Group #20287705 — subgroups

Subgroups — #20287705

Order 6784.

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

normal subgroups only

This group has 360 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	#16811347	C₂	1	2	yes	maximal, complemented
2	#16811008	C₂	1	2	yes	maximal
3	#16810847	C₂	1	4	yes	maximal, complemented
4	#16810825	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×Dic₅₃	C₂	1	2	yes	maximal, complemented
7	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11719924	C₂²	1	4	yes	—
10	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
11	#13133212	—	2	4	no	—
12	#13133206	C₂²	1	4	yes	—
13	#13133197	—	2	2	no	—
14	Q₈×Dic₅₃	C₂²	1	4	yes	—
15	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	C₄×D₂₁₂	C₂²	1	2	yes	—
17	#13133175	—	2	2	no	—
18	#13133151	C₂²	1	1	yes	characteristic
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	#13133146	C₂²	1	2	yes	—
21	#13133141	C₂²	1	4	yes	—
22	#13133139	C₂²	1	2	yes	—
23	C₈×Dic₅₃	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
25	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
26	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
29	C₂×D₂₁₂	C₂³	1	2	yes	—
30	C₂×C₄×D₅₃	—	2	2	no	—
31	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
32	Q₁₆×C₅₃	—	2	4	no	—
33	Q₁₆×C₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#11719924	—	4	4	no	—
40	#11719921	—	2	2	no	—
41	#11719921	—	2	4	no	—
42	#11719921	—	2	4	no	—
43	C₄⋊Dic₅₃	—	2	4	no	—
44	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
45	C₄⋊Dic₅₃	C₂³	1	2	yes	—
46	#11719919	D₄	1	2	yes	—
47	C₄×Dic₅₃	—	2	4	no	—
48	C₄×Dic₅₃	C₂³	1	2	yes	—
49	C₂×#88322	C₂³	1	2	yes	—
50	D₄₂₄	—	2	2	no	—
51	C₂²×D₅₃	—	2	2	no	—
52	Q₈×C₅₃	—	2	4	no	—
53	Q₈×C₅₃	—	2	4	no	—
54	C₂×C₄×C₅₃	—	2	4	no	—
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₅₃	—	2	2	no	—
59	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
60	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
61	D₂₁₂	—	2	2	no	—
62	D₂₁₂	—	2	2	no	—
63	C₄×D₅₃	—	4	4	no	—
64	Dic₁₀₆	—	2	2	no	—
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	#41	1	2	yes	—
67	#88322	—	2	2	no	—
68	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₅₃	—	4	2	no	—
70	C₂×D₅₃	—	4	2	no	—
71	C₂×D₅₃	—	4	2	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	4	4	no	—
74	C₄×C₅₃	#135	1	2	yes	—
75	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
76	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
77	Dic₅₃	—	2	2	no	—
78	Dic₅₃	—	2	2	no	—
79	Dic₅₃	—	4	2	no	—
80	#3393	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	#550	1	1	yes	characteristic
82	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
83	C₂×C₅₃	#578	1	1	yes	characteristic
84	D₅₃	—	8	2	no	—
85	#556	—	53	2	no	—
86	#495	—	53	2	no	—
87	#494	D₅₃	1	1	yes	characteristic, complemented
88	#464	—	53	4	no	—
89	#461	—	53	2	no	—
90	C₈×Q₈	—	53	1	no	—
91	C₄×Q₁₆	—	53	2	no	—
92	C₄×D₈	—	53	1	no	—
93	C₅₃	#3393	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
95	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
96	C₂×SD₁₆	—	53	4	no	—
97	C₂×D₈	—	53	1	no	—
98	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
99	#126	—	106	4	no	—
100	#124	—	53	4	no	—
101	C₂²⋊Q₈	—	106	2	no	—
102	C₄×Q₈	—	53	1	no	—
103	C₄×Q₈	—	53	4	no	—
104	C₄×D₄	—	53	2	no	—
105	#117	—	106	2	no	—
106	#107	—	53	1	no	—
107	C₄⋊C₈	—	53	2	no	—
108	C₄⋊C₈	—	53	1	no	—
109	#103	—	53	4	no	—
110	#102	—	53	2	no	—
111	C₄×C₈	—	53	2	no	—
112	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
113	C₂×Q₈	—	53	1	no	—
114	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
115	C₂×D₄	—	53	2	no	—
116	C₂²×C₄	—	106	2	no	—
117	Q₁₆	—	2	4	no	—
118	Q₁₆	—	4	2	no	—
119	SD₁₆	—	212	4	no	—
120	D₈	—	106	2	no	—
121	C₂×C₈	—	53	2	no	—
122	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	53	1	no	—
127	C₄⋊C₄	—	53	2	no	—
128	C₄⋊C₄	—	53	2	no	—
129	C₄⋊C₄	—	106	4	no	—
130	C₂²⋊C₄	—	106	4	no	—
131	C₂²⋊C₄	—	106	4	no	—
132	C₂²⋊C₄	—	106	2	no	—
133	C₄²	—	53	2	no	—
134	C₄²	—	106	4	no	—
135	C₄²	C₂²×D₅₃	1	1	yes	characteristic
136	C₂³	—	106	2	no	—
137	Q₈	—	2	4	no	—
138	Q₈	—	2	4	no	—
139	Q₈	—	106	2	no	—
140	D₄	—	106	2	no	—
141	D₄	—	106	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	53	2	no	—
144	C₂×C₄	—	53	2	no	—
145	C₂×C₄	—	106	2	no	—
146	C₂×C₄	—	212	4	no	—
147	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
148	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	106	2	no	—
152	C₈	#11719947	1	2	yes	—
153	C₂²	—	212	2	no	—
154	C₂²	—	212	2	no	—
155	C₂²	—	212	2	no	—
156	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	4	no	—
159	C₄	—	106	2	no	—
160	C₄	—	106	2	no	—
161	C₄	—	212	2	no	—
162	C₄	#13133240	1	2	yes	—
163	C₄	C₂×#11719947	1	1	yes	characteristic
164	C₄	#13133262	1	1	yes	characteristic
165	C₂	—	424	2	no	—
166	C₂	#16811293	1	1	yes	characteristic
167	C₂	C₂×#13133240	1	1	yes	characteristic
168	C₂	#16811436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811347

C₂

yes

maximal, complemented

#16811008

C₂

yes

maximal

#16810847

C₂

yes

maximal, complemented

#16810825

C₂

yes

maximal, complemented

C₈×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

Q₁₆×Dic₅₃

C₂

yes

maximal, complemented

C₄×D₄₂₄

C₂

yes

maximal, characteristic

C₅₃×#494

C₂

yes

maximal, characteristic, complemented

C₂×#11719924

C₂²

yes

—

C₂×D₄₂₄

C₂²

yes

characteristic

#13133212

—

#13133206

C₂²

yes

—

#13133197

—

Q₈×Dic₅₃

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×D₂₁₂

C₂²

yes

—

#13133175

—

#13133151

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

#13133146

C₂²

yes

—

#13133141

C₂²

yes

—

#13133139

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

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×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

D₄₂₄

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×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

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

—

D₂₁₂

—

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

#135

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#556

—

#495

—

#494

D₅₃

yes

characteristic, complemented

#464

—

#461

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×D₈

—

C₅₃

#3393

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#126

—

106

—

100

#124

—

101

C₂²⋊Q₈

—

106

—

102

C₄×Q₈

—

103

C₄×Q₈

—

104

C₄×D₄

—

105

#117

—

106

—

106

#107

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

#103

—

110

#102

—

111

C₄×C₈

—

112

C₄×C₈

C₂×D₅₃

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

C₂²×D₅₃

yes

—

115

C₂×D₄

—

116

C₂²×C₄

—

106

—

117

Q₁₆

—

118

Q₁₆

—

119

SD₁₆

—

212

—

120

D₈

—

106

—

121

C₂×C₈

—

122

C₂×C₈

C₂²×D₅₃

yes

characteristic

123

C₂×C₈

C₂²×D₅₃

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

106

—

130

C₂²⋊C₄

—

106

—

131

C₂²⋊C₄

—

106

—

132

C₂²⋊C₄

—

106

—

133

C₄²

—

134

C₄²

—

106

—

135

C₄²

C₂²×D₅₃

yes

characteristic

136

C₂³

—

106

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

106

—

140

D₄

—

106

—

141

D₄

—

106

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

106

—

146

C₂×C₄

—

212

—

147

C₂×C₄

D₄×D₅₃

yes

characteristic

148

C₂×C₄

D₄×D₅₃

yes

characteristic

149

C₂×C₄

C₂³×D₅₃

yes

characteristic

150

C₈

—

151

C₈

—

106

—

152

C₈

#11719947

yes

—

153

C₂²

—

212

—

154

C₂²

—

212

—

155

C₂²

—

212

—

156

C₂²

C₂×D₄×D₅₃

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

106

—

160

C₄

—

106

—

161

C₄

—

212

—

162

C₄

#13133240

yes

—

163

C₄

C₂×#11719947

yes

characteristic

164

C₄

#13133262

yes

characteristic

165

C₂

—

424

—

166

C₂

#16811293

yes

characteristic

167

C₂

C₂×#13133240

yes

characteristic

168

C₂

#16811436

yes

characteristic

	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	#3393
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
1. #16811347	C₂	semidirect
3. #16810847	C₂	semidirect
4. #16810825	C₂	semidirect
5. C₈×Dic₁₀₆	C₂	semidirect
6. Q₁₆×Dic₅₃	C₂	semidirect
8. C₅₃×#494	C₂	semidirect
87. #494	D₅₃	semidirect
93. C₅₃	#3393	semidirect

H ≅ G/N

type

1. #16811347

C₂

semidirect

3. #16810847

C₂

semidirect

4. #16810825

C₂

semidirect

5. C₈×Dic₁₀₆

C₂

semidirect

6. Q₁₆×Dic₅₃

C₂

semidirect

8. C₅₃×#494

C₂

semidirect

87. #494

D₅₃

semidirect

93. C₅₃

#3393

semidirect

Subgroups — #20287705

Conjugacy classes of subgroups

Complements