Group #20290600 — subgroups

Subgroups — #20290600

Order 6784.

252 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 72 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	D₅₃⋊#108	C₂	1	2	yes	maximal, complemented
2	#16811609	C₂	1	2	yes	maximal
3	#16811492	C₂	1	2	yes	maximal, complemented
4	C₂×#13133225	C₂	1	1	yes	maximal, characteristic
5	D₅₃⋊M₁₆	C₂²	1	4	yes	—
6	#13133288	—	2	2	no	—
7	#13133287	—	2	2	no	—
8	#13133278	—	2	2	no	—
9	#13133277	—	2	2	no	—
10	C₂×#11719945	C₂²	1	1	yes	characteristic
11	#13133225	—	2	2	no	—
12	#13133225	—	2	2	no	—
13	C₂×#11719912	C₄	1	1	yes	characteristic
14	C₂×#11719912	C₂²	1	1	yes	characteristic
15	C₂²×#88322	C₄	1	1	yes	characteristic
16	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
17	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₅₃	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₅₃	Q₈	1	1	yes	characteristic
22	#11719945	—	2	2	no	—
23	#11719945	—	2	2	no	—
24	C₂×D₂₁₂	Q₈	1	1	yes	characteristic
25	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
28	#11719940	—	2	4	no	—
29	#11719936	—	2	4	no	—
30	D₅₃⋊C₈	—	2	4	no	—
31	M₁₆×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#88322	—	2	1	no	—
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×C₄	1	1	yes	characteristic
38	#11719912	—	2	2	no	—
39	#11719912	—	2	2	no	—
40	C₈×D₅₃	—	2	1	no	—
41	C₈×D₅₃	—	2	1	no	—
42	C₈×D₅₃	D₄	1	2	yes	—
43	C₈×D₅₃	D₄	1	2	yes	—
44	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×D₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²⋊D₅₃	—	2	2	no	—
52	C₂²⋊D₅₃	—	2	2	no	—
53	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
55	D₂₁₂	—	2	2	no	—
56	C₄×D₅₃	—	2	1	no	—
57	C₄×D₅₃	—	2	1	no	—
58	C₄×D₅₃	C₂×D₄	1	2	yes	—
59	C₄×D₅₃	#41	1	2	yes	—
60	Dic₁₀₆	—	2	2	no	—
61	#88324	—	4	4	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	#88322	—	2	1	no	—
65	#88322	C₂²⋊C₄	1	1	yes	characteristic
66	#88322	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₅₃	—	2	1	no	—
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	—	4	1	no	—
72	C₂×D₅₃	—	4	1	no	—
73	C₂×D₅₃	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₅₃	#117	1	1	yes	characteristic
77	Dic₅₃	—	2	1	no	—
78	Dic₅₃	#121	1	2	yes	—
79	#1813	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	—	2	1	no	—
81	C₂×C₅₃	—	4	1	no	—
82	C₂×C₅₃	#380	1	1	yes	characteristic
83	D₅₃	—	2	2	no	—
84	D₅₃	—	4	1	no	—
85	C₂×#131	—	53	1	no	—
86	C₂×#108	—	53	2	no	—
87	#413	—	53	2	no	—
88	#338	—	53	2	no	—
89	C₅₃	#1813	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	53	1	no	—
91	#131	—	106	2	no	—
92	#131	—	106	2	no	—
93	C₂×M₁₆	—	53	1	no	—
94	C₂×M₁₆	—	53	1	no	—
95	C₂×M₁₆	—	53	4	no	—
96	C₂×M₁₆	#6157	1	1	yes	characteristic
97	C₂²×C₈	—	53	1	no	—
98	C₂²×C₈	—	53	1	no	—
99	C₂²×C₈	—	53	1	no	—
100	#108	—	106	2	no	—
101	#108	—	106	2	no	—
102	#101	—	106	2	no	—
103	#100	—	106	2	no	—
104	#41	—	106	2	no	—
105	#41	—	106	2	no	—
106	C₂×Q₈	—	53	1	no	—
107	C₂×D₄	—	53	1	no	—
108	C₂×D₄	—	53	1	no	—
109	C₂×D₄	—	53	1	no	—
110	C₂²×C₄	—	53	1	no	—
111	C₂²×C₄	—	53	1	no	—
112	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	106	2	no	—
115	M₁₆	—	106	2	no	—
116	M₁₆	—	106	4	no	—
117	M₁₆	—	106	4	no	—
118	C₂×C₈	—	53	1	no	—
119	C₂×C₈	—	53	2	no	—
120	C₂×C₈	—	53	2	no	—
121	C₂×C₈	—	53	1	no	—
122	C₂×C₈	—	106	1	no	—
123	C₂×C₈	—	106	1	no	—
124	C₂×C₈	—	106	1	no	—
125	C₂×C₈	—	106	1	no	—
126	C₂×C₈	—	106	4	no	—
127	C₂×C₈	C₂×#6157	1	1	yes	characteristic
128	C₂×C₈	C₂×#6157	1	1	yes	characteristic
129	C₂³	—	53	1	no	—
130	C₂³	—	53	1	no	—
131	C₂³	C₄⋊#6157	1	1	yes	characteristic
132	Q₈	—	106	2	no	—
133	D₄	—	106	2	no	—
134	D₄	—	106	2	no	—
135	D₄	—	106	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	53	1	no	—
139	C₂×C₄	—	53	2	no	—
140	C₂×C₄	—	53	2	no	—
141	C₂×C₄	—	53	1	no	—
142	C₂×C₄	—	106	1	no	—
143	C₂×C₄	—	106	1	no	—
144	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
145	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	53	1	no	—
148	C₈	—	53	1	no	—
149	C₈	—	106	1	no	—
150	C₈	—	212	4	no	—
151	C₈	C₂²⋊#6157	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	53	2	no	—
155	C₂²	—	106	1	no	—
156	C₂²	—	212	1	no	—
157	C₂²	—	212	1	no	—
158	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	53	2	no	—
161	C₄	—	106	1	no	—
162	C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
163	C₄	#13133296	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	4	1	no	—
166	C₂	—	106	2	no	—
167	C₂	—	212	1	no	—
168	C₂	#16811565	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊#108

C₂

yes

maximal, complemented

#16811609

C₂

yes

maximal

#16811492

C₂

yes

maximal, complemented

C₂×#13133225

C₂

yes

maximal, characteristic

D₅₃⋊M₁₆

C₂²

yes

—

—

—

—

—

C₂²

yes

—

—

C₄

yes

characteristic

C₂×#11719912

C₂²

yes

characteristic

C₂²×#88322

C₄

yes

characteristic

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₅₃

D₄

yes

characteristic

C₂×C₂²⋊D₅₃

Q₈

yes

characteristic

#11719945

—

#11719945

—

C₂×D₂₁₂

Q₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

D₄

yes

characteristic

#11719940

—

#11719936

—

D₅₃⋊C₈

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

#11719912

—

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

D₄

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×D₅₃

C₄⋊C₄

yes

characteristic

C₂²×D₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

—

C₄×D₅₃

#41

yes

—

Dic₁₀₆

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

—

#88322

—

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

Dic₅₃

—

Dic₅₃

#121

yes

—

#1813

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#380

yes

characteristic

D₅₃

—

D₅₃

—

—

—

—

—

C₅₃

#1813

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

#6157

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

100

#108

—

106

—

101

#108

—

106

—

102

#101

—

106

—

103

#100

—

106

—

104

#41

—

106

—

105

#41

—

106

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂×#6157

yes

characteristic

113

M₁₆

—

114

M₁₆

—

106

—

115

M₁₆

—

106

—

116

M₁₆

—

106

—

117

M₁₆

—

106

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

106

—

124

C₂×C₈

—

106

—

125

C₂×C₈

—

106

—

126

C₂×C₈

—

106

—

127

C₂×C₈

C₂×#6157

yes

characteristic

128

C₂×C₈

C₂×#6157

yes

characteristic

129

C₂³

—

130

C₂³

—

131

C₂³

C₄⋊#6157

yes

characteristic

132

Q₈

—

106

—

133

D₄

—

106

—

134

D₄

—

106

—

135

D₄

—

106

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

106

—

143

C₂×C₄

—

106

—

144

C₂×C₄

C₄⋊#6157

yes

characteristic

145

C₂×C₄

C₂²×#6157

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

106

—

150

C₈

—

212

—

151

C₈

C₂²⋊#6157

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

106

—

156

C₂²

—

212

—

157

C₂²

—

212

—

158

C₂²

C₂×C₄⋊#6157

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

106

—

162

C₄

C₂×C₂²⋊#6157

yes

characteristic

163

C₄

#13133296

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

—

106

—

167

C₂

—

212

—

168

C₂

#16811565

yes

characteristic

	subgroup	G/H
center	C₄	#13133296
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×M₁₆×C₅₃	C₄
socle	C₂×C₅₃	#380
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	#1813
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

#13133296

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₂×M₁₆×C₅₃

C₄

C₂×C₅₃

C₂×M₁₆

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. D₅₃⋊#108	C₂	semidirect
3. #16811492	C₂	semidirect
89. C₅₃	#1813	semidirect

H ≅ G/N

type

1. D₅₃⋊#108

C₂

semidirect

3. #16811492

C₂

semidirect

89. C₅₃

#1813

semidirect

Subgroups — #20290600

Conjugacy classes of subgroups

Complements