Group #20284366 — subgroups

Subgroups — #20284366

Order 6784.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×#13133223	C₂	1	1	yes	maximal, characteristic
2	#16811197	C₂	1	2	yes	maximal
3	#16810878	C₂	1	4	yes	maximal, complemented
4	#16810861	C₂	1	4	yes	maximal, complemented
5	#16810661	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810632	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719945	C₂²	1	1	yes	characteristic
9	#13133223	—	2	4	no	—
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	C₂×#11719912	C₂²	1	2	yes	—
12	C₂×C₈×D₅₃	C₂²	1	2	yes	—
13	#13133215	—	2	2	no	—
14	#13133206	—	2	2	no	—
15	#13133200	—	2	4	no	—
16	#13133193	C₂²	1	8	yes	—
17	#13133176	C₂²	1	2	yes	—
18	#13133151	C₂²	1	2	yes	—
19	#13133149	C₂²	1	2	yes	—
20	#13133141	C₂²	1	8	yes	—
21	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
22	C₂×Q₁₆×C₅₃	—	2	4	no	—
23	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₅₃	—	2	2	no	—
27	C₂×Q₈×C₅₃	—	2	4	no	—
28	C₂×Q₈×C₅₃	—	2	4	no	—
29	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₂²⋊D₅₃	D₄	1	2	yes	—
33	#11719945	—	2	2	no	—
34	#11719945	—	2	2	no	—
35	C₂×D₂₁₂	D₄	1	1	yes	characteristic
36	C₂×C₄×D₅₃	C₂³	1	2	yes	—
37	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
38	Q₁₆×C₅₃	—	2	4	no	—
39	Q₁₆×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	D₄	1	4	yes	—
41	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₅₃	—	2	2	no	—
44	#11719921	—	2	8	no	—
45	C₄⋊Dic₅₃	C₂³	1	4	yes	—
46	#11719919	—	2	8	no	—
47	C₄×Dic₅₃	—	2	2	no	—
48	#11719917	—	2	2	no	—
49	C₂×#88322	C₂³	1	2	yes	—
50	#11719912	—	2	4	no	—
51	C₈×D₅₃	—	2	4	no	—
52	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×D₅₃	C₂×D₄	1	2	yes	—
54	Q₈×C₅₃	—	2	4	no	—
55	Q₈×C₅₃	—	2	4	no	—
56	Q₈×C₅₃	—	4	4	no	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	—	4	4	no	—
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅₃	#41	1	2	yes	—
62	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
63	C₂²⋊D₅₃	—	2	4	no	—
64	C₂×Dic₅₃	—	2	4	no	—
65	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
66	D₂₁₂	—	2	2	no	—
67	C₄×D₅₃	—	2	2	no	—
68	C₄×D₅₃	—	2	2	no	—
69	Dic₁₀₆	—	2	2	no	—
70	C₈×C₅₃	C₂×D₄	1	2	yes	—
71	C₈×C₅₃	C₂×D₄	1	2	yes	—
72	#88322	—	2	2	no	—
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	#121	1	2	yes	—
75	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×D₅₃	—	2	2	no	—
77	C₂×D₅₃	—	2	4	no	—
78	C₄×C₅₃	—	4	4	no	—
79	C₄×C₅₃	#121	1	2	yes	—
80	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
81	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
82	Dic₅₃	—	2	2	no	—
83	Dic₅₃	—	4	4	no	—
84	#3059	—	53	1	no	maximal, Sylow
85	C₂×C₅₃	—	2	2	no	—
86	C₂×C₅₃	C₂×#121	1	1	yes	characteristic
87	C₂×C₅₃	#578	1	2	yes	—
88	D₅₃	—	4	2	no	—
89	C₂²×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
90	C₂×#131	—	53	1	no	—
91	#536	—	53	2	no	—
92	#470	—	53	4	no	—
93	#467	—	53	4	no	—
94	#427	—	53	1	no	—
95	#419	—	53	2	no	—
96	C₅₃	#3059	1	1	yes	characteristic, complemented, Sylow
97	C₂×#41	—	53	1	no	—
98	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
99	C₂×Q₁₆	—	2	4	no	—
100	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
101	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
102	#131	—	106	4	no	—
103	C₂×M₁₆	—	53	1	no	—
104	C₂×M₁₆	—	53	2	no	—
105	C₂²×C₈	—	53	2	no	—
106	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
107	C₄⋊Q₈	—	106	2	no	—
108	#124	—	106	2	no	—
109	#123	—	106	4	no	—
110	C₂²⋊Q₈	—	53	8	no	—
111	#117	—	53	2	no	—
112	#107	—	53	2	no	—
113	#106	—	53	2	no	—
114	#103	—	53	8	no	—
115	#41	—	106	2	no	—
116	#41	—	106	2	no	—
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	53	1	no	—
121	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
122	C₂×D₄	—	53	2	no	—
123	C₂×D₄	—	53	1	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	53	2	no	—
126	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
127	Q₁₆	—	2	4	no	—
128	Q₁₆	—	2	4	no	—
129	M₁₆	—	106	2	no	—
130	M₁₆	—	106	4	no	—
131	C₂×C₈	—	53	2	no	—
132	C₂×C₈	—	106	4	no	—
133	C₂×C₈	C₂²⋊D₅₃	1	4	yes	—
134	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
136	C₄⋊C₄	—	53	4	no	—
137	C₄⋊C₄	—	106	8	no	—
138	C₂²⋊C₄	—	106	8	no	—
139	C₂²⋊C₄	—	106	2	no	—
140	C₄²	—	106	2	no	—
141	C₂³	—	53	2	no	—
142	C₂³	D₄×D₅₃	1	1	yes	characteristic
143	Q₈	—	2	4	no	—
144	Q₈	—	2	4	no	—
145	Q₈	—	4	4	no	—
146	Q₈	—	106	2	no	—
147	D₄	—	106	4	no	—
148	D₄	—	106	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	—	53	2	no	—
152	C₂×C₄	—	106	2	no	—
153	C₂×C₄	—	106	2	no	—
154	C₂×C₄	—	106	4	no	—
155	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
156	C₂×C₄	#11719947	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
158	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
159	C₈	—	106	2	no	—
160	C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
161	C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	106	2	no	—
164	C₂²	—	106	4	no	—
165	C₂²	#13133188	1	2	yes	—
166	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
167	C₄	—	4	4	no	—
168	C₄	—	106	2	no	—
169	C₄	—	212	4	no	—
170	C₄	#13133188	1	2	yes	—
171	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
172	C₄	C₂×#11719947	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	212	2	no	—
175	C₂	C₂×#13133188	1	1	yes	characteristic
176	C₂	#16811436	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133223

C₂

yes

maximal, characteristic

#16811197

C₂

yes

maximal

#16810878

C₂

yes

maximal, complemented

#16810861

C₂

yes

maximal, complemented

#16810661

C₂

yes

maximal, characteristic, complemented

#16810632

C₂

yes

maximal, complemented

C₂²×Q₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

characteristic

#13133223

—

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719912

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

—

—

—

C₂²

yes

—

#13133176

C₂²

yes

—

#13133151

C₂²

yes

—

#13133149

C₂²

yes

—

#13133141

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

D₄

yes

—

#11719945

—

#11719945

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

—

C₂×Dic₁₀₆

D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719921

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

C₂³

yes

—

#11719912

—

C₈×D₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

C₂×D₄

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₂×D₄

yes

—

C₈×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#121

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#3059

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#121

yes

characteristic

C₂×C₅₃

#578

yes

—

D₅₃

—

C₂²×Q₁₆

D₅₃

yes

characteristic, complemented

—

—

—

—

—

—

C₅₃

#3059

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

C₂×D₅₃

yes

—

C₂×Q₁₆

—

100

C₂×Q₁₆

C₂×D₅₃

yes

—

101

C₂×Q₁₆

C₂×D₅₃

yes

—

102

#131

—

106

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×D₅₃

yes

characteristic

107

C₄⋊Q₈

—

106

—

108

#124

—

106

—

109

#123

—

106

—

110

C₂²⋊Q₈

—

111

#117

—

112

#107

—

113

#106

—

114

#103

—

115

#41

—

106

—

116

#41

—

106

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

C₂²×D₅₃

yes

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²×D₅₃

yes

characteristic

127

Q₁₆

—

128

Q₁₆

—

129

M₁₆

—

106

—

130

M₁₆

—

106

—

131

C₂×C₈

—

132

C₂×C₈

—

106

—

133

C₂×C₈

C₂²⋊D₅₃

yes

—

134

C₂×C₈

C₂²×D₅₃

yes

characteristic

135

C₂×C₈

C₂²×D₅₃

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

—

106

—

138

C₂²⋊C₄

—

106

—

139

C₂²⋊C₄

—

106

—

140

C₄²

—

106

—

141

C₂³

—

142

C₂³

D₄×D₅₃

yes

characteristic

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

106

—

147

D₄

—

106

—

148

D₄

—

106

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

106

—

153

C₂×C₄

—

106

—

154

C₂×C₄

—

106

—

155

C₂×C₄

D₄×D₅₃

yes

characteristic

156

C₂×C₄

#11719947

yes

—

157

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

158

C₂×C₄

C₂³×D₅₃

yes

characteristic

159

C₈

—

106

—

160

C₈

C₂×C₂²⋊D₅₃

yes

—

161

C₈

C₂×C₂²⋊D₅₃

yes

—

162

C₂²

—

163

C₂²

—

106

—

164

C₂²

—

106

—

165

C₂²

#13133188

yes

—

166

C₂²

C₂×D₄×D₅₃

yes

characteristic

167

C₄

—

168

C₄

—

106

—

169

C₄

—

212

—

170

C₄

#13133188

yes

—

171

C₄

C₂²×C₂²⋊D₅₃

yes

characteristic

172

C₄

C₂×#11719947

yes

characteristic

173

C₂

—

174

C₂

—

212

—

175

C₂

C₂×#13133188

yes

characteristic

176

C₂

#16811436

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂²×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₂²×Q₁₆	D₅₃
hypocenter	C₅₃	#3059
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂²×Q₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₂²×Q₁₆

D₅₃

hypocenter

C₅₃

#3059

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
3. #16810878	C₂	semidirect
4. #16810861	C₂	semidirect
5. #16810661	C₂	semidirect
6. #16810632	C₂	semidirect
7. C₂²×Q₁₆×C₅₃	C₂	semidirect
89. C₂²×Q₁₆	D₅₃	semidirect
96. C₅₃	#3059	semidirect

H ≅ G/N

type

3. #16810878

C₂

semidirect

4. #16810861

C₂

semidirect

5. #16810661

C₂

semidirect

6. #16810632

C₂

semidirect

7. C₂²×Q₁₆×C₅₃

C₂

semidirect

89. C₂²×Q₁₆

D₅₃

semidirect

96. C₅₃

#3059

semidirect

Subgroups — #20284366

Conjugacy classes of subgroups

Complements