Group #20283432 — subgroups

Subgroups — #20283432

Order 6784.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13133225	C₂	1	2	yes	maximal, complemented
2	#16811173	C₂	1	1	yes	maximal, characteristic
3	#16810693	C₂	1	8	yes	maximal, complemented
4	#16810591	C₂	1	2	yes	maximal, complemented
5	C₂×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11719945	C₂²	1	2	yes	—
8	#13133225	—	2	8	no	—
9	C₂×#11719912	C₂²	1	4	yes	—
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₅₃	C₂²	1	4	yes	—
13	#13133217	C₄	1	1	yes	characteristic
14	#13133214	C₄	1	1	yes	characteristic
15	#13133205	C₄	1	2	yes	—
16	#13133191	C₄	1	4	yes	—
17	C₄×D₂₁₂	C₂²	1	4	yes	—
18	#13133129	C₂²	1	8	yes	—
19	#13133128	C₂²	1	4	yes	—
20	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₅₃	C₂×C₄	1	4	yes	—
26	#11719945	—	2	4	no	—
27	#11719945	—	2	4	no	—
28	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
29	C₂×D₂₁₂	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₅₃	C₂³	1	4	yes	—
31	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
32	M₁₆×C₅₃	—	2	4	no	—
33	C₂×C₈×C₅₃	C₂³	1	4	yes	—
34	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	C₂³	1	2	yes	—
37	#11719921	C₂×C₄	1	8	yes	—
38	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	D₄	1	4	yes	complemented
41	C₂×#88322	C₂³	1	2	yes	—
42	C₂×#88322	C₂³	1	2	yes	—
43	#11719912	—	2	8	no	—
44	C₈×D₅₃	—	2	8	no	—
45	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₅₃	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₃	#41	1	2	yes	—
53	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₅₃	—	2	8	no	—
55	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
56	D₂₁₂	—	2	4	no	—
57	D₂₁₂	—	4	4	no	—
58	C₄×D₅₃	—	2	4	no	—
59	C₄×D₅₃	—	2	4	no	—
60	Dic₁₀₆	—	2	4	no	—
61	C₈×C₅₃	—	2	4	no	—
62	#88322	—	2	2	no	—
63	#88322	C₂×D₄	1	4	yes	—
64	C₂²×C₅₃	—	2	2	no	—
65	C₂²×C₅₃	C₄×D₄	1	2	yes	—
66	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₅₃	—	2	4	no	—
68	C₂×D₅₃	—	2	8	no	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	C₄×D₄	1	2	yes	—
72	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
74	Dic₅₃	—	2	4	no	—
75	#2932	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	—	2	2	no	—
77	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂×C₅₃	#568	1	2	yes	—
79	D₅₃	—	4	4	no	—
80	C₂×#131	—	53	2	no	—
81	#532	—	53	1	no	—
82	#435	—	53	8	no	—
83	#423	D₅₃	1	1	yes	characteristic, complemented
84	#408	—	53	2	no	—
85	C₂×#97	—	53	1	no	—
86	C₅₃	#2932	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	—	53	2	no	—
88	#131	—	106	8	no	—
89	C₂×M₁₆	—	53	4	no	—
90	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
91	C₂²×C₈	—	53	1	no	—
92	C₂²×C₈	—	53	1	no	—
93	C₂²×C₈	—	53	4	no	—
94	C₄⋊Q₈	—	53	1	no	—
95	#127	—	53	1	no	—
96	#124	—	53	2	no	—
97	#121	—	53	4	no	—
98	C₄×D₄	—	53	4	no	—
99	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
100	C₄⋊C₈	C₂×D₅₃	1	4	yes	—
101	C₂²⋊C₈	—	53	8	no	—
102	#97	—	53	4	no	—
103	#41	—	106	4	no	—
104	#41	—	106	4	no	—
105	C₂×Q₈	—	53	2	no	—
106	C₂×D₄	—	53	4	no	—
107	C₂×D₄	—	53	2	no	—
108	C₂×D₄	—	53	4	no	—
109	C₂²×C₄	—	53	4	no	—
110	C₂²×C₄	C₄×D₅₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	106	8	no	—
114	C₂×C₈	—	53	4	no	—
115	C₂×C₈	—	53	2	no	—
116	C₂×C₈	—	53	2	no	—
117	C₂×C₈	—	106	8	no	—
118	C₂×C₈	—	106	2	no	—
119	C₂×C₈	C₂²×D₅₃	1	4	yes	—
120	C₄⋊C₄	—	53	4	no	—
121	C₂²⋊C₄	—	53	8	no	—
122	C₄²	C₄×D₅₃	1	1	yes	characteristic
123	C₄²	C₄×D₅₃	1	1	yes	characteristic
124	C₄²	C₂²×D₅₃	1	2	yes	—
125	C₂³	—	53	4	no	—
126	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
127	Q₈	—	106	4	no	—
128	D₄	—	106	8	no	—
129	D₄	—	106	4	no	—
130	D₄	—	212	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	53	4	no	—
133	C₂×C₄	—	106	4	no	—
134	C₂×C₄	—	106	4	no	—
135	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄	D₄×D₅₃	1	2	yes	—
139	C₂×C₄	#11719949	1	2	yes	—
140	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	53	4	no	—
143	C₈	—	106	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	106	4	no	—
146	C₂²	—	106	8	no	—
147	C₂²	#13133179	1	2	yes	—
148	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	106	4	no	—
152	C₄	#13133179	1	2	yes	—
153	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
154	C₄	C₂×#11719949	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	212	4	no	—
157	C₂	C₂×#13133179	1	1	yes	characteristic
158	C₂	#16811389	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133225

C₂

yes

maximal, complemented

#16811173

C₂

yes

maximal, characteristic

#16810693

C₂

yes

maximal, complemented

#16810591

C₂

yes

maximal, complemented

C₂×#13133128

C₂

yes

maximal, characteristic, complemented

C₅₃×#423

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

—

#13133225

—

C₂×#11719912

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

—

#13133217

C₄

yes

characteristic

#13133214

C₄

yes

characteristic

#13133205

C₄

yes

—

#13133191

C₄

yes

—

C₄×D₂₁₂

C₂²

yes

—

#13133129

C₂²

yes

—

#13133128

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

C₂×C₄

yes

—

#11719945

—

#11719945

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

#11719921

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

D₄

yes

complemented

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

#11719912

—

C₈×D₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

C₂×D₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

#2932

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×D₄

yes

characteristic

C₂×C₅₃

#568

yes

—

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

#408

—

C₂×#97

—

C₅₃

#2932

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

100

C₄⋊C₈

C₂×D₅₃

yes

—

101

C₂²⋊C₈

—

102

#97

—

103

#41

—

106

—

104

#41

—

106

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₅₃

yes

—

111

C₂²×C₄

C₂²×D₅₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

106

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

106

—

118

C₂×C₈

—

106

—

119

C₂×C₈

C₂²×D₅₃

yes

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₄×D₅₃

yes

characteristic

123

C₄²

C₄×D₅₃

yes

characteristic

124

C₄²

C₂²×D₅₃

yes

—

125

C₂³

—

126

C₂³

C₂×C₄×D₅₃

yes

characteristic

127

Q₈

—

106

—

128

D₄

—

106

—

129

D₄

—

106

—

130

D₄

—

212

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

106

—

135

C₂×C₄

C₂×C₄×D₅₃

yes

—

136

C₂×C₄

C₂×C₄×D₅₃

yes

—

137

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

138

C₂×C₄

D₄×D₅₃

yes

—

139

C₂×C₄

#11719949

yes

—

140

C₂×C₄

C₂³×D₅₃

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

106

—

144

C₂²

—

145

C₂²

—

106

—

146

C₂²

—

106

—

147

C₂²

#13133179

yes

—

148

C₂²

C₂²×C₄×D₅₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

106

—

152

C₄

#13133179

yes

—

153

C₄

C₂×D₄×D₅₃

yes

characteristic

154

C₄

C₂×#11719949

yes

characteristic

155

C₂

—

156

C₂

—

212

—

157

C₂

C₂×#13133179

yes

characteristic

158

C₂

#16811389

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#423	C₂
hypercenter	#423	D₅₃
hypocenter	C₅₃	#2932
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#423

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. C₂×#13133225	C₂	semidirect
3. #16810693	C₂	semidirect
4. #16810591	C₂	semidirect
5. C₂×#13133128	C₂	semidirect
6. C₅₃×#423	C₂	semidirect
40. C₂×#88322	D₄	semidirect
83. #423	D₅₃	semidirect
86. C₅₃	#2932	semidirect

H ≅ G/N

type

1. C₂×#13133225

C₂

semidirect

3. #16810693

C₂

semidirect

4. #16810591

C₂

semidirect

5. C₂×#13133128

C₂

semidirect

6. C₅₃×#423

C₂

semidirect

40. C₂×#88322

D₄

semidirect

83. #423

D₅₃

semidirect

86. C₅₃

#2932

semidirect

Subgroups — #20283432

Conjugacy classes of subgroups

Complements