Group #27277087 — subgroups

Subgroups — #27277087

Order 13568.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20283645	C₂	1	1	yes	maximal, characteristic
2	#20273476	C₂	1	2	yes	maximal
3	#20273352	C₂	1	2	yes	maximal
4	#20272626	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1307	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	1	yes	characteristic
7	#16810750	—	2	2	no	—
8	#16810747	—	2	2	no	—
9	#16810680	C₂²	1	2	yes	—
10	#16810651	C₄	1	2	yes	—
11	#16810581	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810375	—	2	2	no	—
14	#16810373	—	2	2	no	—
15	#16810364	—	2	4	no	—
16	C₅₃×#432	C₂²	1	2	yes	—
17	C₅₃×#423	C₂²	1	1	yes	characteristic
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	#13133217	Q₈	1	2	yes	—
21	#13133209	D₄	1	2	yes	—
22	#13133196	—	2	2	no	—
23	C₄×Dic₁₀₆	—	2	2	no	—
24	#13133176	C₂×C₄	1	2	yes	—
25	C₄⋊#88322	—	2	2	no	—
26	#13133146	—	2	4	no	—
27	C₂²⋊#88322	—	2	2	no	—
28	#13133128	—	2	2	no	—
29	#13133127	—	2	2	no	—
30	C₄×#88322	D₄	1	2	yes	—
31	C₄×#88322	D₄	1	2	yes	—
32	C₈×Dic₅₃	—	2	2	no	—
33	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₂²⋊C₈×C₅₃	—	2	2	no	—
38	#97×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	2	yes	—
41	C₂×Dic₁₀₆	—	2	2	no	—
42	M₁₆×C₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	4	2	no	—
45	C₄²×C₅₃	C₂×D₄	1	2	yes	—
46	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²⋊Dic₅₃	—	2	2	no	—
49	C₄⋊Dic₅₃	—	2	2	no	—
50	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
51	#11719919	—	2	4	no	—
52	C₄×Dic₅₃	—	2	2	no	—
53	C₂×#88322	—	2	2	no	—
54	C₂×#88322	—	4	1	no	—
55	C₂×#88322	—	4	2	no	—
56	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×#88322	C₂²⋊C₄	1	2	yes	—
58	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
59	C₂³×C₅₃	#117	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#121	1	2	yes	—
66	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
67	C₂×Dic₅₃	—	2	4	no	—
68	Dic₁₀₆	—	4	4	no	—
69	C₈×C₅₃	—	4	2	no	—
70	C₈×C₅₃	—	8	2	no	—
71	#88322	—	2	1	no	—
72	#88322	—	2	2	no	—
73	#88322	—	8	2	no	—
74	#88322	#101	1	2	yes	—
75	#8631	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	—	2	1	no	—
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	#380	1	1	yes	characteristic
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
83	C₄×C₅₃	#413	1	1	yes	characteristic
84	Dic₅₃	—	4	4	no	—
85	#2960	—	53	1	no	—
86	#1568	—	53	2	no	—
87	#1537	—	53	2	no	—
88	#1307	—	53	1	no	—
89	#1307	D₅₃	1	1	yes	characteristic
90	C₂×C₅₃	—	4	1	no	—
91	C₂×C₅₃	#1801	1	1	yes	characteristic
92	C₂×C₅₃	#1814	1	2	yes	—
93	#533	—	53	1	no	—
94	#446	—	106	2	no	—
95	C₈×Q₈	—	106	2	no	—
96	#432	—	53	2	no	—
97	#432	C₂×D₅₃	1	2	yes	—
98	#424	—	53	2	no	—
99	#423	C₂×D₅₃	1	1	yes	characteristic
100	#405	—	53	2	no	—
101	C₂×C₄×C₈	—	53	1	no	—
102	#335	—	106	2	no	—
103	#334	—	106	2	no	—
104	#330	—	106	4	no	—
105	C₅₃	#8631	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	—	53	1	no	—
109	C₄⋊Q₈	—	53	2	no	—
110	#125	—	53	2	no	—
111	C₂²⋊Q₈	—	106	2	no	—
112	C₄×Q₈	—	106	2	no	—
113	#117	—	53	2	no	—
114	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	106	4	no	—
118	C₄⋊C₈	—	106	2	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	106	2	no	—
121	#97	—	2	2	no	—
122	#97	—	106	2	no	—
123	#97	—	106	2	no	—
124	C₄×C₈	—	53	2	no	—
125	C₄×C₈	—	53	2	no	—
126	C₄×C₈	—	106	2	no	—
127	C₂×Q₈	—	106	2	no	—
128	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
129	C₂²×C₄	#11719945	1	2	yes	—
130	M₁₆	—	4	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	53	1	no	—
134	C₂×C₈	—	53	1	no	—
135	C₂×C₈	—	53	2	no	—
136	C₂×C₈	—	106	2	no	—
137	C₂×C₈	—	212	1	no	—
138	C₂×C₈	—	212	2	no	—
139	C₄⋊C₄	—	53	4	no	—
140	C₄⋊C₄	—	106	4	no	—
141	C₄⋊C₄	—	106	2	no	—
142	C₂²⋊C₄	—	106	2	no	—
143	C₄²	—	106	2	no	—
144	C₄²	D₄×D₅₃	1	2	yes	—
145	C₄²	D₄×D₅₃	1	1	yes	characteristic
146	C₄²	Q₈×D₅₃	1	1	yes	characteristic
147	C₂³	#13133174	1	1	yes	characteristic
148	Q₈	—	212	4	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	106	4	no	—
153	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
155	C₂×C₄	#13133190	1	2	yes	—
156	C₂×C₄	#13133195	1	2	yes	—
157	C₈	—	4	2	no	—
158	C₈	—	8	2	no	—
159	C₈	—	53	2	no	—
160	C₈	—	106	1	no	—
161	C₈	—	106	2	no	—
162	C₈	—	424	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	#16810482	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	212	4	no	—
170	C₄	#101×D₅₃	1	1	yes	characteristic
171	C₄	#16810604	1	1	yes	characteristic
172	C₂	—	4	1	no	—
173	C₂	#20274710	1	1	yes	characteristic
174	C₂	#20274768	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283645

C₂

yes

maximal, characteristic

#20273476

C₂

yes

maximal

#20273352

C₂

yes

maximal

#20272626

C₂

yes

maximal, characteristic

C₅₃×#1307

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

—

—

C₂²

yes

—

#16810651

C₄

yes

—

#16810581

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

—

—

—

C₅₃×#432

C₂²

yes

—

C₅₃×#423

C₂²

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

#13133217

Q₈

yes

—

#13133209

D₄

yes

—

#13133196

—

C₄×Dic₁₀₆

—

#13133176

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

—

C₄×#88322

D₄

yes

—

C₈×Dic₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₄⋊C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂²⋊C₄

yes

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

yes

—

#8631

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#380

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1801

yes

characteristic

C₂×C₅₃

#1814

yes

—

#533

—

#446

—

106

—

C₈×Q₈

—

106

—

#432

—

#432

C₂×D₅₃

yes

—

#424

—

#423

C₂×D₅₃

yes

characteristic

100

#405

—

101

C₂×C₄×C₈

—

102

#335

—

106

—

103

#334

—

106

—

104

#330

—

106

—

105

C₅₃

#8631

yes

characteristic, complemented, Sylow

106

C₂×M₁₆

C₄×D₅₃

yes

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₄⋊Q₈

—

110

#125

—

111

C₂²⋊Q₈

—

106

—

112

C₄×Q₈

—

106

—

113

#117

—

114

C₂×C₄²

C₂²×D₅₃

yes

characteristic

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

106

—

118

C₄⋊C₈

—

106

—

119

C₂²⋊C₈

—

120

C₂²⋊C₈

—

106

—

121

#97

—

122

#97

—

106

—

123

#97

—

106

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

106

—

127

C₂×Q₈

—

106

—

128

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

129

C₂²×C₄

#11719945

yes

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

106

—

137

C₂×C₈

—

212

—

138

C₂×C₈

—

212

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

106

—

141

C₄⋊C₄

—

106

—

142

C₂²⋊C₄

—

106

—

143

C₄²

—

106

—

144

C₄²

D₄×D₅₃

yes

—

145

C₄²

D₄×D₅₃

yes

characteristic

146

C₄²

Q₈×D₅₃

yes

characteristic

147

C₂³

#13133174

yes

characteristic

148

Q₈

—

212

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

106

—

153

C₂×C₄

C₂²⋊C₄×D₅₃

yes

characteristic

154

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

155

C₂×C₄

#13133190

yes

—

156

C₂×C₄

#13133195

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

106

—

161

C₈

—

106

—

162

C₈

—

424

—

163

C₂²

—

164

C₂²

—

165

C₂²

#16810482

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

212

—

170

C₄

#101×D₅₃

yes

characteristic

171

C₄

#16810604

yes

characteristic

172

C₂

—

173

C₂

#20274710

yes

characteristic

174

C₂

#20274768

yes

—

	subgroup	G/H
center	C₂²	#16810482
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1307	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1307	D₅₃
hypocenter	C₅₃	#8631
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#16810482

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1307

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

N	H ≅ G/N	type
105. C₅₃	#8631	semidirect

H ≅ G/N

type

105. C₅₃

#8631

semidirect

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

subgroup	index
#20283645	2
#20273476	2
#20273352	2
#20272626	2
C₅₃×#1307	2
#8631	53

subgroup

index

C₅₃×#1307

Subgroups — #27277087

Conjugacy classes of subgroups

Complements

Maximal subgroups