139

C₂³

#13133139

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

424

—

143

C₂×C₄

—

424

—

144

C₂×C₄

—

212

—

145

C₂×C₄

#13133124

yes

characteristic

146

C₂×C₄

#13133129

yes

—

147

C₂×C₄

#13133139

yes

characteristic

148

C₂×C₄

#13133142

yes

characteristic

149

C₂×C₄

#13133142

yes

characteristic

150

C₂×C₄

#13133145

yes

—

151

C₂×C₄

#13133149

yes

characteristic

152

C₂×C₄

#13133151

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

—

424

—

156

C₈

C₈×Dic₅₃

yes

characteristic

157

C₈

#13133127

yes

characteristic

158

C₈

#13133146

yes

—

159

C₂²

—

160

C₂²

—

161

C₂²

#16810352

yes

—

162

C₂²

#16810386

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

424

—

167

C₄

#16810355

yes

—

168

C₄

#16810377

yes

characteristic

169

C₄

#16810383

yes

characteristic

170

C₄

#16810434

yes

—

171

C₂

—

172

C₂

#20272555

yes

characteristic

173

C₂

#20272821

yes

characteristic

174

C₂

#20272830

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133139
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#102
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#7175
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂×C₄

#13133139

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₄×M₃₂×C₅₃

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×C₈×C₅₃

C₂²

hypercenter

C₄×M₃₂

D₅₃

hypocenter

C₅₃

#7175

LCS

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
8. #16810376	C₄	semidirect
12. C₅₃×#346	C₄	semidirect
17. #13133149	C₈	semidirect
96. #346	Dic₅₃	semidirect
101. C₅₃	#7175	semidirect

H ≅ G/N

type

8. #16810376

C₄

semidirect

12. C₅₃×#346

C₄

semidirect

17. #13133149

C₈

semidirect

96. #346

Dic₅₃

semidirect

101. C₅₃

#7175

semidirect

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

subgroup	index
C₄×#13133149	2
#20273463	2
C₄×M₃₂×C₅₃	2
#7175	53

subgroup

index

C₄×#13133149

#20273463

C₄×M₃₂×C₅₃

#7175

Subgroups — #27271019

Conjugacy classes of subgroups

Complements

Maximal subgroups