Group #20285371 — subgroups

Subgroups — #20285371

Order 6784.

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

normal subgroups only

This group has 338 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	#16811234	C₂	1	2	yes	maximal
2	#16811025	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810991	C₂	1	2	yes	maximal, complemented
4	#16810878	C₂	1	2	yes	maximal
5	#16810861	C₂	1	2	yes	maximal
6	#16810730	C₂	1	2	yes	maximal, complemented
7	Q₁₆×Dic₅₃	C₂	1	2	yes	maximal, complemented
8	#16810577	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719912	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#13133212	—	2	4	no	—
13	#13133209	C₂²	1	2	yes	—
14	#13133207	—	2	4	no	—
15	#13133193	C₂²	1	4	yes	—
16	Q₈×Dic₅₃	C₂²	1	4	yes	—
17	#13133174	C₂²	1	1	yes	characteristic
18	#13133151	C₂²	1	2	yes	—
19	#13133149	C₂²	1	2	yes	—
20	#13133141	C₂²	1	4	yes	—
21	#13133141	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	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
26	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
28	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
30	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
31	Q₁₆×C₅₃	—	2	4	no	—
32	Q₁₆×C₅₃	—	2	4	no	—
33	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#11719921	—	2	4	no	—
38	#11719921	D₄	1	2	yes	—
39	C₄⋊Dic₅₃	—	2	4	no	—
40	C₄⋊Dic₅₃	C₂³	1	4	yes	—
41	#11719919	—	2	4	no	—
42	#11719919	—	2	4	no	—
43	#11719919	D₄	1	2	yes	—
44	C₄×Dic₅₃	—	2	4	no	—
45	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
46	C₂×#88322	C₂³	1	1	yes	characteristic
47	C₂×#88322	C₂³	1	1	yes	characteristic
48	#11719912	—	2	2	no	—
49	C₈×D₅₃	—	2	2	no	—
50	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₅₃	—	2	4	no	—
52	Q₈×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	—	2	4	no	—
54	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	4	no	—
57	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
59	C₄×D₅₃	—	2	1	no	—
60	C₄×D₅₃	—	2	1	no	—
61	C₈×C₅₃	#41	1	2	yes	—
62	C₈×C₅₃	#41	1	2	yes	—
63	#88322	—	2	1	no	—
64	#88322	—	2	1	no	—
65	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₅₃	—	2	1	no	—
67	C₂×D₅₃	—	2	2	no	—
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	4	4	no	—
70	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
71	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
72	Dic₅₃	—	2	1	no	—
73	Dic₅₃	—	2	2	no	—
74	Dic₅₃	—	4	4	no	—
75	#3177	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	#532	1	1	yes	characteristic
77	C₂×C₅₃	#578	1	2	yes	—
78	D₅₃	—	4	1	no	—
79	#541	—	53	2	no	—
80	#499	—	53	1	no	—
81	#494	D₅₃	1	1	yes	characteristic, complemented
82	#491	—	53	2	no	—
83	#470	—	53	2	no	—
84	#467	—	53	2	no	—
85	#441	—	53	2	no	—
86	C₄×Q₁₆	—	53	2	no	—
87	#405	—	53	1	no	—
88	C₅₃	#3177	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
90	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
91	C₂×M₁₆	—	53	1	no	—
92	C₂²×C₈	—	53	1	no	—
93	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
94	#126	—	106	4	no	—
95	#125	—	53	2	no	—
96	#125	—	106	4	no	—
97	C₂²⋊Q₈	—	53	4	no	—
98	C₄×Q₈	—	53	4	no	—
99	#117	—	53	1	no	—
100	#107	—	53	2	no	—
101	#106	—	53	2	no	—
102	#103	—	53	4	no	—
103	#103	—	53	4	no	—
104	#97	—	53	1	no	—
105	#97	—	53	1	no	—
106	C₄×C₈	—	53	1	no	—
107	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
108	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
109	C₂²×C₄	—	53	1	no	—
110	Q₁₆	—	2	4	no	—
111	Q₁₆	—	2	4	no	—
112	M₁₆	—	106	2	no	—
113	C₂×C₈	—	53	1	no	—
114	C₂×C₈	—	53	1	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	53	2	no	—
120	C₄⋊C₄	—	53	4	no	—
121	C₄⋊C₄	—	106	4	no	—
122	C₄⋊C₄	—	106	4	no	—
123	C₄⋊C₄	—	106	4	no	—
124	C₂²⋊C₄	—	53	2	no	—
125	C₂²⋊C₄	—	106	4	no	—
126	C₄²	—	53	1	no	—
127	C₄²	—	106	4	no	—
128	C₄²	C₂²×D₅₃	1	1	yes	characteristic
129	C₂³	—	53	1	no	—
130	Q₈	—	2	4	no	—
131	Q₈	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	53	2	no	—
134	C₂×C₄	—	53	1	no	—
135	C₂×C₄	—	106	1	no	—
136	C₂×C₄	—	106	1	no	—
137	C₂×C₄	—	106	4	no	—
138	C₂×C₄	D₄×D₅₃	1	2	yes	—
139	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
140	C₈	—	106	1	no	—
141	C₈	—	106	1	no	—
142	C₈	#11719947	1	2	yes	—
143	C₈	#11719947	1	2	yes	—
144	C₂²	—	106	1	no	—
145	C₂²	—	106	2	no	—
146	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	106	1	no	—
150	C₄	—	106	2	no	—
151	C₄	—	212	4	no	—
152	C₄	C₂×#11719947	1	1	yes	characteristic
153	C₄	C₂×#11719947	1	1	yes	characteristic
154	C₂	—	212	1	no	—
155	C₂	#16811171	1	1	yes	characteristic
156	C₂	#16811436	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811234

C₂

yes

maximal

#16811025

C₂

yes

maximal, characteristic, complemented

#16810991

C₂

yes

maximal, complemented

#16810878

C₂

yes

maximal

#16810861

C₂

yes

maximal

#16810730

C₂

yes

maximal, complemented

Q₁₆×Dic₅₃

C₂

yes

maximal, complemented

#16810577

C₂

yes

maximal, characteristic

C₅₃×#494

C₂

yes

maximal, characteristic, complemented

C₂×#11719912

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133212

—

#13133209

C₂²

yes

—

#13133207

—

#13133193

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

—

#13133174

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133149

C₂²

yes

—

#13133141

C₂²

yes

—

#13133141

C₂²

yes

—

#13133128

C₂²

yes

characteristic

#13133127

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

#11719921

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

#11719919

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

#11719912

—

C₈×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

#41

yes

—

C₈×C₅₃

#41

yes

—

#88322

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

#541

—

#499

—

#494

D₅₃

yes

characteristic, complemented

—

—

—

—

C₄×Q₁₆

—

#405

—

C₅₃

#3177

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#126

—

106

—

#125

—

#125

—

106

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

100

#107

—

101

#106

—

102

#103

—

103

#103

—

104

#97

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

C₂×D₅₃

yes

characteristic

108

C₂×Q₈

C₂²×D₅₃

yes

—

109

C₂²×C₄

—

110

Q₁₆

—

111

Q₁₆

—

112

M₁₆

—

106

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

106

—

116

C₂×C₈

C₂²×D₅₃

yes

characteristic

117

C₂×C₈

C₂²×D₅₃

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

—

106

—

123

C₄⋊C₄

—

106

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

106

—

126

C₄²

—

127

C₄²

—

106

—

128

C₄²

C₂²×D₅₃

yes

characteristic

129

C₂³

—

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

106

—

138

C₂×C₄

D₄×D₅₃

yes

—

139

C₂×C₄

C₂³×D₅₃

yes

characteristic

140

C₈

—

106

—

141

C₈

—

106

—

142

C₈

#11719947

yes

—

143

C₈

#11719947

yes

—

144

C₂²

—

106

—

145

C₂²

—

106

—

146

C₂²

C₂×D₄×D₅₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

106

—

150

C₄

—

106

—

151

C₄

—

212

—

152

C₄

C₂×#11719947

yes

characteristic

153

C₄

C₂×#11719947

yes

characteristic

154

C₂

—

212

—

155

C₂

#16811171

yes

characteristic

156

C₂

#16811436

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#494	D₅₃
hypocenter	C₅₃	#3177
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#494

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #16811025	C₂	semidirect
3. #16810991	C₂	semidirect
6. #16810730	C₂	semidirect
7. Q₁₆×Dic₅₃	C₂	semidirect
9. C₅₃×#494	C₂	semidirect
81. #494	D₅₃	semidirect
88. C₅₃	#3177	semidirect

H ≅ G/N

type

2. #16811025

C₂

semidirect

3. #16810991

C₂

semidirect

6. #16810730

C₂

semidirect

7. Q₁₆×Dic₅₃

C₂

semidirect

9. C₅₃×#494

C₂

semidirect

81. #494

D₅₃

semidirect

88. C₅₃

#3177

semidirect

Subgroups — #20285371

Conjugacy classes of subgroups

Complements