Group #20274069 — subgroups

Subgroups — #20274069

Order 6784.

204 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16810999	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810819	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810815	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810688	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₂₄	—	2	1	no	—
9	C₂×#11719922	—	2	1	no	—
10	C₂×#11719922	—	2	1	no	—
11	C₂×#11719922	—	2	1	no	—
12	C₂×C₈×D₅₃	—	2	1	no	—
13	#13133214	C₂²	1	1	yes	characteristic
14	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
15	C₄⋊#88322	C₂²	1	1	yes	characteristic
16	#13133139	—	2	1	no	—
17	#13133138	—	2	1	no	—
18	#13133129	—	2	1	no	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	#127×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×D₂₁₂	—	2	1	no	—
29	C₂×D₂₁₂	D₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#11719922	—	4	2	no	—
35	#11719922	—	4	2	no	—
36	#11719922	—	4	2	no	—
37	#11719921	—	2	1	no	—
38	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	D₄	1	1	yes	characteristic
41	C₂×#88322	D₄	1	1	yes	characteristic
42	D₄₂₄	—	4	2	no	—
43	C₈×D₅₃	—	4	2	no	—
44	C₂³×C₅₃	—	4	1	no	—
45	C₂²×D₅₃	—	2	1	no	—
46	C₂²×D₅₃	—	4	1	no	—
47	D₄×C₅₃	—	4	1	no	—
48	D₄×C₅₃	—	4	1	no	—
49	D₄×C₅₃	—	4	1	no	—
50	D₄×C₅₃	—	4	1	no	—
51	D₄×C₅₃	—	8	1	no	—
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×Dic₅₃	—	2	1	no	—
56	D₂₁₂	—	2	1	no	—
57	D₂₁₂	—	4	1	no	—
58	D₂₁₂	—	4	1	no	—
59	D₂₁₂	—	4	1	no	—
60	D₂₁₂	—	4	1	no	—
61	D₂₁₂	—	8	1	no	—
62	D₂₁₂	D₈	1	2	yes	complemented
63	C₄×D₅₃	—	4	1	no	—
64	C₄×D₅₃	—	4	1	no	—
65	C₈×C₅₃	—	4	1	no	—
66	C₈×C₅₃	—	4	1	no	—
67	#88322	—	2	1	no	—
68	#88322	—	4	1	no	—
69	#88322	D₈	1	2	yes	complemented
70	C₂²×C₅₃	—	8	1	no	—
71	C₂²×C₅₃	—	8	1	no	—
72	C₂²×C₅₃	—	8	1	no	—
73	C₂²×C₅₃	#121	1	1	yes	characteristic
74	C₂×D₅₃	—	2	2	no	—
75	C₂×D₅₃	—	4	1	no	—
76	C₂×D₅₃	—	4	1	no	—
77	C₂×D₅₃	—	8	1	no	—
78	C₂×D₅₃	—	8	1	no	—
79	C₂×D₅₃	—	8	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
82	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
83	C₄×C₅₃	#135	1	1	yes	characteristic
84	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	Dic₅₃	—	4	1	no	—
86	#1667	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	—	16	1	no	—
88	C₂×C₅₃	#459	1	1	yes	characteristic
89	C₂×C₅₃	#466	1	1	yes	characteristic
90	C₂×C₅₃	#472	1	1	yes	characteristic
91	D₅₃	—	4	2	no	—
92	D₅₃	—	16	1	no	—
93	#493	—	53	1	no	—
94	#459	—	53	1	no	—
95	#459	—	53	1	no	—
96	C₈×D₄	—	53	1	no	—
97	#335	—	53	1	no	—
98	#331	—	53	1	no	—
99	#331	D₅₃	1	1	yes	characteristic, complemented
100	C₅₃	#1667	1	1	yes	characteristic, complemented, Sylow
101	C₂×D₈	—	106	1	no	—
102	C₂×D₈	—	106	1	no	—
103	C₂×D₈	—	106	1	no	—
104	C₂×D₈	—	106	1	no	—
105	C₂²×C₈	—	106	1	no	—
106	#127	—	53	1	no	—
107	#127	C₂×D₅₃	1	1	yes	characteristic
108	C₄×D₄	—	53	1	no	—
109	C₄⋊C₈	—	53	1	no	—
110	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
112	#102	—	106	1	no	—
113	#102	—	106	1	no	—
114	C₂²⋊C₈	—	106	1	no	—
115	C₄×C₈	—	53	1	no	—
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	—	2	1	no	—
119	C₂×D₄	—	53	1	no	—
120	C₂×D₄	—	106	1	no	—
121	C₂×D₄	—	106	1	no	—
122	C₂×D₄	—	106	1	no	—
123	C₂²×C₄	—	106	1	no	—
124	D₈	—	212	2	no	—
125	D₈	—	212	2	no	—
126	D₈	—	212	2	no	—
127	D₈	—	212	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	53	1	no	—
131	C₂×C₈	—	53	1	no	—
132	C₂×C₈	—	106	1	no	—
133	C₂×C₈	—	212	2	no	—
134	C₄⋊C₄	—	53	1	no	—
135	C₂²⋊C₄	—	106	1	no	—
136	C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	C₂³	—	4	1	no	—
138	C₂³	—	106	1	no	—
139	C₂³	—	212	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	4	1	no	—
142	D₄	—	4	1	no	—
143	D₄	—	4	1	no	—
144	D₄	—	8	1	no	—
145	D₄	—	53	2	no	—
146	D₄	—	106	1	no	—
147	D₄	—	424	1	no	—
148	D₄	—	212	1	no	—
149	D₄	—	212	1	no	—
150	D₄	—	212	1	no	—
151	D₄	—	212	1	no	—
152	C₂×C₄	—	106	1	no	—
153	C₂×C₄	—	212	1	no	—
154	C₂×C₄	—	212	1	no	—
155	C₂×C₄	#11719945	1	1	yes	characteristic
156	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
157	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
158	C₈	—	4	1	no	—
159	C₈	—	4	1	no	—
160	C₈	—	53	2	no	—
161	C₈	—	106	1	no	—
162	C₈	—	212	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	—	106	2	no	—
167	C₂²	—	424	1	no	—
168	C₂²	—	424	1	no	—
169	C₂²	—	424	1	no	—
170	C₂²	—	212	1	no	—
171	C₂²	—	212	1	no	—
172	C₂²	#13133190	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	212	1	no	—
175	C₄	D₈×D₅₃	1	1	yes	characteristic
176	C₄	D₈×D₅₃	1	1	yes	characteristic
177	C₄	#13133238	1	1	yes	characteristic
178	C₄	#13133244	1	1	yes	characteristic
179	C₂	—	16	1	no	—
180	C₂	—	848	1	no	—
181	C₂	—	212	2	no	—
182	C₂	#16810817	1	1	yes	characteristic
183	C₂	#16810858	1	1	yes	characteristic
184	C₂	#16810892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810999

C₂

yes

maximal, characteristic, complemented

#16810819

C₂

yes

maximal, characteristic, complemented

#16810815

C₂

yes

maximal, characteristic, complemented

#16810688

C₂

yes

maximal, characteristic, complemented

#16810375

C₂

yes

maximal, characteristic, complemented

#16810366

C₂

yes

maximal, characteristic, complemented

C₅₃×#331

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

—

C₂×#11719922

—

C₂×#11719922

—

C₂×#11719922

—

C₂×C₈×D₅₃

—

#13133214

C₂²

yes

characteristic

C₄×D₂₁₂

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

D₄₂₄

—

C₈×D₅₃

—

C₂³×C₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

D₈

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

D₈

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1667

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

#493

—

#459

—

#459

—

C₈×D₄

—

#335

—

#331

—

#331

D₅₃

yes

characteristic, complemented

100

C₅₃

#1667

yes

characteristic, complemented, Sylow

101

C₂×D₈

—

106

—

102

C₂×D₈

—

106

—

103

C₂×D₈

—

106

—

104

C₂×D₈

—

106

—

105

C₂²×C₈

—

106

—

106

#127

—

107

#127

C₂×D₅₃

yes

characteristic

108

C₄×D₄

—

109

C₄⋊C₈

—

110

C₄⋊C₈

C₂×D₅₃

yes

characteristic

111

C₄⋊C₈

C₂×D₅₃

yes

characteristic

112

#102

—

106

—

113

#102

—

106

—

114

C₂²⋊C₈

—

106

—

115

C₄×C₈

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

106

—

121

C₂×D₄

—

106

—

122

C₂×D₄

—

106

—

123

C₂²×C₄

—

106

—

124

D₈

—

212

—

125

D₈

—

212

—

126

D₈

—

212

—

127

D₈

—

212

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

106

—

133

C₂×C₈

—

212

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

106

—

136

C₄²

C₂²×D₅₃

yes

characteristic

137

C₂³

—

138

C₂³

—

106

—

139

C₂³

—

212

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

106

—

147

D₄

—

424

—

148

D₄

—

212

—

149

D₄

—

212

—

150

D₄

—

212

—

151

D₄

—

212

—

152

C₂×C₄

—

106

—

153

C₂×C₄

—

212

—

154

C₂×C₄

—

212

—

155

C₂×C₄

#11719945

yes

characteristic

156

C₂×C₄

D₄×D₅₃

yes

characteristic

157

C₂×C₄

D₄×D₅₃

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

106

—

162

C₈

—

212

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

106

—

167

C₂²

—

424

—

168

C₂²

—

424

—

169

C₂²

—

424

—

170

C₂²

—

212

—

171

C₂²

—

212

—

172

C₂²

#13133190

yes

characteristic

173

C₄

—

174

C₄

—

212

—

175

C₄

D₈×D₅₃

yes

characteristic

176

C₄

D₈×D₅₃

yes

characteristic

177

C₄

#13133238

yes

characteristic

178

C₄

#13133244

yes

characteristic

179

C₂

—

180

C₂

—

848

—

181

C₂

—

212

—

182

C₂

#16810817

yes

characteristic

183

C₂

#16810858

yes

characteristic

184

C₂

#16810892

yes

characteristic

	subgroup	G/H
center	C₂²	#13133190
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#331	C₂
socle	C₂²×C₅₃	#121
hypercenter	#331	D₅₃
hypocenter	C₅₃	#1667
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#13133190

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#331

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #16810999	C₂	semidirect
2. #16810819	C₂	semidirect
3. #16810815	C₂	semidirect
4. #16810688	C₂	semidirect
5. #16810375	C₂	semidirect
6. #16810366	C₂	semidirect
7. C₅₃×#331	C₂	semidirect
62. D₂₁₂	D₈	semidirect
69. #88322	D₈	semidirect
99. #331	D₅₃	semidirect
100. C₅₃	#1667	semidirect

H ≅ G/N

type

1. #16810999

C₂

semidirect

2. #16810819

C₂

semidirect

3. #16810815

C₂

semidirect

4. #16810688

C₂

semidirect

5. #16810375

C₂

semidirect

6. #16810366

C₂

semidirect

7. C₅₃×#331

C₂

semidirect

62. D₂₁₂

D₈

semidirect

69. #88322

D₈

semidirect

99. #331

D₅₃

semidirect

100. C₅₃

#1667

semidirect

Subgroups — #20274069

Conjugacy classes of subgroups

Complements