Group #20276124 — subgroups

Subgroups — #20276124

Order 6784.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂×#13133140	C₂	1	2	yes	maximal
2	#16810529	C₂	1	1	yes	maximal, characteristic
3	#16810385	C₂	1	2	yes	maximal
4	#16810378	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133140	—	2	4	no	—
10	#13133124	—	2	1	no	—
11	#13133124	—	2	1	no	—
12	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
13	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	—	2	1	no	—
21	C₂²×Dic₅₃	—	2	1	no	—
22	C₂×Dic₁₀₆	—	2	2	no	—
23	C₂×Dic₁₀₆	—	2	2	no	—
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×Dic₁₀₆	D₄	1	2	yes	—
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₄⋊C₄×C₅₃	—	4	2	no	—
29	C₄⋊C₄×C₅₃	—	4	2	no	—
30	C₄²×C₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	2	1	no	—
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
44	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	2	yes	—
49	C₂×Dic₅₃	—	2	2	no	—
50	C₂×Dic₅₃	—	4	1	no	—
51	C₂×Dic₅₃	—	4	1	no	—
52	C₂×Dic₅₃	—	4	1	no	—
53	C₂×Dic₅₃	—	4	1	no	—
54	C₂×Dic₅₃	—	4	1	no	—
55	Dic₁₀₆	—	2	4	no	—
56	Dic₁₀₆	—	4	2	no	—
57	#88322	—	4	2	no	—
58	C₂²×C₅₃	#121	1	1	yes	characteristic
59	C₂²×C₅₃	#123	1	1	yes	characteristic
60	C₂²×C₅₃	#124	1	1	yes	characteristic
61	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₅₃	#137	1	1	yes	characteristic
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	8	2	no	—
68	C₄×C₅₃	#123	1	2	yes	—
69	C₄×C₅₃	#126	1	2	yes	—
70	Dic₅₃	—	4	2	no	—
71	Dic₅₃	—	8	1	no	—
72	#2072	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	#397	1	1	yes	characteristic
74	C₂×C₅₃	#460	1	1	yes	characteristic
75	C₂×C₅₃	#462	1	1	yes	characteristic
76	C₂×C₅₃	#483	1	1	yes	characteristic
77	C₂×C₅₃	#484	1	1	yes	characteristic
78	C₂×C₅₃	#487	1	1	yes	characteristic
79	C₂×C₅₃	#489	1	1	yes	characteristic
80	C₂×#103	—	53	2	no	—
81	#391	—	53	1	no	—
82	#384	D₅₃	1	1	yes	characteristic
83	#340	—	53	2	no	—
84	#336	—	53	1	no	—
85	C₅₃	#2072	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	53	1	no	—
87	C₂²×C₈	—	53	2	no	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	53	1	no	—
91	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
92	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
93	#103	—	106	4	no	—
94	#95	—	106	1	no	—
95	#95	—	106	1	no	—
96	C₂×Q₈	—	53	2	no	—
97	C₂×Q₈	—	106	1	no	—
98	C₂×Q₈	—	106	2	no	—
99	C₂×Q₈	—	106	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	106	1	no	—
102	C₂²×C₄	—	106	1	no	—
103	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	C₂×C₈	—	106	2	no	—
107	C₂×C₈	—	106	2	no	—
108	C₂×C₈	—	106	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	106	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
116	Q₈	—	106	4	no	—
117	Q₈	—	212	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	106	2	no	—
126	C₂×C₄	—	212	1	no	—
127	C₂×C₄	—	212	1	no	—
128	C₂×C₄	—	212	1	no	—
129	C₂×C₄	—	212	1	no	—
130	C₂×C₄	—	212	1	no	—
131	C₂×C₄	#11719923	1	2	yes	—
132	C₂×C₄	#11719925	1	2	yes	—
133	C₂×C₄	#11719945	1	2	yes	—
134	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
135	C₂×C₄	#11719947	1	1	yes	characteristic
136	C₂×C₄	#11719949	1	2	yes	—
137	C₈	—	212	2	no	—
138	C₂²	#13133188	1	1	yes	characteristic
139	C₂²	#13133200	1	1	yes	characteristic
140	C₂²	#13133206	1	1	yes	characteristic
141	C₂²	C₂×#11719923	1	1	yes	characteristic
142	C₂²	C₂×#11719925	1	1	yes	characteristic
143	C₂²	#13133243	1	1	yes	characteristic
144	C₂²	#13133248	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	8	2	no	—
148	C₄	—	424	1	no	—
149	C₄	—	212	2	no	—
150	C₄	#13133202	1	2	yes	—
151	C₄	#13133212	1	2	yes	—
152	C₂	#16810549	1	1	yes	characteristic
153	C₂	#16810820	1	1	yes	characteristic
154	C₂	#16810830	1	1	yes	characteristic
155	C₂	#16810953	1	1	yes	characteristic
156	C₂	#16810958	1	1	yes	characteristic
157	C₂	#16810975	1	1	yes	characteristic
158	C₂	#16810985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133140

C₂

yes

maximal

#16810529

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

#16810378

C₂

yes

maximal, characteristic

C₅₃×#384

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#123

yes

—

C₄×C₅₃

#126

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#340

—

#336

—

C₅₃

#2072

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

106

—

#95

—

106

—

#95

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

100

C₂²×C₄

—

101

C₂²×C₄

—

106

—

102

C₂²×C₄

—

106

—

103

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

104

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

105

C₂²×C₄

C₂²×D₅₃

yes

characteristic

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

106

—

114

C₄²

—

115

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

116

Q₈

—

106

—

117

Q₈

—

212

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

212

—

127

C₂×C₄

—

212

—

128

C₂×C₄

—

212

—

129

C₂×C₄

—

212

—

130

C₂×C₄

—

212

—

131

C₂×C₄

#11719923

yes

—

132

C₂×C₄

#11719925

yes

—

133

C₂×C₄

#11719945

yes

—

134

C₂×C₄

D₄×D₅₃

yes

characteristic

135

C₂×C₄

#11719947

yes

characteristic

136

C₂×C₄

#11719949

yes

—

137

C₈

—

212

—

138

C₂²

#13133188

yes

characteristic

139

C₂²

#13133200

yes

characteristic

140

C₂²

#13133206

yes

characteristic

141

C₂²

C₂×#11719923

yes

characteristic

142

C₂²

C₂×#11719925

yes

characteristic

143

C₂²

#13133243

yes

characteristic

144

C₂²

#13133248

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

424

—

149

C₄

—

212

—

150

C₄

#13133202

yes

—

151

C₄

#13133212

yes

—

152

C₂

#16810549

yes

characteristic

153

C₂

#16810820

yes

characteristic

154

C₂

#16810830

yes

characteristic

155

C₂

#16810953

yes

characteristic

156

C₂

#16810958

yes

characteristic

157

C₂

#16810975

yes

characteristic

158

C₂

#16810985

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#384	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#384	D₅₃
hypocenter	C₅₃	#2072
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#384

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
85. C₅₃	#2072	semidirect

H ≅ G/N

type

85. C₅₃

#2072

semidirect

Subgroups — #20276124

Conjugacy classes of subgroups

Complements