Group #20285366 — subgroups

Subgroups — #20285366

Order 6784.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16811229	C₂	1	2	yes	maximal, complemented
2	#16811000	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810989	C₂	1	2	yes	maximal, complemented
4	#16810877	C₂	1	2	yes	maximal, complemented
5	#16810860	C₂	1	2	yes	maximal, complemented
6	#16810734	C₂	1	2	yes	maximal, complemented
7	#16810705	C₂	1	2	yes	maximal, complemented
8	#16810577	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄₂₄	C₂²	1	2	yes	—
11	C₂×D₄₂₄	C₂²	1	2	yes	—
12	C₂×#11719912	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133214	C₂²	1	2	yes	—
15	#13133202	—	2	4	no	—
16	#13133192	—	2	4	no	—
17	#13133192	C₂²	1	4	yes	—
18	#13133179	C₂²	1	4	yes	—
19	#13133174	C₂²	1	1	yes	characteristic
20	#13133137	C₂²	1	4	yes	—
21	#13133137	C₂²	1	4	yes	—
22	#13133128	C₂²	1	1	yes	characteristic
23	#13133127	C₂²	1	1	yes	characteristic
24	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
25	#125×C₅₃	C₂²	1	2	yes	—
26	#107×C₅₃	C₂²	1	2	yes	—
27	#106×C₅₃	C₂²	1	2	yes	—
28	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×D₂₁₂	—	2	4	no	—
30	C₂×D₂₁₂	—	2	4	no	—
31	C₂×D₂₁₂	C₂³	1	4	yes	—
32	C₂×C₄×D₅₃	—	2	4	no	—
33	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#11719921	—	2	4	no	—
40	#11719921	—	2	4	no	—
41	#11719921	D₄	1	2	yes	—
42	#11719919	D₄	1	2	yes	—
43	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
44	C₂×#88322	C₂³	1	1	yes	characteristic
45	C₂×#88322	C₂³	1	1	yes	characteristic
46	D₄₂₄	—	2	4	no	—
47	D₄₂₄	—	2	4	no	—
48	#11719912	—	2	2	no	—
49	C₈×D₅₃	—	2	2	no	—
50	C₂²×D₅₃	—	2	4	no	—
51	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
56	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
57	D₂₁₂	—	2	4	no	—
58	D₂₁₂	—	2	4	no	—
59	D₂₁₂	—	8	4	no	—
60	D₂₁₂	—	8	4	no	—
61	C₄×D₅₃	—	2	1	no	—
62	C₄×D₅₃	—	2	1	no	—
63	C₄×D₅₃	—	8	4	no	—
64	C₈×C₅₃	#41	1	2	yes	—
65	C₈×C₅₃	#41	1	2	yes	—
66	#88322	—	2	1	no	—
67	#88322	—	2	1	no	—
68	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₅₃	—	2	1	no	—
70	C₂×D₅₃	—	2	2	no	—
71	C₂×D₅₃	—	4	4	no	—
72	C₂×D₅₃	—	4	4	no	—
73	C₂×D₅₃	—	4	4	no	—
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	—	4	4	no	—
76	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
77	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
78	Dic₅₃	—	2	1	no	—
79	Dic₅₃	—	2	2	no	—
80	#3176	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	#532	1	1	yes	characteristic
82	C₂×C₅₃	#576	1	2	yes	—
83	D₅₃	—	4	1	no	—
84	D₅₃	—	8	4	no	—
85	#540	—	53	2	no	—
86	#499	D₅₃	1	1	yes	characteristic, complemented
87	#493	—	53	1	no	—
88	#490	—	53	2	no	—
89	#469	—	53	2	no	—
90	#466	—	53	2	no	—
91	#442	—	53	2	no	—
92	C₄×D₈	—	53	2	no	—
93	#405	—	53	1	no	—
94	C₅₃	#3176	1	1	yes	characteristic, complemented, Sylow
95	C₂×D₈	—	53	2	no	—
96	C₂×D₈	—	53	2	no	—
97	C₂×M₁₆	—	53	1	no	—
98	C₂²×C₈	—	53	1	no	—
99	#127	—	53	2	no	—
100	#125	C₂×D₅₃	1	2	yes	—
101	#123	—	106	4	no	—
102	#121	—	53	4	no	—
103	#121	—	106	4	no	—
104	C₄×D₄	—	53	4	no	—
105	#117	—	53	1	no	—
106	#107	C₂×D₅₃	1	2	yes	—
107	#106	C₂×D₅₃	1	2	yes	—
108	#102	—	53	4	no	—
109	#102	—	53	4	no	—
110	#97	—	53	1	no	—
111	#97	—	53	1	no	—
112	C₄×C₈	—	53	1	no	—
113	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	C₂×D₄	—	53	4	no	—
115	C₂×D₄	—	106	4	no	—
116	C₂×D₄	—	106	4	no	—
117	C₂²×C₄	—	53	1	no	—
118	C₂²×C₄	—	106	4	no	—
119	D₈	—	106	4	no	—
120	D₈	—	106	4	no	—
121	M₁₆	—	106	2	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	53	1	no	—
124	C₂×C₈	—	106	2	no	—
125	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	53	2	no	—
129	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
130	C₂²⋊C₄	—	53	2	no	—
131	C₂²⋊C₄	—	106	4	no	—
132	C₂²⋊C₄	—	106	4	no	—
133	C₄²	—	53	1	no	—
134	C₄²	C₂²×D₅₃	1	1	yes	characteristic
135	C₂³	—	53	1	no	—
136	C₂³	—	106	4	no	—
137	D₄	—	106	4	no	—
138	D₄	—	106	4	no	—
139	D₄	—	424	4	no	—
140	D₄	—	424	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	53	2	no	—
143	C₂×C₄	—	53	1	no	—
144	C₂×C₄	—	106	1	no	—
145	C₂×C₄	—	106	1	no	—
146	C₂×C₄	—	424	4	no	—
147	C₂×C₄	D₄×D₅₃	1	2	yes	—
148	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
149	C₈	—	106	1	no	—
150	C₈	—	106	1	no	—
151	C₈	#11719949	1	2	yes	—
152	C₈	#11719949	1	2	yes	—
153	C₂²	—	106	2	no	—
154	C₂²	—	106	1	no	—
155	C₂²	—	212	4	no	—
156	C₂²	—	212	4	no	—
157	C₂²	—	212	4	no	—
158	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	4	no	—
161	C₄	—	106	2	no	—
162	C₄	—	106	1	no	—
163	C₄	C₂×#11719949	1	1	yes	characteristic
164	C₄	C₂×#11719949	1	1	yes	characteristic
165	C₂	—	424	4	no	—
166	C₂	—	212	1	no	—
167	C₂	#16811174	1	1	yes	characteristic
168	C₂	#16811425	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811229

C₂

yes

maximal, complemented

#16811000

C₂

yes

maximal, characteristic, complemented

#16810989

C₂

yes

maximal, complemented

#16810877

C₂

yes

maximal, complemented

#16810860

C₂

yes

maximal, complemented

#16810734

C₂

yes

maximal, complemented

#16810705

C₂

yes

maximal, complemented

#16810577

C₂

yes

maximal, characteristic, complemented

C₅₃×#499

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₂²

yes

—

C₂×D₄₂₄

C₂²

yes

—

C₂×#11719912

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133214

C₂²

yes

—

#13133202

—

#13133192

—

#13133192

C₂²

yes

—

#13133179

C₂²

yes

—

#13133174

C₂²

yes

characteristic

#13133137

C₂²

yes

—

#13133137

C₂²

yes

—

#13133128

C₂²

yes

characteristic

#13133127

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

#125×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

#106×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

—

D₄

yes

—

#11719919

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

D₄₂₄

—

D₄₂₄

—

#11719912

—

C₈×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

#41

yes

—

C₈×C₅₃

#41

yes

—

#88322

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

D₅₃

—

#540

—

#499

D₅₃

yes

characteristic, complemented

—

—

—

—

—

C₄×D₈

—

#405

—

C₅₃

#3176

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

#127

—

100

#125

C₂×D₅₃

yes

—

101

#123

—

106

—

102

#121

—

103

#121

—

106

—

104

C₄×D₄

—

105

#117

—

106

#107

C₂×D₅₃

yes

—

107

#106

C₂×D₅₃

yes

—

108

#102

—

109

#102

—

110

#97

—

111

#97

—

112

C₄×C₈

—

113

C₄×C₈

C₂×D₅₃

yes

characteristic

114

C₂×D₄

—

115

C₂×D₄

—

106

—

116

C₂×D₄

—

106

—

117

C₂²×C₄

—

118

C₂²×C₄

—

106

—

119

D₈

—

106

—

120

D₈

—

106

—

121

M₁₆

—

106

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

106

—

125

C₂×C₈

C₂²×D₅₃

yes

characteristic

126

C₂×C₈

C₂²×D₅₃

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

C₂²×D₅₃

yes

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

106

—

132

C₂²⋊C₄

—

106

—

133

C₄²

—

134

C₄²

C₂²×D₅₃

yes

characteristic

135

C₂³

—

136

C₂³

—

106

—

137

D₄

—

106

—

138

D₄

—

106

—

139

D₄

—

424

—

140

D₄

—

424

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

106

—

145

C₂×C₄

—

106

—

146

C₂×C₄

—

424

—

147

C₂×C₄

D₄×D₅₃

yes

—

148

C₂×C₄

C₂³×D₅₃

yes

characteristic

149

C₈

—

106

—

150

C₈

—

106

—

151

C₈

#11719949

yes

—

152

C₈

#11719949

yes

—

153

C₂²

—

106

—

154

C₂²

—

106

—

155

C₂²

—

212

—

156

C₂²

—

212

—

157

C₂²

—

212

—

158

C₂²

C₂×D₄×D₅₃

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

106

—

162

C₄

—

106

—

163

C₄

C₂×#11719949

yes

characteristic

164

C₄

C₂×#11719949

yes

characteristic

165

C₂

—

424

—

166

C₂

—

212

—

167

C₂

#16811174

yes

characteristic

168

C₂

#16811425

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#499	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#499	D₅₃
hypocenter	C₅₃	#3176
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#499

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
1. #16811229	C₂	semidirect
2. #16811000	C₂	semidirect
3. #16810989	C₂	semidirect
4. #16810877	C₂	semidirect
5. #16810860	C₂	semidirect
6. #16810734	C₂	semidirect
7. #16810705	C₂	semidirect
8. #16810577	C₂	semidirect
9. C₅₃×#499	C₂	semidirect
86. #499	D₅₃	semidirect
94. C₅₃	#3176	semidirect

H ≅ G/N

type

1. #16811229

C₂

semidirect

2. #16811000

C₂

semidirect

3. #16810989

C₂

semidirect

4. #16810877

C₂

semidirect

5. #16810860

C₂

semidirect

6. #16810734

C₂

semidirect

7. #16810705

C₂

semidirect

8. #16810577

C₂

semidirect

9. C₅₃×#499

C₂

semidirect

86. #499

D₅₃

semidirect

94. C₅₃

#3176

semidirect

Subgroups — #20285366

Conjugacy classes of subgroups

Complements