Group #20276185 — subgroups

Subgroups — #20276185

Order 6784.

184 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16810671	C₂	1	1	yes	maximal, characteristic
2	#16810633	C₂	1	1	yes	maximal, characteristic
3	#16810631	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133135	C₂	1	1	yes	maximal, characteristic
5	#16810395	C₂	1	1	yes	maximal, characteristic
6	#16810379	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#344	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	#13133176	C₂²	1	1	yes	characteristic
12	#13133152	—	2	2	no	—
13	#13133142	—	2	2	no	—
14	#13133140	—	2	2	no	—
15	#13133135	—	2	2	no	—
16	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	#117×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	—	2	1	no	—
21	C₂×Dic₁₀₆	—	2	1	no	—
22	C₂×Dic₁₀₆	—	2	1	no	—
23	C₂×Dic₁₀₆	—	2	1	no	—
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×Dic₁₀₆	—	2	1	no	—
26	C₂×Dic₁₀₆	D₄	1	2	yes	—
27	M₁₆×C₅₃	—	2	1	no	—
28	M₁₆×C₅₃	—	2	1	no	—
29	M₁₆×C₅₃	—	2	1	no	—
30	M₁₆×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂²⋊C₄×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	—	2	1	no	—
36	C₂²⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	C₄×Dic₅₃	—	2	1	no	—
39	#11719917	—	2	1	no	—
40	#11719917	—	2	1	no	—
41	#11719917	—	2	1	no	—
42	#11719917	—	2	1	no	—
43	C₂×#88322	—	2	1	no	—
44	C₂×#88322	D₄	1	2	yes	—
45	C₂³×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂×Dic₅₃	—	2	2	no	—
54	C₂×Dic₅₃	—	4	1	no	—
55	C₂×Dic₅₃	—	4	1	no	—
56	C₂×Dic₅₃	—	4	1	no	—
57	Dic₁₀₆	—	2	2	no	—
58	Dic₁₀₆	—	2	2	no	—
59	Dic₁₀₆	—	4	1	no	—
60	Dic₁₀₆	—	4	1	no	—
61	C₈×C₅₃	—	4	1	no	—
62	C₈×C₅₃	—	4	1	no	—
63	#88322	—	2	2	no	—
64	#88322	—	4	1	no	—
65	C₂²×C₅₃	—	4	1	no	—
66	C₂²×C₅₃	#123	1	1	yes	characteristic
67	C₂²×C₅₃	#124	1	1	yes	characteristic
68	C₂²×C₅₃	#126	1	1	yes	characteristic
69	C₄×C₅₃	—	8	1	no	—
70	C₄×C₅₃	#121	1	1	yes	characteristic
71	C₄×C₅₃	#123	1	1	yes	characteristic
72	C₄×C₅₃	#123	1	1	yes	characteristic
73	C₄×C₅₃	#126	1	1	yes	characteristic
74	Dic₅₃	—	4	2	no	—
75	Dic₅₃	—	8	1	no	—
76	#2082	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	#397	1	1	yes	characteristic
81	#430	—	53	1	no	—
82	#419	—	53	1	no	—
83	#419	—	53	1	no	—
84	C₂×#101	—	53	1	no	—
85	#344	—	53	1	no	—
86	#344	D₅₃	1	1	yes	characteristic
87	H(3,ℤ/4)	—	53	1	no	—
88	C₅₃	#2082	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	53	1	no	—
90	C₂×M₁₆	—	53	1	no	—
91	C₂×M₁₆	—	53	1	no	—
92	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
93	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
94	#117	—	53	1	no	—
95	#117	C₂×D₅₃	1	1	yes	characteristic
96	#108	—	106	2	no	—
97	#103	—	106	2	no	—
98	#103	—	106	2	no	—
99	#101	—	106	2	no	—
100	C₂×Q₈	—	53	2	no	—
101	C₂×Q₈	—	106	1	no	—
102	C₂×Q₈	—	106	1	no	—
103	C₂×Q₈	—	106	1	no	—
104	C₂×Q₈	—	106	1	no	—
105	C₂×Q₈	—	106	1	no	—
106	C₂²×C₄	—	106	1	no	—
107	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	—	2	1	no	—
110	M₁₆	—	2	1	no	—
111	M₁₆	—	2	1	no	—
112	M₁₆	—	106	1	no	—
113	M₁₆	—	106	1	no	—
114	M₁₆	—	106	1	no	—
115	M₁₆	—	106	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	53	2	no	—
119	C₂×C₈	—	106	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	106	1	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	106	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	106	1	no	—
126	C₂³	#11719945	1	1	yes	characteristic
127	Q₈	—	106	2	no	—
128	Q₈	—	106	2	no	—
129	Q₈	—	212	1	no	—
130	Q₈	—	212	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	—	212	1	no	—
134	C₂×C₄	—	212	1	no	—
135	C₂×C₄	—	212	1	no	—
136	C₂×C₄	#11719945	1	1	yes	characteristic
137	C₂×C₄	#11719945	1	1	yes	characteristic
138	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
139	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
140	C₂×C₄	#11719947	1	1	yes	characteristic
141	C₂×C₄	#11719949	1	1	yes	characteristic
142	C₈	—	4	1	no	—
143	C₈	—	4	1	no	—
144	C₈	—	106	2	no	—
145	C₈	—	212	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	#13133202	1	1	yes	characteristic
148	C₂²	#13133203	1	1	yes	characteristic
149	C₂²	#13133211	1	1	yes	characteristic
150	C₄	—	8	1	no	—
151	C₄	—	424	1	no	—
152	C₄	—	212	2	no	—
153	C₄	#13133190	1	1	yes	characteristic
154	C₄	#13133199	1	1	yes	characteristic
155	C₄	#13133202	1	1	yes	characteristic
156	C₄	#13133212	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#16810552	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810671

C₂

yes

maximal, characteristic

#16810633

C₂

yes

maximal, characteristic

#16810631

C₂

yes

maximal, characteristic

C₂×#13133135

C₂

yes

maximal, characteristic

#16810395

C₂

yes

maximal, characteristic

#16810379

C₂

yes

maximal, characteristic

C₅₃×#344

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

#13133176

C₂²

yes

—

—

—

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

D₄

yes

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#2082

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

—

—

—

D₅₃

yes

characteristic

H(3,ℤ/4)

—

C₅₃

#2082

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂×M₁₆

C₂×D₅₃

yes

characteristic

#117

—

#117

C₂×D₅₃

yes

characteristic

#108

—

106

—

#103

—

106

—

#103

—

106

—

#101

—

106

—

100

C₂×Q₈

—

101

C₂×Q₈

—

106

—

102

C₂×Q₈

—

106

—

103

C₂×Q₈

—

106

—

104

C₂×Q₈

—

106

—

105

C₂×Q₈

—

106

—

106

C₂²×C₄

—

106

—

107

C₂²×C₄

C₂²×D₅₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

106

—

113

M₁₆

—

106

—

114

M₁₆

—

106

—

115

M₁₆

—

106

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

106

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

106

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

106

—

124

C₄²

—

125

C₄²

—

106

—

126

C₂³

#11719945

yes

characteristic

127

Q₈

—

106

—

128

Q₈

—

106

—

129

Q₈

—

212

—

130

Q₈

—

212

—

131

C₂×C₄

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

212

—

134

C₂×C₄

—

212

—

135

C₂×C₄

—

212

—

136

C₂×C₄

#11719945

yes

characteristic

137

C₂×C₄

#11719945

yes

characteristic

138

C₂×C₄

D₄×D₅₃

yes

characteristic

139

C₂×C₄

D₄×D₅₃

yes

characteristic

140

C₂×C₄

#11719947

yes

characteristic

141

C₂×C₄

#11719949

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

106

—

145

C₈

—

212

—

146

C₂²

—

147

C₂²

#13133202

yes

characteristic

148

C₂²

#13133203

yes

characteristic

149

C₂²

#13133211

yes

characteristic

150

C₄

—

151

C₄

—

424

—

152

C₄

—

212

—

153

C₄

#13133190

yes

characteristic

154

C₄

#13133199

yes

characteristic

155

C₄

#13133202

yes

characteristic

156

C₄

#13133212

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#16810552

yes

characteristic

	subgroup	G/H
center	C₂	#16810552
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#344	C₂
socle	C₂×C₅₃	#397
hypercenter	#344	D₅₃
hypocenter	C₅₃	#2082
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₂

#16810552

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#344

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
88. C₅₃	#2082	semidirect

H ≅ G/N

type

88. C₅₃

#2082

semidirect

Subgroups — #20276185

Conjugacy classes of subgroups

Complements