Group #20275836 — subgroups

Subgroups — #20275836

Order 6784.

342 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 73 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×#13133247	C₂	1	2	yes	maximal
2	C₂²×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133135	C₂	1	2	yes	maximal, complemented
4	#16810388	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#408	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
7	#13133247	—	2	4	no	—
8	#13133247	—	2	4	no	—
9	C₂×Dic₂₁₂	—	2	2	no	—
10	C₂×Dic₂₁₂	—	2	4	no	—
11	C₂×#11719925	—	2	2	no	—
12	C₂×#11719923	—	2	2	no	—
13	C₂×#11719917	C₂²	1	2	yes	—
14	#13133135	—	2	4	no	—
15	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₅₃	—	2	2	no	—
19	#41×C₅₃	—	2	2	no	—
20	#41×C₅₃	—	2	2	no	—
21	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×D₄×C₅₃	—	2	1	no	—
23	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	—	2	1	no	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₃	—	2	2	no	—
27	C₂×Dic₁₀₆	—	2	2	no	—
28	C₂×Dic₁₀₆	—	2	2	no	—
29	C₂×Dic₁₀₆	—	2	2	no	—
30	C₂×Dic₁₀₆	—	2	2	no	—
31	C₂×Dic₁₀₆	—	2	2	no	—
32	C₂×Dic₁₀₆	D₄	1	4	yes	—
33	M₁₆×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₂×C₈×C₅₃	D₄	1	2	yes	—
37	#11719925	—	2	8	no	—
38	#11719923	—	2	8	no	—
39	#11719917	—	2	2	no	—
40	#11719917	D₄	1	4	yes	—
41	C₂×#88322	—	2	2	no	—
42	Dic₂₁₂	—	2	8	no	—
43	C₂³×C₅₃	—	2	1	no	—
44	C₂³×C₅₃	#41	1	1	yes	characteristic
45	Q₈×C₅₃	—	2	2	no	—
46	D₄×C₅₃	—	2	2	no	—
47	D₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₅₃	—	2	4	no	—
56	C₂×Dic₅₃	—	4	2	no	—
57	C₂×Dic₅₃	—	4	2	no	—
58	Dic₁₀₆	—	2	4	no	—
59	Dic₁₀₆	—	2	4	no	—
60	Dic₁₀₆	—	4	2	no	—
61	Dic₁₀₆	—	4	2	no	—
62	C₈×C₅₃	—	2	2	no	—
63	C₈×C₅₃	—	4	1	no	—
64	#88322	—	2	4	no	—
65	C₂²×C₅₃	—	2	2	no	—
66	C₂²×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	—	4	2	no	—
68	C₂²×C₅₃	#121	1	2	yes	—
69	C₂²×C₅₃	#121	1	1	yes	characteristic
70	C₄×C₅₃	—	4	1	no	—
71	C₄×C₅₃	#120	1	1	yes	characteristic
72	C₄×C₅₃	#120	1	2	yes	—
73	C₄×C₅₃	#127	1	1	yes	characteristic
74	Dic₅₃	—	4	4	no	—
75	#2020	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	—	2	2	no	—
77	C₂×C₅₃	—	8	1	no	—
78	C₂×C₅₃	#392	1	1	yes	characteristic
79	C₂×C₅₃	#473	1	2	yes	—
80	C₂×#137	—	53	2	no	—
81	C₂²×Q₁₆	—	53	1	no	—
82	C₂×#101	—	53	2	no	—
83	#408	D₅₃	1	1	yes	characteristic
84	#341	—	53	1	no	—
85	C₅₃	#2020	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
87	C₂²×Q₈	—	53	2	no	—
88	#137	—	106	4	no	—
89	#137	—	106	4	no	—
90	C₂×Q₁₆	—	106	4	no	—
91	C₂×Q₁₆	—	106	2	no	—
92	C₂×Q₁₆	—	106	2	no	—
93	C₂×SD₁₆	—	106	2	no	—
94	C₂×M₁₆	—	53	2	no	—
95	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
97	#101	—	106	4	no	—
98	C₂²⋊C₈	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	—	53	4	no	—
102	C₂×Q₈	—	106	2	no	—
103	C₂×Q₈	—	106	2	no	—
104	C₂×Q₈	—	106	2	no	—
105	C₂×Q₈	—	106	2	no	—
106	C₂×Q₈	—	106	2	no	—
107	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	D₂₁₂	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	106	2	no	—
112	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
113	Q₁₆	—	106	8	no	—
114	Q₁₆	—	106	8	no	—
115	SD₁₆	—	106	8	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	—	53	4	no	—
118	M₁₆	—	106	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	106	2	no	—
122	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
123	C₂³	—	2	1	no	—
124	C₂³	#11719945	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	106	4	no	—
127	Q₈	—	106	4	no	—
128	Q₈	—	212	2	no	—
129	Q₈	—	212	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	4	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	106	4	no	—
136	C₂×C₄	—	212	2	no	—
137	C₂×C₄	—	212	2	no	—
138	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
139	C₂×C₄	D₄×D₅₃	1	2	yes	—
140	C₂×C₄	D₄×D₅₃	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	4	1	no	—
144	C₈	—	106	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	#13133190	1	2	yes	—
149	C₂²	#13133191	1	1	yes	characteristic
150	C₄	—	4	1	no	—
151	C₄	—	212	4	no	—
152	C₄	#13133185	1	1	yes	characteristic
153	C₄	#13133187	1	2	yes	—
154	C₄	#13133213	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	8	1	no	—
157	C₂	#16810532	1	1	yes	characteristic
158	C₂	#16810898	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133247

C₂

yes

maximal

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic, complemented

C₂×#13133135

C₂

yes

maximal, complemented

#16810388

C₂

yes

maximal, characteristic, complemented

C₅₃×#408

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

#13133247

—

#13133247

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×#11719925

—

C₂×#11719923

—

C₂×#11719917

C₂²

yes

—

#13133135

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

—

—

—

D₄

yes

—

C₂×#88322

—

Dic₂₁₂

—

C₂³×C₅₃

—

C₂³×C₅₃

#41

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#120

yes

—

C₄×C₅₃

#127

yes

characteristic

Dic₅₃

—

#2020

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#392

yes

characteristic

C₂×C₅₃

#473

yes

—

C₂×#137

—

C₂²×Q₁₆

—

C₂×#101

—

#408

D₅₃

yes

characteristic

#341

—

C₅₃

#2020

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

C₂²×Q₈

—

#137

—

106

—

#137

—

106

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×SD₁₆

—

106

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#101

—

106

—

C₂²⋊C₈

—

#41

—

100

#41

—

101

C₂×Q₈

—

102

C₂×Q₈

—

106

—

103

C₂×Q₈

—

106

—

104

C₂×Q₈

—

106

—

105

C₂×Q₈

—

106

—

106

C₂×Q₈

—

106

—

107

C₂×Q₈

D₂₁₂

yes

characteristic

108

C₂×D₄

—

109

C₂×D₄

D₂₁₂

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

—

106

—

112

C₂²×C₄

C₂²×D₅₃

yes

characteristic

113

Q₁₆

—

106

—

114

Q₁₆

—

106

—

115

SD₁₆

—

106

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

106

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

106

—

122

C₂×C₈

C₂²⋊D₅₃

yes

—

123

C₂³

—

124

C₂³

#11719945

yes

characteristic

125

Q₈

—

126

Q₈

—

106

—

127

Q₈

—

106

—

128

Q₈

—

212

—

129

Q₈

—

212

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

212

—

137

C₂×C₄

—

212

—

138

C₂×C₄

C₂×D₂₁₂

yes

characteristic

139

C₂×C₄

D₄×D₅₃

yes

—

140

C₂×C₄

D₄×D₅₃

yes

—

141

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

106

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#13133190

yes

—

149

C₂²

#13133191

yes

characteristic

150

C₄

—

151

C₄

—

212

—

152

C₄

#13133185

yes

characteristic

153

C₄

#13133187

yes

—

154

C₄

#13133213

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#16810532

yes

characteristic

158

C₂

#16810898

yes

—

	subgroup	G/H
center	C₂²	#13133191
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#408	C₂
socle	C₂²×C₅₃	#121
hypercenter	#408	D₅₃
hypocenter	C₅₃	#2020
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#13133191

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#408

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. C₂²×Dic₂₁₂	C₂	semidirect
3. C₂×#13133135	C₂	semidirect
4. #16810388	C₂	semidirect
85. C₅₃	#2020	semidirect

H ≅ G/N

type

2. C₂²×Dic₂₁₂

C₂

semidirect

3. C₂×#13133135

C₂

semidirect

4. #16810388

C₂

semidirect

85. C₅₃

#2020

semidirect

Subgroups — #20275836

Conjugacy classes of subgroups

Complements