Group #20275433 — subgroups

Subgroups — #20275433

Order 6784.

260 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133146	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810377	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
10	C₄⋊#88322	—	2	2	no	—
11	#13133146	—	2	4	no	—
12	#13133124	C₄	1	2	yes	—
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	—	2	1	no	—
24	C₂×C₈×C₅₃	—	2	1	no	—
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	D₄	1	2	yes	—
27	C₂×C₈×C₅₃	Q₈	1	2	yes	—
28	C₄²×C₅₃	D₄	1	2	yes	—
29	C₄²×C₅₃	Q₈	1	2	yes	—
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	C₈	1	4	yes	complemented
32	C₄×Dic₅₃	—	2	2	no	—
33	C₄×Dic₅₃	—	2	2	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	—	2	2	no	—
37	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	2	no	—
51	C₂×Dic₅₃	—	2	2	no	—
52	C₂×Dic₅₃	—	2	2	no	—
53	C₂×Dic₅₃	—	2	2	no	—
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
56	C₈×C₅₃	—	2	2	no	—
57	C₈×C₅₃	—	2	2	no	—
58	#88322	—	4	2	no	—
59	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₅₃	#131	1	1	yes	characteristic
65	C₂²×C₅₃	#131	1	1	yes	characteristic
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	C₄⋊C₈	1	4	yes	complemented
68	C₄×C₅₃	#121	1	1	yes	characteristic
69	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₄×C₅₃	#125	1	1	yes	characteristic
71	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
72	Dic₅₃	—	2	4	no	—
73	Dic₅₃	—	4	2	no	—
74	#1939	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	#389	1	1	yes	characteristic
76	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₅₃	#424	1	1	yes	characteristic
78	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
79	C₂×C₅₃	#436	1	1	yes	characteristic
80	C₂×C₅₃	C₈×Q₈	1	1	yes	characteristic
81	C₂×C₅₃	#446	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	53	2	no	—
83	C₂×C₄⋊C₈	—	53	1	no	—
84	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
85	C₄×C₄⋊C₄	—	53	1	no	—
86	#336	—	53	2	no	—
87	C₅₃	#1939	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	53	2	no	—
89	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	53	1	no	—
92	C₂×C₄⋊C₄	—	53	1	no	—
93	C₂×C₄²	—	53	2	no	—
94	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
95	C₄⋊C₈	—	106	4	no	—
96	C₄⋊C₈	—	106	2	no	—
97	C₄×C₈	—	2	2	no	—
98	#95	—	53	2	no	—
99	C₂²×C₄	—	53	2	no	—
100	C₂²×C₄	—	53	2	no	—
101	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
102	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	106	2	no	—
109	C₂×C₈	—	106	2	no	—
110	C₂×C₈	—	106	2	no	—
111	C₂×C₈	Dic₁₀₆	1	2	yes	—
112	C₂×C₈	D₂₁₂	1	2	yes	—
113	C₄⋊C₄	—	53	4	no	—
114	C₄⋊C₄	—	106	2	no	—
115	C₄²	—	106	2	no	—
116	C₄²	—	106	2	no	—
117	C₄²	Dic₁₀₆	1	2	yes	—
118	C₄²	C₂²⋊D₅₃	1	2	yes	—
119	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	53	4	no	—
123	C₂×C₄	—	106	2	no	—
124	C₂×C₄	—	106	2	no	—
125	C₂×C₄	—	106	2	no	—
126	C₂×C₄	—	106	2	no	—
127	C₂×C₄	—	106	2	no	—
128	C₂×C₄	C₈×D₅₃	1	2	yes	—
129	C₂×C₄	#11719912	1	2	yes	—
130	C₂×C₄	#11719919	1	2	yes	—
131	C₂×C₄	#11719919	1	2	yes	—
132	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
133	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
134	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
135	C₂×C₄	#11719945	1	1	yes	characteristic
136	C₂×C₄	#11719945	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	212	2	no	—
141	C₂²	C₂×#11719919	1	1	yes	characteristic
142	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
143	C₂²	C₄×Dic₁₀₆	1	1	yes	characteristic
144	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
145	C₂²	C₂×#11719912	1	1	yes	characteristic
146	C₂²	#13133223	1	1	yes	characteristic
147	C₂²	#13133223	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	106	4	no	—
150	C₄	—	212	2	no	—
151	C₄	#13133146	1	4	yes	complemented
152	C₄	#13133191	1	1	yes	characteristic
153	C₄	#13133196	1	1	yes	characteristic
154	C₄	#13133209	1	1	yes	characteristic
155	C₄	#13133217	1	1	yes	characteristic
156	C₂	#16810521	1	1	yes	characteristic
157	C₂	C₂×#13133146	1	1	yes	characteristic
158	C₂	#16810649	1	1	yes	characteristic
159	C₂	C₈×D₂₁₂	1	1	yes	characteristic
160	C₂	#16810702	1	1	yes	characteristic
161	C₂	C₈×Dic₁₀₆	1	1	yes	characteristic
162	C₂	#16810753	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₂×#13133146

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄⋊#88322

—

#13133146

—

#13133124

C₄

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₈

yes

complemented

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

characteristic

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₈

yes

complemented

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#125

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₅₃

#424

yes

characteristic

C₂×C₅₃

C₈×D₄

yes

characteristic

C₂×C₅₃

#436

yes

characteristic

C₂×C₅₃

C₈×Q₈

yes

characteristic

C₂×C₅₃

#446

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₅₃

yes

characteristic

C₄×C₄⋊C₄

—

#336

—

C₅₃

#1939

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄×C₈

—

#95

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₅₃

yes

characteristic

102

C₂²×C₄

C₄×D₅₃

yes

characteristic

103

C₂²×C₄

C₂²×D₅₃

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

106

—

111

C₂×C₈

Dic₁₀₆

yes

—

112

C₂×C₈

D₂₁₂

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

106

—

115

C₄²

—

106

—

116

C₄²

—

106

—

117

C₄²

Dic₁₀₆

yes

—

118

C₄²

C₂²⋊D₅₃

yes

—

119

C₂³

C₂×C₄×D₅₃

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

C₈×D₅₃

yes

—

129

C₂×C₄

#11719912

yes

—

130

C₂×C₄

#11719919

yes

—

131

C₂×C₄

#11719919

yes

—

132

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

133

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

134

C₂×C₄

C₂×D₂₁₂

yes

characteristic

135

C₂×C₄

#11719945

yes

characteristic

136

C₂×C₄

#11719945

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

212

—

141

C₂²

C₂×#11719919

yes

characteristic

142

C₂²

C₄×D₂₁₂

yes

characteristic

143

C₂²

C₄×Dic₁₀₆

yes

characteristic

144

C₂²

C₂×C₈×D₅₃

yes

characteristic

145

C₂²

C₂×#11719912

yes

characteristic

146

C₂²

#13133223

yes

characteristic

147

C₂²

#13133223

yes

characteristic

148

C₄

—

149

C₄

—

106

—

150

C₄

—

212

—

151

C₄

#13133146

yes

complemented

152

C₄

#13133191

yes

characteristic

153

C₄

#13133196

yes

characteristic

154

C₄

#13133209

yes

characteristic

155

C₄

#13133217

yes

characteristic

156

C₂

#16810521

yes

characteristic

157

C₂

C₂×#13133146

yes

characteristic

158

C₂

#16810649

yes

characteristic

159

C₂

C₈×D₂₁₂

yes

characteristic

160

C₂

#16810702

yes

characteristic

161

C₂

C₈×Dic₁₀₆

yes

characteristic

162

C₂

#16810753

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1939
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂²×C₈

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂×C₄×C₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
31. C₄⋊Dic₅₃	C₈	semidirect
67. C₄×C₅₃	C₄⋊C₈	semidirect
87. C₅₃	#1939	semidirect
151. C₄	#13133146	semidirect

H ≅ G/N

type

31. C₄⋊Dic₅₃

C₈

semidirect

67. C₄×C₅₃

C₄⋊C₈

semidirect

87. C₅₃

#1939

semidirect

151. C₄

#13133146

semidirect

Subgroups — #20275433

Conjugacy classes of subgroups

Complements