Group #20287147 — subgroups

Subgroups — #20287147

Order 6784.

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

normal subgroups only

This group has 340 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	#16811346	C₂	1	2	yes	maximal
2	#16811028	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810963	C₂	1	4	yes	maximal, complemented
4	#16810908	C₂	1	2	yes	maximal, complemented
5	#16810861	C₂	1	2	yes	maximal, complemented
6	C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
7	#16810660	C₂	1	2	yes	maximal, complemented
8	C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₈×D₅₃	C₂²	1	2	yes	—
10	#13133217	C₂²	1	2	yes	—
11	#13133212	—	2	4	no	—
12	#13133205	—	2	2	no	—
13	#13133193	C₂²	1	4	yes	—
14	C₄×Dic₁₀₆	—	2	2	no	—
15	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
16	#13133175	C₂²	1	4	yes	—
17	#13133151	C₂²	1	2	yes	—
18	#13133151	C₂²	1	2	yes	—
19	#13133148	C₂²	1	4	yes	—
20	C₄⋊#88322	C₂²	1	1	yes	characteristic
21	#13133141	C₂²	1	4	yes	—
22	#13133129	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
25	#107×C₅₃	C₂²	1	1	yes	characteristic
26	#103×C₅₃	C₂²	1	2	yes	—
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
29	C₂×D₂₁₂	D₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	C₂³	1	2	yes	—
31	C₂×Dic₁₀₆	—	2	2	no	—
32	Q₁₆×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
37	C₄²×C₅₃	—	2	2	no	—
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	C₄⋊Dic₅₃	—	2	2	no	—
43	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
44	C₄⋊Dic₅₃	C₂³	1	4	yes	—
45	#11719919	—	2	4	no	—
46	C₄×Dic₅₃	—	2	4	no	—
47	C₂×#88322	C₂³	1	2	yes	—
48	C₈×D₅₃	—	2	4	no	—
49	C₂²×D₅₃	C₂×D₄	1	2	yes	—
50	Q₈×C₅₃	—	2	2	no	—
51	Q₈×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	4	no	—
58	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
59	D₂₁₂	—	2	2	no	—
60	C₄×D₅₃	—	2	2	no	—
61	C₄×D₅₃	—	2	2	no	—
62	Dic₁₀₆	—	4	4	no	—
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	#41	1	2	yes	—
65	#88322	—	2	2	no	—
66	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₅₃	—	2	2	no	—
68	C₂×D₅₃	—	2	2	no	—
69	C₂×D₅₃	—	2	2	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	4	2	no	—
72	C₄×C₅₃	—	4	2	no	—
73	C₄×C₅₃	#135	1	2	yes	—
74	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
75	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
76	Dic₅₃	—	2	2	no	—
77	Dic₅₃	—	4	4	no	—
78	#3342	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	#547	1	1	yes	characteristic
80	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
81	C₂×C₅₃	#578	1	1	yes	characteristic
82	D₅₃	—	4	2	no	—
83	#556	—	53	2	no	—
84	#500	—	53	1	no	—
85	#485	—	53	4	no	—
86	#475	—	53	2	no	—
87	#467	—	53	2	no	—
88	C₄×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
89	C₈×D₄	—	53	1	no	—
90	#427	—	53	2	no	—
91	C₅₃	#3342	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
93	C₂²×C₈	—	53	2	no	—
94	C₄⋊Q₈	—	53	2	no	—
95	#126	—	106	4	no	—
96	#124	—	106	2	no	—
97	C₂²⋊Q₈	—	53	4	no	—
98	C₄×Q₈	—	106	2	no	—
99	C₄×Q₈	C₂×D₅₃	1	2	yes	—
100	C₄×D₄	—	53	1	no	—
101	#117	—	53	4	no	—
102	#107	—	53	2	no	—
103	#107	—	53	2	no	—
104	#107	C₂×D₅₃	1	1	yes	characteristic
105	#106	—	53	4	no	—
106	C₄⋊C₈	—	53	1	no	—
107	#103	—	53	4	no	—
108	#103	C₂×D₅₃	1	2	yes	—
109	C₂²⋊C₈	—	53	2	no	—
110	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
111	C₂×Q₈	—	106	2	no	—
112	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
113	C₂×D₄	—	53	1	no	—
114	C₂²×C₄	—	53	2	no	—
115	Q₁₆	—	2	2	no	—
116	C₂×C₈	—	53	2	no	—
117	C₂×C₈	—	106	4	no	—
118	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	53	1	no	—
122	C₄⋊C₄	—	53	4	no	—
123	C₄⋊C₄	—	106	4	no	—
124	C₄⋊C₄	—	106	2	no	—
125	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
126	C₂²⋊C₄	—	53	2	no	—
127	C₂²⋊C₄	—	106	4	no	—
128	C₂²⋊C₄	—	106	4	no	—
129	C₄²	—	2	2	no	—
130	C₄²	—	106	4	no	—
131	C₄²	C₂²×D₅₃	1	1	yes	characteristic
132	C₂³	—	53	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	2	2	no	—
135	Q₈	—	212	4	no	—
136	D₄	—	106	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	53	2	no	—
140	C₂×C₄	—	106	2	no	—
141	C₂×C₄	—	106	2	no	—
142	C₂×C₄	—	106	4	no	—
143	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
144	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	106	2	no	—
148	C₈	#11719945	1	2	yes	—
149	C₂²	—	106	2	no	—
150	C₂²	—	106	2	no	—
151	C₂²	—	106	2	no	—
152	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	2	no	—
155	C₄	—	4	2	no	—
156	C₄	—	106	2	no	—
157	C₄	—	212	4	no	—
158	C₄	#13133240	1	2	yes	—
159	C₄	C₂×#11719945	1	1	yes	characteristic
160	C₄	#13133264	1	1	yes	characteristic
161	C₂	—	212	2	no	—
162	C₂	#16811278	1	1	yes	characteristic
163	C₂	C₂×#13133240	1	1	yes	characteristic
164	C₂	#16811437	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811346

C₂

yes

maximal

#16811028

C₂

yes

maximal, characteristic, complemented

#16810963

C₂

yes

maximal, complemented

#16810908

C₂

yes

maximal, complemented

#16810861

C₂

yes

maximal, complemented

C₈×D₂₁₂

C₂

yes

maximal, characteristic

#16810660

C₂

yes

maximal, complemented

C₄×Q₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

C₂²

yes

—

#13133217

C₂²

yes

—

#13133212

—

#13133205

—

#13133193

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×D₂₁₂

C₂²

yes

characteristic

#13133175

C₂²

yes

—

#13133151

C₂²

yes

—

#13133151

C₂²

yes

—

#13133148

C₂²

yes

—

C₄⋊#88322

C₂²

yes

characteristic

#13133141

C₂²

yes

—

#13133129

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

characteristic

#103×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

C₂³

yes

—

C₈×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

—

—

C₄×Q₁₆

D₅₃

yes

characteristic, complemented

C₈×D₄

—

#427

—

C₅₃

#3342

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

—

#126

—

106

—

#124

—

106

—

C₂²⋊Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

—

100

C₄×D₄

—

101

#117

—

102

#107

—

103

#107

—

104

#107

C₂×D₅₃

yes

characteristic

105

#106

—

106

C₄⋊C₈

—

107

#103

—

108

#103

C₂×D₅₃

yes

—

109

C₂²⋊C₈

—

110

C₄×C₈

C₂×D₅₃

yes

characteristic

111

C₂×Q₈

—

106

—

112

C₂×Q₈

C₂²×D₅₃

yes

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

Q₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

106

—

118

C₂×C₈

C₂²×D₅₃

yes

characteristic

119

C₂×C₈

C₂²×D₅₃

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

—

106

—

125

C₄⋊C₄

C₂²×D₅₃

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

106

—

128

C₂²⋊C₄

—

106

—

129

C₄²

—

130

C₄²

—

106

—

131

C₄²

C₂²×D₅₃

yes

characteristic

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

212

—

136

D₄

—

106

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

106

—

141

C₂×C₄

—

106

—

142

C₂×C₄

—

106

—

143

C₂×C₄

D₄×D₅₃

yes

characteristic

144

C₂×C₄

D₄×D₅₃

yes

characteristic

145

C₂×C₄

C₂³×D₅₃

yes

characteristic

146

C₈

—

147

C₈

—

106

—

148

C₈

#11719945

yes

—

149

C₂²

—

106

—

150

C₂²

—

106

—

151

C₂²

—

106

—

152

C₂²

C₂×D₄×D₅₃

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

106

—

157

C₄

—

212

—

158

C₄

#13133240

yes

—

159

C₄

C₂×#11719945

yes

characteristic

160

C₄

#13133264

yes

characteristic

161

C₂

—

212

—

162

C₂

#16811278

yes

characteristic

163

C₂

C₂×#13133240

yes

characteristic

164

C₂

#16811437

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#3342
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₄×Q₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₄×Q₁₆

D₅₃

hypocenter

C₅₃

#3342

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
2. #16811028	C₂	semidirect
3. #16810963	C₂	semidirect
4. #16810908	C₂	semidirect
5. #16810861	C₂	semidirect
7. #16810660	C₂	semidirect
8. C₄×Q₁₆×C₅₃	C₂	semidirect
88. C₄×Q₁₆	D₅₃	semidirect
91. C₅₃	#3342	semidirect

H ≅ G/N

type

2. #16811028

C₂

semidirect

3. #16810963

C₂

semidirect

4. #16810908

C₂

semidirect

5. #16810861

C₂

semidirect

7. #16810660

C₂

semidirect

8. C₄×Q₁₆×C₅₃

C₂

semidirect

88. C₄×Q₁₆

D₅₃

semidirect

91. C₅₃

#3342

semidirect

Subgroups — #20287147

Conjugacy classes of subgroups

Complements