Group #27270846 — subgroups

Subgroups — #27270846

Order 13568.

294 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20272797	C₂	1	2	yes	maximal, complemented
2	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133157	—	2	2	no	—
4	C₂×#127×C₅₃	C₄	1	1	yes	characteristic
5	C₂×#102×C₅₃	—	2	2	no	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#384	C₄	1	1	yes	characteristic
8	#13133157	—	2	4	no	—
9	#13133157	—	4	2	no	—
10	C₂×#11719908	—	4	2	no	—
11	C₂²×D₄×C₅₃	—	2	1	no	—
12	C₂²×D₄×C₅₃	—	4	1	no	—
13	C₂²×C₈×C₅₃	D₄	1	2	yes	—
14	#127×C₅₃	—	2	1	no	—
15	#127×C₅₃	—	4	1	no	—
16	#127×C₅₃	C₈	1	2	yes	—
17	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	#102×C₅₃	—	4	2	no	—
21	#102×C₅₃	—	4	2	no	—
22	C₄×C₈×C₅₃	—	4	1	no	—
23	C₂⁴×C₅₃	—	4	1	no	—
24	C₂×D₄×C₅₃	—	2	2	no	—
25	C₂×D₄×C₅₃	—	4	2	no	—
26	C₂×D₄×C₅₃	—	4	2	no	—
27	C₂×D₄×C₅₃	—	4	1	no	—
28	C₂×D₄×C₅₃	—	4	2	no	—
29	C₂×D₄×C₅₃	—	4	2	no	—
30	C₂×D₄×C₅₃	—	8	2	no	—
31	C₂×D₄×C₅₃	—	8	2	no	—
32	C₂²×C₄×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	—	4	1	no	—
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	D₈	1	2	yes	—
38	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
39	C₄⋊C₄×C₅₃	—	4	2	no	—
40	C₄⋊C₄×C₅₃	—	8	2	no	—
41	C₄²×C₅₃	—	2	1	no	—
42	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
44	#11719908	—	4	4	no	—
45	C₂³×C₅₃	—	4	2	no	—
46	C₂³×C₅₃	—	8	2	no	—
47	C₂³×C₅₃	—	8	2	no	—
48	C₂³×C₅₃	—	8	1	no	—
49	C₂³×C₅₃	—	8	1	no	—
50	C₂³×C₅₃	#100	1	1	yes	characteristic
51	D₄×C₅₃	—	4	4	no	—
52	D₄×C₅₃	—	8	4	no	—
53	D₄×C₅₃	—	8	4	no	—
54	D₄×C₅₃	—	8	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	8	2	no	—
60	C₂×C₄×C₅₃	—	8	1	no	—
61	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#102	1	2	yes	—
64	C₂×C₄×C₅₃	#104	1	2	yes	—
65	C₈×C₅₃	—	2	2	no	—
66	C₈×C₅₃	—	2	2	no	—
67	#7107	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	—	8	4	no	—
71	C₂²×C₅₃	—	8	2	no	—
72	C₂²×C₅₃	—	16	2	no	—
73	C₂²×C₅₃	—	16	1	no	—
74	C₂²×C₅₃	—	16	1	no	—
75	C₂²×C₅₃	#323	1	1	yes	characteristic
76	C₂²×C₅₃	#329	1	1	yes	characteristic
77	C₂²×C₅₃	#331	1	1	yes	characteristic
78	C₄×C₅₃	—	2	2	no	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	16	1	no	—
81	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
82	C₄×C₅₃	#327	1	2	yes	—
83	#1957	D₅₃	1	1	yes	characteristic
84	#1376	—	53	2	no	—
85	C₂×C₅₃	—	2	1	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	16	2	no	—
88	C₂×C₅₃	#1270	1	1	yes	characteristic
89	C₂×C₅₃	#1357	1	1	yes	characteristic
90	C₂×C₅₃	#1359	1	1	yes	characteristic
91	C₂×#127	Dic₅₃	1	1	yes	characteristic
92	C₂×M₃₂	—	106	2	no	—
93	C₂×#102	—	2	2	no	—
94	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
95	#384	Dic₅₃	1	1	yes	characteristic
96	C₅₃	#7107	1	1	yes	characteristic, complemented, Sylow
97	C₂²×D₄	—	2	1	no	—
98	C₂²×D₄	—	4	1	no	—
99	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
100	#127	—	2	1	no	—
101	#127	—	4	1	no	—
102	#127	#88322	1	2	yes	—
103	C₂×C₄⋊C₄	—	2	1	no	—
104	C₂×C₄⋊C₄	—	4	1	no	—
105	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
106	M₃₂	—	106	4	no	—
107	M₃₂	—	212	2	no	—
108	C₂×C₁₆	—	212	2	no	—
109	#102	—	4	2	no	—
110	#102	—	4	2	no	—
111	C₄×C₈	—	4	1	no	—
112	C₂⁴	—	4	1	no	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	4	2	no	—
115	C₂×D₄	—	4	2	no	—
116	C₂×D₄	—	4	1	no	—
117	C₂×D₄	—	4	2	no	—
118	C₂×D₄	—	4	2	no	—
119	C₂×D₄	—	8	2	no	—
120	C₂×D₄	—	8	2	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	4	1	no	—
123	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	#11719922	1	2	yes	—
127	C₂×C₈	#11719923	1	2	yes	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	8	2	no	—
130	C₄²	—	2	1	no	—
131	C₄²	C₂×#88322	1	1	yes	characteristic
132	C₄²	#11719917	1	1	yes	characteristic
133	C₁₆	—	212	4	no	—
134	C₂³	—	4	2	no	—
135	C₂³	—	8	2	no	—
136	C₂³	—	8	2	no	—
137	C₂³	—	8	1	no	—
138	C₂³	—	8	1	no	—
139	C₂³	#13133134	1	1	yes	characteristic
140	D₄	—	4	4	no	—
141	D₄	—	8	4	no	—
142	D₄	—	8	4	no	—
143	D₄	—	8	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
151	C₂×C₄	#13133132	1	1	yes	characteristic
152	C₂×C₄	#13133138	1	2	yes	—
153	C₂×C₄	#13133144	1	2	yes	—
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	8	4	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	16	2	no	—
161	C₂²	—	16	1	no	—
162	C₂²	—	16	1	no	—
163	C₂²	#16810347	1	1	yes	characteristic
164	C₂²	#16810363	1	1	yes	characteristic
165	C₂²	#16810367	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	16	1	no	—
169	C₄	D₄⋊#88322	1	2	yes	—
170	C₄	#16810357	1	2	yes	—
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	16	2	no	—
174	C₂	#20272536	1	1	yes	characteristic
175	C₂	#20272752	1	1	yes	characteristic
176	C₂	#20272756	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272797

C₂

yes

maximal, complemented

C₅₃×#1957

C₂

yes

maximal, characteristic

C₂×#13133157

—

C₂×#127×C₅₃

C₄

yes

characteristic

C₂×#102×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₄

yes

—

—

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

#127×C₅₃

—

#127×C₅₃

—

#127×C₅₃

C₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#102×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#100

yes

characteristic

D₄×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

C₂²⋊C₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#7107

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

D₄⋊C₈

yes

—

C₄×C₅₃

#327

yes

—

#1957

D₅₃

yes

characteristic

#1376

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

Dic₅₃

yes

characteristic

C₂×M₃₂

—

106

—

C₂×#102

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#384

Dic₅₃

yes

characteristic

C₅₃

#7107

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

100

#127

—

101

#127

—

102

#127

#88322

yes

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

106

M₃₂

—

106

—

107

M₃₂

—

212

—

108

C₂×C₁₆

—

212

—

109

#102

—

110

#102

—

111

C₄×C₈

—

112

C₂⁴

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

#11719922

yes

—

127

C₂×C₈

#11719923

yes

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄²

—

131

C₄²

C₂×#88322

yes

characteristic

132

C₄²

#11719917

yes

characteristic

133

C₁₆

—

212

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

#13133134

yes

characteristic

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

C₂²⋊#88322

yes

characteristic

151

C₂×C₄

#13133132

yes

characteristic

152

C₂×C₄

#13133138

yes

—

153

C₂×C₄

#13133144

yes

—

154

C₈

—

155

C₈

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#16810347

yes

characteristic

164

C₂²

#16810363

yes

characteristic

165

C₂²

#16810367

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

D₄⋊#88322

yes

—

170

C₄

#16810357

yes

—

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

#20272536

yes

characteristic

175

C₂

#20272752

yes

characteristic

176

C₂

#20272756

yes

characteristic

	subgroup	G/H
center	C₂²	#16810367
commutator	C₄²×C₅₃	C₂×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1957	C₂
socle	C₂²×C₅₃	#331
prime radical	C₂×#127×C₅₃	C₄
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1957	D₅₃
hypocenter	C₅₃	#7107
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1957

subgroup

G/H

center

C₂²

#16810367

commutator

C₄²×C₅₃

C₂×C₈

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₅₃×#1957

C₂

socle

C₂²×C₅₃

#331

prime radical

C₂×#127×C₅₃

C₄

cosocle

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1957

N	H ≅ G/N	type
1. #20272797	C₂	semidirect
96. C₅₃	#7107	semidirect

H ≅ G/N

type

1. #20272797

C₂

semidirect

96. C₅₃

#7107

semidirect

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

subgroup	index
#20272797	2
C₅₃×#1957	2
#7107	53

subgroup

index

#20272797

C₅₃×#1957

#7107

Subgroups — #27270846

Conjugacy classes of subgroups

Complements

Maximal subgroups