Group #20275643 — subgroups

Subgroups — #20275643

Order 6784.

428 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 129 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, complemented
2	C₂×#13133138	C₂	1	4	yes	maximal, complemented
3	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
7	C₂×C₄×Dic₅₃	—	2	1	no	—
8	C₄⋊#88322	—	2	2	no	—
9	#13133138	—	2	8	no	—
10	#13133124	—	2	1	no	—
11	C₂²×D₄×C₅₃	—	2	1	no	—
12	C₂²×D₄×C₅₃	—	2	1	no	—
13	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
14	#127×C₅₃	—	2	2	no	—
15	#127×C₅₃	C₄	1	4	yes	complemented
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	2	4	no	—
19	C₂×D₄×C₅₃	—	2	4	no	—
20	C₂×D₄×C₅₃	—	2	4	no	—
21	C₂×D₄×C₅₃	—	2	2	no	—
22	C₂×D₄×C₅₃	—	2	4	no	—
23	C₂×D₄×C₅₃	—	4	4	no	—
24	C₂×D₄×C₅₃	—	4	2	no	—
25	C₂×D₄×C₅₃	—	4	2	no	—
26	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₃	—	2	2	no	—
31	C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₄²×C₅₃	D₄	1	2	yes	—
33	C₄⋊Dic₅₃	—	2	2	no	—
34	C₄⋊Dic₅₃	D₄	1	4	yes	complemented
35	C₄×Dic₅₃	—	4	2	no	—
36	C₂×#88322	—	2	4	no	—
37	C₂×#88322	—	2	2	no	—
38	C₂³×C₅₃	—	2	4	no	—
39	C₂³×C₅₃	—	4	4	no	—
40	C₂³×C₅₃	—	4	2	no	—
41	C₂³×C₅₃	—	4	2	no	—
42	C₂³×C₅₃	—	4	4	no	—
43	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
44	D₄×C₅₃	—	2	8	no	—
45	D₄×C₅₃	—	4	8	no	—
46	D₄×C₅₃	—	4	8	no	—
47	D₄×C₅₃	—	4	2	no	—
48	D₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₅₃	D₈	1	2	yes	—
53	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
54	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#41	1	2	yes	—
58	C₂×Dic₅₃	—	2	4	no	—
59	C₂×Dic₅₃	—	4	2	no	—
60	C₂×Dic₅₃	—	4	2	no	—
61	#88322	—	4	2	no	—
62	C₂²×C₅₃	—	4	4	no	—
63	C₂²×C₅₃	—	4	4	no	—
64	C₂²×C₅₃	—	4	4	no	—
65	C₂²×C₅₃	—	8	4	no	—
66	C₂²×C₅₃	—	8	4	no	—
67	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₅₃	#121	1	2	yes	—
69	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
70	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	#102	1	4	yes	complemented
74	C₄×C₅₃	C₄×D₄	1	2	yes	—
75	C₄×C₅₃	#124	1	1	yes	characteristic
76	C₄×C₅₃	#127	1	1	yes	characteristic
77	Dic₅₃	—	4	4	no	—
78	#1982	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	—	8	4	no	—
80	C₂×C₅₃	#390	1	1	yes	characteristic
81	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
82	C₂×C₅₃	#418	1	1	yes	characteristic
83	C₂×C₅₃	#459	1	2	yes	—
84	C₂×C₅₃	#460	1	2	yes	—
85	C₂×#127	D₅₃	1	1	yes	characteristic
86	C₂×C₄⋊C₈	—	53	1	no	—
87	C₂×#102	—	53	4	no	—
88	C₄×C₄⋊C₄	—	53	1	no	—
89	C₅₃	#1982	1	1	yes	characteristic, complemented, Sylow
90	C₂²×D₄	—	2	1	no	—
91	C₂²×D₄	—	2	1	no	—
92	C₂²×D₄	C₂×D₅₃	1	2	yes	—
93	C₂²×C₈	—	53	2	no	—
94	#127	—	2	2	no	—
95	#127	Dic₅₃	1	4	yes	complemented
96	C₂×C₄⋊C₄	—	53	2	no	—
97	C₂×C₄²	—	106	1	no	—
98	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
99	C₄⋊C₈	—	106	2	no	—
100	#102	—	106	8	no	—
101	#95	—	106	1	no	—
102	C₂⁴	—	2	2	no	—
103	C₂×D₄	—	2	4	no	—
104	C₂×D₄	—	2	4	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	4	no	—
107	C₂×D₄	—	2	4	no	—
108	C₂×D₄	—	4	4	no	—
109	C₂×D₄	—	4	2	no	—
110	C₂×D₄	—	4	2	no	—
111	C₂×D₄	C₂×Dic₅₃	1	4	yes	—
112	C₂²×C₄	—	106	2	no	—
113	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
116	C₂×C₈	—	106	4	no	—
117	C₂×C₈	—	106	2	no	—
118	C₄⋊C₄	—	53	4	no	—
119	C₄⋊C₄	—	106	2	no	—
120	C₄²	—	212	2	no	—
121	C₄²	C₂×Dic₅₃	1	2	yes	—
122	C₄²	C₂²⋊D₅₃	1	2	yes	—
123	C₂³	—	2	4	no	—
124	C₂³	—	4	4	no	—
125	C₂³	—	4	2	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	4	4	no	—
128	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
129	D₄	—	2	8	no	—
130	D₄	—	4	8	no	—
131	D₄	—	4	8	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	106	4	no	—
136	C₂×C₄	—	212	2	no	—
137	C₂×C₄	—	212	2	no	—
138	C₂×C₄	#11719922	1	2	yes	—
139	C₂×C₄	#11719923	1	2	yes	—
140	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
141	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
142	C₂×C₄	D₄×D₅₃	1	2	yes	—
143	C₂×C₄	#11719947	1	2	yes	—
144	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
146	C₈	—	212	2	no	—
147	C₂²	—	4	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	4	no	—
150	C₂²	—	8	4	no	—
151	C₂²	—	8	4	no	—
152	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
153	C₂²	#13133188	1	2	yes	—
154	C₂²	C₂×#11719922	1	1	yes	characteristic
155	C₂²	C₂×#11719923	1	1	yes	characteristic
156	C₂²	#13133243	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	212	4	no	—
159	C₄	#13133138	1	4	yes	complemented
160	C₄	D₄×Dic₅₃	1	2	yes	—
161	C₄	#13133204	1	1	yes	characteristic
162	C₄	#13133213	1	1	yes	characteristic
163	C₂	—	8	4	no	—
164	C₂	#16810526	1	1	yes	characteristic
165	C₂	C₂×#13133138	1	1	yes	characteristic
166	C₂	#16810628	1	1	yes	characteristic
167	C₂	#16810815	1	2	yes	—
168	C₂	#16810820	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic, complemented

C₂×#13133138

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

—

C₄⋊#88322

—

#13133138

—

#13133124

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

C₂²

yes

—

#127×C₅₃

—

#127×C₅₃

C₄

yes

complemented

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

complemented

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

—

C₄×C₅₃

#102

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1982

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#460

yes

—

C₂×#127

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#102

—

C₄×C₄⋊C₄

—

C₅₃

#1982

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

C₂×D₅₃

yes

—

C₂²×C₈

—

#127

—

#127

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

100

#102

—

106

—

101

#95

—

106

—

102

C₂⁴

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂×Dic₅₃

yes

—

112

C₂²×C₄

—

106

—

113

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

114

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

115

C₂²×C₄

C₂²×D₅₃

yes

characteristic

116

C₂×C₈

—

106

—

117

C₂×C₈

—

106

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

106

—

120

C₄²

—

212

—

121

C₄²

C₂×Dic₅₃

yes

—

122

C₄²

C₂²⋊D₅₃

yes

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

129

D₄

—

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

212

—

137

C₂×C₄

—

212

—

138

C₂×C₄

#11719922

yes

—

139

C₂×C₄

#11719923

yes

—

140

C₂×C₄

C₂²⋊Dic₅₃

yes

—

141

C₂×C₄

C₂²⋊Dic₅₃

yes

—

142

C₂×C₄

D₄×D₅₃

yes

—

143

C₂×C₄

#11719947

yes

—

144

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

145

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

146

C₈

—

212

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

153

C₂²

#13133188

yes

—

154

C₂²

C₂×#11719922

yes

characteristic

155

C₂²

C₂×#11719923

yes

characteristic

156

C₂²

#13133243

yes

—

157

C₄

—

158

C₄

—

212

—

159

C₄

#13133138

yes

complemented

160

C₄

D₄×Dic₅₃

yes

—

161

C₄

#13133204

yes

characteristic

162

C₄

#13133213

yes

characteristic

163

C₂

—

164

C₂

#16810526

yes

characteristic

165

C₂

C₂×#13133138

yes

characteristic

166

C₂

#16810628

yes

characteristic

167

C₂

#16810815

yes

—

168

C₂

#16810820

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#127×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#127	D₅₃
hypocenter	C₅₃	#1982
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#127×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×C₄⋊#88322	C₂	semidirect
2. C₂×#13133138	C₂	semidirect
3. C₄×C₄⋊Dic₅₃	C₂	semidirect
15. #127×C₅₃	C₄	semidirect
34. C₄⋊Dic₅₃	D₄	semidirect
73. C₄×C₅₃	#102	semidirect
89. C₅₃	#1982	semidirect
95. #127	Dic₅₃	semidirect
159. C₄	#13133138	semidirect

H ≅ G/N

type

1. C₂×C₄⋊#88322

C₂

semidirect

2. C₂×#13133138

C₂

semidirect

3. C₄×C₄⋊Dic₅₃

C₂

semidirect

15. #127×C₅₃

C₄

semidirect

34. C₄⋊Dic₅₃

D₄

semidirect

73. C₄×C₅₃

#102

semidirect

89. C₅₃

#1982

semidirect

95. #127

Dic₅₃

semidirect

159. C₄

#13133138

semidirect

Subgroups — #20275643

Conjugacy classes of subgroups

Complements