Group #20273224 — subgroups

Subgroups — #20273224

Order 6784.

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

normal subgroups only

This group has 272 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	#16811173	C₂	1	1	yes	maximal, characteristic
2	#16810469	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810358	C₂	1	4	yes	maximal, complemented
4	C₅₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11719945	—	2	1	no	—
6	#13133217	C₄	1	1	yes	characteristic, complemented
7	#13133214	C₄	1	1	yes	characteristic, complemented
8	#13133205	—	2	1	no	—
9	#13133191	—	2	1	no	—
10	#13133191	C₂²	1	2	yes	—
11	C₄×D₂₁₂	—	2	1	no	—
12	C₄×D₂₁₂	C₄	1	2	yes	complemented
13	C₂×C₄×Dic₅₃	—	2	1	no	—
14	#13133124	—	2	1	no	—
15	#13133124	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₃	—	2	1	no	—
18	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	#97×C₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₂²⋊D₅₃	—	2	2	no	—
24	C₂²×Dic₅₃	—	2	2	no	—
25	#11719945	—	4	4	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₅₃	—	2	2	no	—
30	C₂×Dic₁₀₆	—	2	1	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₄²×C₅₃	D₄	1	1	yes	characteristic
36	#11719921	—	2	2	no	—
37	#11719921	—	2	2	no	—
38	C₄⋊Dic₅₃	—	2	1	no	—
39	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
40	C₄×Dic₅₃	—	2	4	no	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²⋊D₅₃	—	4	4	no	—
53	C₂×Dic₅₃	—	2	4	no	—
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	—	4	2	no	—
56	C₂×Dic₅₃	—	4	2	no	—
57	D₂₁₂	—	4	2	no	—
58	D₂₁₂	—	4	2	no	—
59	D₂₁₂	—	4	2	no	—
60	C₄×D₅₃	—	4	4	no	—
61	Dic₁₀₆	—	4	2	no	—
62	C₈×C₅₃	—	4	2	no	—
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
67	C₂×D₅₃	—	4	2	no	—
68	C₂×D₅₃	—	4	2	no	—
69	C₂×D₅₃	—	4	2	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	#104	1	2	yes	—
76	Dic₅₃	—	4	4	no	—
77	Dic₅₃	—	4	2	no	—
78	#1505	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	—	2	2	no	—
80	C₂×C₅₃	#410	1	1	yes	characteristic
81	C₂×C₅₃	#418	1	1	yes	characteristic
82	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
83	D₅₃	—	8	2	no	—
84	#532	—	53	1	no	—
85	#432	D₅₃	1	1	yes	characteristic, complemented
86	#376	—	53	1	no	—
87	#327	—	53	4	no	—
88	C₅₃	#1505	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	—	106	1	no	—
90	C₄⋊Q₈	—	53	1	no	—
91	#127	—	53	1	no	—
92	#124	—	106	1	no	—
93	#121	—	53	2	no	—
94	#121	—	106	1	no	—
95	C₄×D₄	—	53	2	no	—
96	C₄×D₄	—	106	1	no	—
97	C₂×C₄²	—	106	1	no	—
98	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
99	C₄⋊C₈	—	2	1	no	—
100	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
101	#97	—	2	1	no	—
102	#95	—	53	2	no	—
103	#95	—	106	1	no	—
104	#41	—	212	4	no	—
105	C₂×Q₈	—	106	1	no	—
106	C₂×D₄	—	53	2	no	—
107	C₂×D₄	—	106	2	no	—
108	C₂×D₄	—	106	1	no	—
109	C₂×D₄	—	106	1	no	—
110	C₂²×C₄	—	106	2	no	—
111	C₂²×C₄	—	106	2	no	—
112	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
113	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₄⋊C₄	—	53	2	no	—
117	C₄⋊C₄	—	106	1	no	—
118	C₂²⋊C₄	—	106	2	no	—
119	C₂²⋊C₄	—	106	2	no	—
120	C₄²	—	106	4	no	—
121	C₄²	C₄×D₅₃	1	1	yes	characteristic
122	C₄²	C₄×D₅₃	1	1	yes	characteristic
123	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
124	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂³	—	106	2	no	—
126	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
127	Q₈	—	212	2	no	—
128	D₄	—	212	2	no	—
129	D₄	—	212	2	no	—
130	D₄	—	212	4	no	—
131	D₄	—	212	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	106	4	no	—
136	C₂×C₄	—	106	2	no	—
137	C₂×C₄	—	212	4	no	—
138	C₂×C₄	—	212	2	no	—
139	C₂×C₄	—	212	2	no	—
140	C₂×C₄	#11719921	1	1	yes	characteristic
141	C₂×C₄	#11719921	1	1	yes	characteristic
142	C₂×C₄	#11719921	1	1	yes	characteristic
143	C₂×C₄	#11719921	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
146	C₈	—	4	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	212	2	no	—
150	C₂²	—	212	2	no	—
151	C₂²	—	212	2	no	—
152	C₂²	C₂×#11719921	1	1	yes	characteristic
153	C₂²	#13133245	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	212	4	no	—
160	C₄	—	212	2	no	—
161	C₄	#13133143	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	—	424	2	no	—
164	C₂	#16810596	1	1	yes	characteristic
165	C₂	#16810629	1	1	yes	characteristic
166	C₂	C₂×#13133143	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811173

C₂

yes

maximal, characteristic

#16810469

C₂

yes

maximal, characteristic, complemented

#16810358

C₂

yes

maximal, complemented

C₅₃×#432

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

—

#13133217

C₄

yes

characteristic, complemented

#13133214

C₄

yes

characteristic, complemented

#13133205

—

#13133191

—

#13133191

C₂²

yes

—

C₄×D₂₁₂

—

C₄×D₂₁₂

C₄

yes

complemented

C₂×C₄×Dic₅₃

—

#13133124

—

#13133124

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

C₂²

yes

—

#97×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

#11719945

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#104

yes

—

Dic₅₃

—

Dic₅₃

—

#1505

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#532

—

#432

D₅₃

yes

characteristic, complemented

#376

—

#327

—

C₅₃

#1505

yes

characteristic, complemented, Sylow

C₂×#41

—

106

—

C₄⋊Q₈

—

#127

—

#124

—

106

—

#121

—

#121

—

106

—

C₄×D₄

—

C₄×D₄

—

106

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

100

C₂²⋊C₈

C₂×D₅₃

yes

—

101

#97

—

102

#95

—

103

#95

—

106

—

104

#41

—

212

—

105

C₂×Q₈

—

106

—

106

C₂×D₄

—

107

C₂×D₄

—

106

—

108

C₂×D₄

—

106

—

109

C₂×D₄

—

106

—

110

C₂²×C₄

—

106

—

111

C₂²×C₄

—

106

—

112

C₂²×C₄

D₂₁₂

yes

characteristic

113

C₂²×C₄

D₂₁₂

yes

characteristic

114

C₂²×C₄

C₂²×D₅₃

yes

characteristic

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₂²⋊C₄

—

106

—

119

C₂²⋊C₄

—

106

—

120

C₄²

—

106

—

121

C₄²

C₄×D₅₃

yes

characteristic

122

C₄²

C₄×D₅₃

yes

characteristic

123

C₄²

C₂²⋊D₅₃

yes

characteristic

124

C₄²

C₂²⋊D₅₃

yes

characteristic

125

C₂³

—

106

—

126

C₂³

C₂×D₂₁₂

yes

characteristic

127

Q₈

—

212

—

128

D₄

—

212

—

129

D₄

—

212

—

130

D₄

—

212

—

131

D₄

—

212

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

212

—

138

C₂×C₄

—

212

—

139

C₂×C₄

—

212

—

140

C₂×C₄

#11719921

yes

characteristic

141

C₂×C₄

#11719921

yes

characteristic

142

C₂×C₄

#11719921

yes

characteristic

143

C₂×C₄

#11719921

yes

characteristic

144

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

145

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

212

—

150

C₂²

—

212

—

151

C₂²

—

212

—

152

C₂²

C₂×#11719921

yes

characteristic

153

C₂²

#13133245

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

212

—

160

C₄

—

212

—

161

C₄

#13133143

yes

—

162

C₂

—

163

C₂

—

424

—

164

C₂

#16810596

yes

characteristic

165

C₂

#16810629

yes

characteristic

166

C₂

C₂×#13133143

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#432	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#432	D₅₃
hypocenter	C₅₃	#1505
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×#11719921

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#432

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
2. #16810469	C₂	semidirect
3. #16810358	C₂	semidirect
4. C₅₃×#432	C₂	semidirect
6. #13133217	C₄	semidirect
7. #13133214	C₄	semidirect
12. C₄×D₂₁₂	C₄	semidirect
85. #432	D₅₃	semidirect
88. C₅₃	#1505	semidirect

H ≅ G/N

type

2. #16810469

C₂

semidirect

3. #16810358

C₂

semidirect

4. C₅₃×#432

C₂

semidirect

6. #13133217

C₄

semidirect

7. #13133214

C₄

semidirect

12. C₄×D₂₁₂

C₄

semidirect

85. #432

D₅₃

semidirect

88. C₅₃

#1505

semidirect

Subgroups — #20273224

Conjugacy classes of subgroups

Complements