Group #20273327 — subgroups

Subgroups — #20273327

Order 6784.

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

normal subgroups only

This group has 256 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	C₂×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810362	C₂	1	4	yes	maximal, complemented
4	C₅₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11719945	C₄	1	2	yes	—
6	C₂²×#88322	—	2	1	no	—
7	#13133217	C₄	1	1	yes	characteristic
8	#13133214	C₄	1	1	yes	characteristic
9	#13133205	C₂²	1	2	yes	—
10	#13133191	—	2	1	no	—
11	#13133191	—	2	1	no	—
12	C₄×D₂₁₂	—	2	1	no	—
13	C₄×D₂₁₂	—	2	1	no	—
14	#13133128	—	2	1	no	—
15	#13133128	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₃	—	2	1	no	—
18	C₂²⋊C₈×C₅₃	—	2	1	no	—
19	#97×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₅₃	—	2	2	no	—
24	#11719945	—	4	4	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₅₃	—	2	2	no	—
29	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#11719921	—	2	2	no	—
36	#11719921	—	2	2	no	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	C₄⋊Dic₅₃	—	2	1	no	—
39	C₂×#88322	—	2	4	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂³×C₅₃	C₂²⋊C₄	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	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²⋊D₅₃	—	4	4	no	—
53	C₂×Dic₅₃	—	2	2	no	—
54	D₂₁₂	—	4	2	no	—
55	D₂₁₂	—	4	2	no	—
56	D₂₁₂	—	4	2	no	—
57	C₄×D₅₃	—	4	4	no	—
58	Dic₁₀₆	—	4	2	no	—
59	C₈×C₅₃	—	4	2	no	—
60	#88322	—	2	4	no	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	#100	1	2	yes	—
64	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₅₃	—	4	2	no	—
66	C₂×D₅₃	—	4	2	no	—
67	C₂×D₅₃	—	4	2	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	#104	1	2	yes	—
74	Dic₅₃	—	4	2	no	—
75	#1530	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	—	2	2	no	—
77	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
79	C₂×C₅₃	#421	1	1	yes	characteristic
80	D₅₃	—	8	2	no	—
81	#532	—	53	1	no	—
82	#432	D₅₃	1	1	yes	characteristic, complemented
83	C₂×#97	—	53	1	no	—
84	#329	—	53	4	no	—
85	C₅₃	#1530	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	53	2	no	—
87	C₂²×C₈	—	106	1	no	—
88	C₄⋊Q₈	—	53	1	no	—
89	#127	—	53	1	no	—
90	#124	—	53	2	no	—
91	#121	—	106	1	no	—
92	#121	—	106	1	no	—
93	C₄×D₄	—	106	1	no	—
94	C₄×D₄	—	106	1	no	—
95	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
96	C₄⋊C₈	—	2	1	no	—
97	C₂²⋊C₈	—	2	1	no	—
98	#97	—	53	2	no	—
99	#97	—	106	1	no	—
100	#97	C₂×D₅₃	1	2	yes	—
101	#41	—	212	4	no	—
102	C₂×Q₈	—	53	2	no	—
103	C₂×D₄	—	53	2	no	—
104	C₂×D₄	—	106	2	no	—
105	C₂×D₄	—	106	1	no	—
106	C₂×D₄	—	106	1	no	—
107	C₂²×C₄	—	106	2	no	—
108	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
109	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	106	4	no	—
113	C₂×C₈	—	106	2	no	—
114	C₄⋊C₄	—	106	1	no	—
115	C₄⋊C₄	—	106	1	no	—
116	C₂²⋊C₄	—	106	2	no	—
117	C₂²⋊C₄	—	106	2	no	—
118	C₄²	C₄×D₅₃	1	1	yes	characteristic
119	C₄²	D₂₁₂	1	1	yes	characteristic
120	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
121	C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	C₂³	—	106	2	no	—
123	C₂³	#11719921	1	1	yes	characteristic
124	Q₈	—	212	2	no	—
125	D₄	—	212	2	no	—
126	D₄	—	212	2	no	—
127	D₄	—	212	4	no	—
128	D₄	—	212	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	—	212	4	no	—
134	C₂×C₄	#11719921	1	1	yes	characteristic
135	C₂×C₄	#11719921	1	1	yes	characteristic
136	C₂×C₄	#11719921	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₈	—	4	2	no	—
141	C₈	—	106	4	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	212	2	no	—
145	C₂²	—	212	2	no	—
146	C₂²	—	212	2	no	—
147	C₂²	#13133133	1	2	yes	—
148	C₂²	C₂×#11719921	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	212	2	no	—
155	C₄	#13133143	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	—	424	2	no	—
158	C₂	C₂×#13133133	1	1	yes	characteristic
159	C₂	C₂×#13133143	1	1	yes	characteristic
160	C₂	#16810641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811173

C₂

yes

maximal, characteristic

C₂×#13133128

C₂

yes

maximal, characteristic, complemented

#16810362

C₂

yes

maximal, complemented

C₅₃×#432

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₄

yes

—

C₂²×#88322

—

#13133217

C₄

yes

characteristic

#13133214

C₄

yes

characteristic

#13133205

C₂²

yes

—

#13133191

—

#13133191

—

C₄×D₂₁₂

—

C₄×D₂₁₂

—

#13133128

—

#13133128

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

#11719945

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

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

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#100

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#104

yes

—

Dic₅₃

—

#1530

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#532

—

#432

D₅₃

yes

characteristic, complemented

C₂×#97

—

#329

—

C₅₃

#1530

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×C₈

—

106

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

106

—

#121

—

106

—

C₄×D₄

—

106

—

C₄×D₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

106

—

100

#97

C₂×D₅₃

yes

—

101

#41

—

212

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

106

—

105

C₂×D₄

—

106

—

106

C₂×D₄

—

106

—

107

C₂²×C₄

—

106

—

108

C₂²×C₄

C₄×D₅₃

yes

characteristic

109

C₂²×C₄

D₂₁₂

yes

characteristic

110

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

—

106

—

116

C₂²⋊C₄

—

106

—

117

C₂²⋊C₄

—

106

—

118

C₄²

C₄×D₅₃

yes

characteristic

119

C₄²

D₂₁₂

yes

characteristic

120

C₄²

C₂²⋊D₅₃

yes

characteristic

121

C₄²

C₂²×D₅₃

yes

characteristic

122

C₂³

—

106

—

123

C₂³

#11719921

yes

characteristic

124

Q₈

—

212

—

125

D₄

—

212

—

126

D₄

—

212

—

127

D₄

—

212

—

128

D₄

—

212

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

212

—

134

C₂×C₄

#11719921

yes

characteristic

135

C₂×C₄

#11719921

yes

characteristic

136

C₂×C₄

#11719921

yes

characteristic

137

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

138

C₂×C₄

C₂×D₂₁₂

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

140

C₈

—

141

C₈

—

106

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

212

—

145

C₂²

—

212

—

146

C₂²

—

212

—

147

C₂²

#13133133

yes

—

148

C₂²

C₂×#11719921

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

212

—

155

C₄

#13133143

yes

—

156

C₂

—

157

C₂

—

424

—

158

C₂

C₂×#13133133

yes

characteristic

159

C₂

C₂×#13133143

yes

characteristic

160

C₂

#16810641

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#432	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#432	D₅₃
hypocenter	C₅₃	#1530
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

C₄²

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. C₂×#13133128	C₂	semidirect
3. #16810362	C₂	semidirect
4. C₅₃×#432	C₂	semidirect
82. #432	D₅₃	semidirect
85. C₅₃	#1530	semidirect

H ≅ G/N

type

2. C₂×#13133128

C₂

semidirect

3. #16810362

C₂

semidirect

4. C₅₃×#432

C₂

semidirect

82. #432

D₅₃

semidirect

85. C₅₃

#1530

semidirect

Subgroups — #20273327

Conjugacy classes of subgroups

Complements