Group #20275489 — subgroups

Subgroups — #20275489

Order 6784.

200 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 85 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×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133152	C₂	1	1	yes	maximal, characteristic
3	#16810646	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133146	C₂	1	1	yes	maximal, characteristic
5	M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810387	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#341	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	#13133153	—	2	2	no	—
13	#13133152	—	2	2	no	—
14	#13133146	—	2	1	no	—
15	#13133146	—	2	1	no	—
16	#13133146	C₄	1	2	yes	—
17	#13133146	C₄	1	2	yes	—
18	#13133127	—	2	1	no	—
19	C₈×Dic₅₃	—	2	1	no	—
20	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic
25	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic
26	M₁₆×C₅₃	—	2	1	no	—
27	M₁₆×C₅₃	—	2	1	no	—
28	M₁₆×C₅₃	—	2	1	no	—
29	M₁₆×C₅₃	Q₈	1	2	yes	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	—	2	1	no	—
33	C₂×C₈×C₅₃	—	2	1	no	—
34	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₅₃	D₄	1	1	yes	characteristic
38	C₄×Dic₅₃	D₄	1	1	yes	characteristic
39	#11719917	—	2	1	no	—
40	#11719917	—	2	1	no	—
41	#11719917	—	2	1	no	—
42	#11719917	D₄	1	2	yes	—
43	C₂×#88322	—	2	1	no	—
44	C₂×#88322	—	2	1	no	—
45	C₂×#88322	—	2	1	no	—
46	C₂×#88322	—	2	1	no	—
47	C₂×#88322	C₂×C₄	1	2	yes	—
48	C₂³×C₅₃	C₂×Q₈	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	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×Dic₅₃	—	2	2	no	—
56	C₂×Dic₅₃	—	4	1	no	—
57	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₈×C₅₃	—	2	2	no	—
62	C₈×C₅₃	—	2	2	no	—
63	C₈×C₅₃	—	4	1	no	—
64	#88322	—	2	2	no	—
65	#88322	—	2	2	no	—
66	#88322	—	4	1	no	—
67	C₂²×C₅₃	—	2	1	no	—
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₅₃	#125	1	1	yes	characteristic
71	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
72	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
73	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
74	C₄×C₅₃	#121	1	1	yes	characteristic
75	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
76	Dic₅₃	—	2	1	no	—
77	Dic₅₃	—	2	1	no	—
78	Dic₅₃	—	2	1	no	—
79	Dic₅₃	#108	1	2	yes	complemented
80	#1950	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	—	2	1	no	—
82	C₂×C₅₃	—	2	1	no	—
83	C₂×C₅₃	#389	1	1	yes	characteristic
84	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
85	C₂×C₅₃	#430	1	1	yes	characteristic
86	C₂×#108	—	53	1	no	—
87	C₂×#108	—	53	1	no	—
88	#423	—	53	1	no	—
89	C₂×C₄⋊C₈	—	53	1	no	—
90	C₄×M₁₆	—	53	1	no	—
91	#341	—	53	1	no	—
92	#341	D₅₃	1	1	yes	characteristic
93	C₅₃	#1950	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₁₆	—	53	1	no	—
95	C₂×M₁₆	—	53	1	no	—
96	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
97	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
98	C₂²×C₈	—	53	1	no	—
99	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
100	C₂×C₄²	—	53	1	no	—
101	#108	—	106	2	no	—
102	#108	—	106	2	no	—
103	C₄⋊C₈	—	53	2	no	—
104	C₄⋊C₈	—	53	2	no	—
105	C₄⋊C₈	—	106	1	no	—
106	C₄⋊C₈	—	106	1	no	—
107	#97	—	106	1	no	—
108	C₄×C₈	—	106	1	no	—
109	C₂²×C₄	—	53	1	no	—
110	C₂²×C₄	—	53	1	no	—
111	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	53	2	no	—
116	M₁₆	—	106	1	no	—
117	M₁₆	—	106	1	no	—
118	M₁₆	—	106	1	no	—
119	M₁₆	Dic₁₀₆	1	2	yes	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	53	2	no	—
125	C₂×C₈	—	106	1	no	—
126	C₂×C₈	—	106	1	no	—
127	C₂×C₈	—	106	1	no	—
128	C₂×C₈	—	106	1	no	—
129	C₂×C₈	C₄×D₅₃	1	2	yes	—
130	C₄²	—	53	1	no	—
131	C₄²	—	53	1	no	—
132	C₄²	—	53	1	no	—
133	C₄²	—	53	1	no	—
134	C₂³	Q₈×D₅₃	1	1	yes	characteristic
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	—	53	1	no	—
138	C₂×C₄	—	53	1	no	—
139	C₂×C₄	—	106	2	no	—
140	C₂×C₄	—	212	1	no	—
141	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
143	C₂×C₄	#11719945	1	1	yes	characteristic
144	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
145	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
146	C₂×C₄	#11719947	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	2	no	—
149	C₈	—	4	1	no	—
150	C₈	—	106	2	no	—
151	C₈	—	106	2	no	—
152	C₈	—	212	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
156	C₂²	#13133207	1	1	yes	characteristic
157	C₂²	#13133216	1	1	yes	characteristic
158	C₄	—	53	2	no	—
159	C₄	—	106	1	no	—
160	C₄	—	106	1	no	—
161	C₄	—	106	1	no	—
162	C₄	#13133177	1	1	yes	characteristic
163	C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
164	C₄	#13133190	1	1	yes	characteristic
165	C₄	#13133194	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	#16810520	1	1	yes	characteristic
169	C₂	#108×D₅₃	1	1	yes	characteristic
170	C₂	#16810670	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133153

C₂

yes

maximal, characteristic

C₂×#13133152

C₂

yes

maximal, characteristic

#16810646

C₂

yes

maximal, characteristic

C₂×#13133146

C₂

yes

maximal, characteristic

M₁₆×Dic₅₃

C₂

yes

maximal, characteristic

#16810387

C₂

yes

maximal, characteristic

C₅₃×#341

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

—

C₄

yes

—

#13133146

C₄

yes

—

#13133127

—

C₈×Dic₅₃

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

—

—

D₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#125

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#341

—

#341

D₅₃

yes

characteristic

C₅₃

#1950

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

100

C₂×C₄²

—

101

#108

—

106

—

102

#108

—

106

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

—

106

C₄⋊C₈

—

106

—

107

#97

—

106

—

108

C₄×C₈

—

106

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₅₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

106

—

117

M₁₆

—

106

—

118

M₁₆

—

106

—

119

M₁₆

Dic₁₀₆

yes

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

106

—

126

C₂×C₈

—

106

—

127

C₂×C₈

—

106

—

128

C₂×C₈

—

106

—

129

C₂×C₈

C₄×D₅₃

yes

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₂³

Q₈×D₅₃

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

212

—

141

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

142

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

143

C₂×C₄

#11719945

yes

characteristic

144

C₂×C₄

D₄×D₅₃

yes

characteristic

145

C₂×C₄

D₄×D₅₃

yes

characteristic

146

C₂×C₄

#11719947

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

106

—

151

C₈

—

106

—

152

C₈

—

212

—

153

C₂²

—

154

C₂²

—

155

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

156

C₂²

#13133207

yes

characteristic

157

C₂²

#13133216

yes

characteristic

158

C₄

—

159

C₄

—

106

—

160

C₄

—

106

—

161

C₄

—

106

—

162

C₄

#13133177

yes

characteristic

163

C₄

C₄×Dic₁₀₆

yes

characteristic

164

C₄

#13133190

yes

characteristic

165

C₄

#13133194

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

#16810520

yes

characteristic

169

C₂

#108×D₅₃

yes

characteristic

170

C₂

#16810670

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#341	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#341	D₅₃
hypocenter	C₅₃	#1950
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#341

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
79. Dic₅₃	#108	semidirect
93. C₅₃	#1950	semidirect

H ≅ G/N

type

79. Dic₅₃

#108

semidirect

93. C₅₃

#1950

semidirect

Subgroups — #20275489

Conjugacy classes of subgroups

Complements