Group #20276178 — subgroups

Subgroups — #20276178

Order 6784.

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

normal subgroups only

This group has 204 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, complemented
2	#16810629	C₂	1	1	yes	maximal, characteristic
3	#16810627	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133133	C₂	1	1	yes	maximal, characteristic
5	#16810395	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#344	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₁₂	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	#13133139	—	2	2	no	—
14	#13133137	—	2	2	no	—
15	#13133133	—	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	C₂³×D₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×D₂₁₂	—	2	1	no	—
22	C₂×D₂₁₂	—	2	1	no	—
23	C₂×D₂₁₂	—	2	1	no	—
24	C₂×D₂₁₂	—	2	1	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	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	C₂²×D₅₃	—	2	2	no	—
47	C₂²×D₅₃	—	4	1	no	—
48	C₂²×D₅₃	—	4	1	no	—
49	C₂²×D₅₃	—	8	2	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
57	C₂×Dic₅₃	—	4	1	no	—
58	D₂₁₂	—	2	2	no	—
59	D₂₁₂	—	2	2	no	—
60	D₂₁₂	—	4	1	no	—
61	D₂₁₂	—	4	1	no	—
62	C₈×C₅₃	—	4	1	no	—
63	C₈×C₅₃	—	4	1	no	—
64	#88322	—	2	2	no	—
65	#88322	—	4	1	no	—
66	C₂²×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	#123	1	1	yes	characteristic
68	C₂²×C₅₃	#124	1	1	yes	characteristic
69	C₂²×C₅₃	#126	1	1	yes	characteristic
70	C₂×D₅₃	—	4	2	no	—
71	C₂×D₅₃	—	8	2	no	—
72	C₂×D₅₃	—	8	2	no	—
73	C₂×D₅₃	—	8	2	no	—
74	C₄×C₅₃	—	8	1	no	—
75	C₄×C₅₃	#121	1	1	yes	characteristic
76	C₄×C₅₃	#123	1	1	yes	characteristic
77	C₄×C₅₃	#123	1	1	yes	characteristic
78	C₄×C₅₃	#126	1	1	yes	characteristic
79	Dic₅₃	—	8	1	no	—
80	#2081	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	—	2	1	no	—
82	C₂×C₅₃	—	2	1	no	—
83	C₂×C₅₃	—	2	1	no	—
84	C₂×C₅₃	#397	1	1	yes	characteristic
85	D₅₃	—	8	2	no	—
86	#430	—	53	1	no	—
87	#418	—	53	1	no	—
88	#418	—	53	1	no	—
89	C₂×#100	—	53	1	no	—
90	#344	—	53	1	no	—
91	#344	D₅₃	1	1	yes	characteristic, complemented
92	H(3,ℤ/4)	—	53	1	no	—
93	C₅₃	#2081	1	1	yes	characteristic, complemented, Sylow
94	C₂²×D₄	—	53	1	no	—
95	C₂×M₁₆	—	53	1	no	—
96	C₂×M₁₆	—	53	1	no	—
97	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
98	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
99	#117	—	53	1	no	—
100	#117	C₂×D₅₃	1	1	yes	characteristic
101	#108	—	106	2	no	—
102	#102	—	106	2	no	—
103	#102	—	106	2	no	—
104	#100	—	106	2	no	—
105	C₂⁴	—	106	1	no	—
106	C₂×D₄	—	53	2	no	—
107	C₂×D₄	—	106	1	no	—
108	C₂×D₄	—	106	1	no	—
109	C₂×D₄	—	106	1	no	—
110	C₂×D₄	—	106	1	no	—
111	C₂×D₄	—	106	1	no	—
112	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	106	1	no	—
118	M₁₆	—	106	1	no	—
119	M₁₆	—	106	1	no	—
120	M₁₆	—	106	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	53	2	no	—
124	C₂×C₈	—	106	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	106	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	106	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	106	1	no	—
131	C₂³	—	106	2	no	—
132	C₂³	—	424	2	no	—
133	C₂³	—	212	1	no	—
134	C₂³	—	212	1	no	—
135	C₂³	#11719945	1	1	yes	characteristic
136	D₄	—	106	2	no	—
137	D₄	—	106	2	no	—
138	D₄	—	212	1	no	—
139	D₄	—	212	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	212	1	no	—
142	C₂×C₄	#11719945	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	C₂×C₄	#11719949	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	106	2	no	—
151	C₈	—	212	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	424	2	no	—
154	C₂²	—	424	2	no	—
155	C₂²	—	424	2	no	—
156	C₂²	—	212	2	no	—
157	C₂²	#13133202	1	1	yes	characteristic
158	C₂²	#13133203	1	1	yes	characteristic
159	C₂²	#13133211	1	1	yes	characteristic
160	C₄	—	8	1	no	—
161	C₄	—	424	1	no	—
162	C₄	#13133190	1	1	yes	characteristic
163	C₄	#13133199	1	1	yes	characteristic
164	C₄	#13133202	1	1	yes	characteristic
165	C₄	#13133212	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	424	2	no	—
170	C₂	#16810552	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810671

C₂

yes

maximal, characteristic, complemented

#16810629

C₂

yes

maximal, characteristic

#16810627

C₂

yes

maximal, characteristic

C₂×#13133133

C₂

yes

maximal, characteristic

#16810395

C₂

yes

maximal, characteristic, complemented

#16810379

C₂

yes

maximal, characteristic, complemented

C₅₃×#344

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

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

C₂³×D₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

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

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

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

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2081

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#397

yes

characteristic

D₅₃

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

H(3,ℤ/4)

—

C₅₃

#2081

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂×M₁₆

C₂×D₅₃

yes

characteristic

#117

—

100

#117

C₂×D₅₃

yes

characteristic

101

#108

—

106

—

102

#102

—

106

—

103

#102

—

106

—

104

#100

—

106

—

105

C₂⁴

—

106

—

106

C₂×D₄

—

107

C₂×D₄

—

106

—

108

C₂×D₄

—

106

—

109

C₂×D₄

—

106

—

110

C₂×D₄

—

106

—

111

C₂×D₄

—

106

—

112

C₂²×C₄

C₂²×D₅₃

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

106

—

118

M₁₆

—

106

—

119

M₁₆

—

106

—

120

M₁₆

—

106

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

106

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

106

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

106

—

129

C₄²

—

130

C₄²

—

106

—

131

C₂³

—

106

—

132

C₂³

—

424

—

133

C₂³

—

212

—

134

C₂³

—

212

—

135

C₂³

#11719945

yes

characteristic

136

D₄

—

106

—

137

D₄

—

106

—

138

D₄

—

212

—

139

D₄

—

212

—

140

C₂×C₄

—

141

C₂×C₄

—

212

—

142

C₂×C₄

#11719945

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

C₂×C₄

#11719949

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

106

—

151

C₈

—

212

—

152

C₂²

—

153

C₂²

—

424

—

154

C₂²

—

424

—

155

C₂²

—

424

—

156

C₂²

—

212

—

157

C₂²

#13133202

yes

characteristic

158

C₂²

#13133203

yes

characteristic

159

C₂²

#13133211

yes

characteristic

160

C₄

—

161

C₄

—

424

—

162

C₄

#13133190

yes

characteristic

163

C₄

#13133199

yes

characteristic

164

C₄

#13133202

yes

characteristic

165

C₄

#13133212

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

424

—

170

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	#2081
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
1. #16810671	C₂	semidirect
5. #16810395	C₂	semidirect
6. #16810379	C₂	semidirect
7. C₅₃×#344	C₂	semidirect
91. #344	D₅₃	semidirect
93. C₅₃	#2081	semidirect

H ≅ G/N

type

1. #16810671

C₂

semidirect

5. #16810395

C₂

semidirect

6. #16810379

C₂

semidirect

7. C₅₃×#344

C₂

semidirect

91. #344

D₅₃

semidirect

93. C₅₃

#2081

semidirect

Subgroups — #20276178

Conjugacy classes of subgroups

Complements