Group #20275478 — subgroups

Subgroups — #20275478

Order 6784.

256 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133148	C₂	1	2	yes	maximal
2	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#16810493	C₂	1	1	yes	maximal, characteristic
4	#16810385	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	—	2	1	no	—
8	C₂×C₄⋊Dic₅₃	—	2	1	no	—
9	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#13133148	—	2	4	no	—
11	C₄⋊#88322	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	—	2	1	no	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#95×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	—	2	1	no	—
18	C₂²×C₄×C₅₃	—	2	1	no	—
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	2	no	—
23	C₄⋊C₄×C₅₃	—	2	1	no	—
24	C₄⋊C₄×C₅₃	—	2	1	no	—
25	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
26	C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
27	C₄²×C₅₃	—	4	2	no	—
28	C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₄²×C₅₃	Q₈	1	2	yes	—
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₄⋊Dic₅₃	—	4	2	no	—
33	C₄⋊Dic₅₃	—	4	2	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	C₂×C₄	1	4	yes	—
37	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
49	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×Dic₅₃	—	4	2	no	—
57	C₂×Dic₅₃	—	4	2	no	—
58	C₂×Dic₅₃	—	4	2	no	—
59	#88322	—	2	4	no	—
60	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₅₃	#121	1	1	yes	characteristic
62	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₅₃	#137	1	1	yes	characteristic
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	#106	1	4	yes	complemented
71	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
72	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
73	C₄×C₅₃	#125	1	1	yes	characteristic
74	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	Dic₅₃	—	8	2	no	—
76	#1948	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	#389	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#106	1	1	yes	characteristic
79	C₂×C₅₃	#428	1	1	yes	characteristic
80	C₂×C₅₃	#460	1	1	yes	characteristic
81	C₂×C₅₃	#461	1	1	yes	characteristic
82	C₂×C₅₃	#475	1	1	yes	characteristic
83	C₂×C₅₃	#476	1	1	yes	characteristic
84	C₂×#106	—	53	2	no	—
85	C₂×C₄⋊C₈	—	53	1	no	—
86	#384	—	53	1	no	—
87	C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
88	#340	—	53	2	no	—
89	C₅₃	#1948	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	53	2	no	—
91	C₂×C₄⋊C₄	—	53	2	no	—
92	C₂×C₄⋊C₄	—	106	1	no	—
93	C₂×C₄⋊C₄	—	106	1	no	—
94	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
98	#106	—	106	4	no	—
99	C₄⋊C₈	—	53	4	no	—
100	#95	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	106	2	no	—
104	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
107	C₂×C₈	—	53	4	no	—
108	C₂×C₈	—	106	2	no	—
109	C₂×C₈	—	106	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	106	2	no	—
113	C₄⋊C₄	—	106	2	no	—
114	C₄⋊C₄	—	212	2	no	—
115	C₄⋊C₄	—	212	2	no	—
116	C₄⋊C₄	Dic₁₀₆	1	2	yes	—
117	C₄⋊C₄	D₂₁₂	1	2	yes	—
118	C₄²	—	4	2	no	—
119	C₄²	Dic₁₀₆	1	2	yes	—
120	C₄²	C₄×D₅₃	1	2	yes	—
121	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	212	2	no	—
131	C₂×C₄	—	212	2	no	—
132	C₂×C₄	—	212	2	no	—
133	C₂×C₄	#11719919	1	2	yes	—
134	C₂×C₄	#11719919	1	2	yes	—
135	C₂×C₄	#11719923	1	2	yes	—
136	C₂×C₄	#11719924	1	2	yes	—
137	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
138	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
140	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
141	C₂×C₄	#11719945	1	1	yes	characteristic
142	C₂×C₄	#11719945	1	1	yes	characteristic
143	C₈	—	106	4	no	—
144	C₂²	C₂×#11719919	1	1	yes	characteristic
145	C₂²	#13133191	1	1	yes	characteristic
146	C₂²	#13133196	1	1	yes	characteristic
147	C₂²	C₂×#11719924	1	1	yes	characteristic
148	C₂²	C₂×#11719923	1	1	yes	characteristic
149	C₂²	#13133242	1	1	yes	characteristic
150	C₂²	#13133247	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	424	2	no	—
155	C₄	#13133148	1	4	yes	complemented
156	C₄	C₄×D₂₁₂	1	1	yes	characteristic
157	C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
158	C₄	#13133209	1	1	yes	characteristic
159	C₄	#13133217	1	1	yes	characteristic
160	C₂	#16810521	1	1	yes	characteristic
161	C₂	C₂×#13133148	1	1	yes	characteristic
162	C₂	#16810666	1	1	yes	characteristic
163	C₂	#16810823	1	1	yes	characteristic
164	C₂	#16810828	1	1	yes	characteristic
165	C₂	#16810908	1	1	yes	characteristic
166	C₂	#16810913	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133148

C₂

yes

maximal

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

#16810493

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

C₄×C₄⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133148

—

C₄⋊#88322

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

Q₈

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

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

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

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

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#106

yes

complemented

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

#125

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₄⋊C₈

—

#384

—

C₄×C₄⋊C₄

D₅₃

yes

characteristic

#340

—

C₅₃

#1948

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#106

—

106

—

C₄⋊C₈

—

100

#95

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

106

—

104

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

106

C₂²×C₄

C₂²×D₅₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

—

212

—

115

C₄⋊C₄

—

212

—

116

C₄⋊C₄

Dic₁₀₆

yes

—

117

C₄⋊C₄

D₂₁₂

yes

—

118

C₄²

—

119

C₄²

Dic₁₀₆

yes

—

120

C₄²

C₄×D₅₃

yes

—

121

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

212

—

131

C₂×C₄

—

212

—

132

C₂×C₄

—

212

—

133

C₂×C₄

#11719919

yes

—

134

C₂×C₄

#11719919

yes

—

135

C₂×C₄

#11719923

yes

—

136

C₂×C₄

#11719924

yes

—

137

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

138

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

139

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

140

C₂×C₄

C₂×D₂₁₂

yes

characteristic

141

C₂×C₄

#11719945

yes

characteristic

142

C₂×C₄

#11719945

yes

characteristic

143

C₈

—

106

—

144

C₂²

C₂×#11719919

yes

characteristic

145

C₂²

#13133191

yes

characteristic

146

C₂²

#13133196

yes

characteristic

147

C₂²

C₂×#11719924

yes

characteristic

148

C₂²

C₂×#11719923

yes

characteristic

149

C₂²

#13133242

yes

characteristic

150

C₂²

#13133247

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

424

—

155

C₄

#13133148

yes

complemented

156

C₄

C₄×D₂₁₂

yes

characteristic

157

C₄

C₄×Dic₁₀₆

yes

characteristic

158

C₄

#13133209

yes

characteristic

159

C₄

#13133217

yes

characteristic

160

C₂

#16810521

yes

characteristic

161

C₂

C₂×#13133148

yes

characteristic

162

C₂

#16810666

yes

characteristic

163

C₂

#16810823

yes

characteristic

164

C₂

#16810828

yes

characteristic

165

C₂

#16810908

yes

characteristic

166

C₂

#16810913

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₄×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#1948
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₄×C₄⋊C₄×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₅₃

hypocenter

C₅₃

#1948

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
11. C₄⋊#88322	C₄	semidirect
70. C₄×C₅₃	#106	semidirect
89. C₅₃	#1948	semidirect
155. C₄	#13133148	semidirect

H ≅ G/N

type

11. C₄⋊#88322

C₄

semidirect

70. C₄×C₅₃

#106

semidirect

89. C₅₃

#1948

semidirect

155. C₄

#13133148

semidirect

Subgroups — #20275478

Conjugacy classes of subgroups

Complements