Group #20273810 — subgroups

Subgroups — #20273810

Order 6784.

210 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16811352	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810969	C₂	1	1	yes	maximal, characteristic
3	#16810820	C₂	1	1	yes	maximal, characteristic
4	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810353	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#460	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719923	—	2	1	no	—
9	#13133217	—	2	1	no	—
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133208	—	2	1	no	—
12	#13133208	—	2	2	no	—
13	Q₈×Dic₅₃	—	2	1	no	—
14	C₄×Dic₁₀₆	—	2	1	no	—
15	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	C₄⋊#88322	C₂²	1	1	yes	characteristic
17	#13133138	—	2	1	no	—
18	#13133138	—	2	2	no	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₅₃	—	2	1	no	—
21	#127×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#102×C₅₃	—	2	1	no	—
25	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₄×C₅₃	—	2	1	no	—
28	C₂×D₄×C₅₃	—	2	1	no	—
29	C₂×Dic₁₀₆	—	2	1	no	—
30	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
31	SD₁₆×C₅₃	—	4	2	no	—
32	C₂×C₈×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₄²×C₅₃	—	2	1	no	—
36	C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#11719923	—	4	2	no	—
38	C₄⋊Dic₅₃	—	2	2	no	—
39	C₄⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
42	C₄⋊Dic₅₃	D₄	1	2	yes	—
43	#11719919	—	2	2	no	—
44	#11719919	—	2	1	no	—
45	C₄×Dic₅₃	—	2	2	no	—
46	C₄×Dic₅₃	—	2	1	no	—
47	C₂×#88322	—	2	1	no	—
48	C₂×#88322	—	2	1	no	—
49	C₂³×C₅₃	—	4	1	no	—
50	Q₈×C₅₃	—	2	1	no	—
51	Q₈×C₅₃	SD₁₆	1	2	yes	complemented
52	D₄×C₅₃	—	4	2	no	—
53	D₄×C₅₃	—	4	1	no	—
54	D₄×C₅₃	—	4	1	no	—
55	D₄×C₅₃	—	4	1	no	—
56	D₄×C₅₃	—	4	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₅₃	—	2	2	no	—
63	C₂×Dic₅₃	—	2	1	no	—
64	C₂×Dic₅₃	—	2	1	no	—
65	Dic₁₀₆	—	2	1	no	—
66	Dic₁₀₆	—	2	1	no	—
67	Dic₁₀₆	—	4	1	no	—
68	Dic₁₀₆	—	4	1	no	—
69	C₈×C₅₃	—	4	1	no	—
70	#88322	—	4	1	no	—
71	#88322	—	4	1	no	—
72	C₂²×C₅₃	—	8	1	no	—
73	C₂²×C₅₃	—	8	1	no	—
74	C₂²×C₅₃	—	8	1	no	—
75	C₂²×C₅₃	#120	1	1	yes	characteristic
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	—	4	1	no	—
79	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₄×C₅₃	#135	1	1	yes	characteristic
81	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₄×C₅₃	#137	1	1	yes	characteristic
83	Dic₅₃	—	4	1	no	—
84	Dic₅₃	—	4	2	no	—
85	Dic₅₃	—	4	1	no	—
86	#1626	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	—	16	1	no	—
88	C₂×C₅₃	#448	1	1	yes	characteristic
89	C₂×C₅₃	#449	1	1	yes	characteristic
90	C₂×C₅₃	#454	1	1	yes	characteristic
91	#557	—	53	1	no	—
92	#486	—	53	1	no	—
93	#460	—	53	1	no	—
94	#460	D₅₃	1	1	yes	characteristic
95	#332	—	53	1	no	—
96	Q₈⋊C₈	—	53	1	no	—
97	Q₈⋊C₈	—	53	1	no	—
98	C₅₃	#1626	1	1	yes	characteristic, complemented, Sylow
99	C₂×SD₁₆	—	2	1	no	—
100	C₂×SD₁₆	—	106	1	no	—
101	C₄⋊Q₈	—	53	1	no	—
102	C₄⋊Q₈	—	106	1	no	—
103	#127	C₂×D₅₃	1	1	yes	characteristic
104	#125	—	106	2	no	—
105	#125	—	106	1	no	—
106	C₄×Q₈	—	53	1	no	—
107	C₄×Q₈	—	106	1	no	—
108	C₄×Q₈	—	106	1	no	—
109	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
110	C₄⋊C₈	—	53	1	no	—
111	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
112	#102	—	2	1	no	—
113	#102	—	106	1	no	—
114	#102	—	106	2	no	—
115	C₄×C₈	—	53	1	no	—
116	C₂×Q₈	—	53	1	no	—
117	C₂×Q₈	—	106	1	no	—
118	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
119	C₂×D₄	—	2	1	no	—
120	C₂×D₄	—	2	1	no	—
121	C₂×D₄	—	2	1	no	—
122	SD₁₆	—	4	2	no	—
123	SD₁₆	—	212	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	106	1	no	—
126	C₂×C₈	—	106	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	53	1	no	—
129	C₄⋊C₄	—	53	2	no	—
130	C₄⋊C₄	—	106	2	no	—
131	C₄⋊C₄	—	106	2	no	—
132	C₄⋊C₄	—	106	2	no	—
133	C₄⋊C₄	—	106	1	no	—
134	C₄⋊C₄	—	106	1	no	—
135	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₄²	—	2	1	no	—
137	C₄²	—	106	2	no	—
138	C₄²	—	106	1	no	—
139	C₄²	C₂²×D₅₃	1	1	yes	characteristic
140	C₂³	—	4	1	no	—
141	Q₈	—	2	1	no	—
142	Q₈	—	106	1	no	—
143	Q₈	—	106	1	no	—
144	Q₈	—	212	1	no	—
145	Q₈	—	212	1	no	—
146	Q₈	#11719923	1	2	yes	complemented
147	D₄	—	4	2	no	—
148	D₄	—	4	1	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	106	1	no	—
155	C₂×C₄	—	106	2	no	—
156	C₂×C₄	—	106	1	no	—
157	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
158	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
159	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
160	C₈	—	4	1	no	—
161	C₈	—	212	1	no	—
162	C₈	—	212	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	#13133185	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	212	1	no	—
171	C₄	—	212	2	no	—
172	C₄	—	212	1	no	—
173	C₄	C₂×#11719923	1	1	yes	characteristic
174	C₄	#13133238	1	1	yes	characteristic
175	C₄	#13133244	1	1	yes	characteristic
176	C₄	#13133247	1	1	yes	characteristic
177	C₂	—	16	1	no	—
178	C₂	#16810761	1	1	yes	characteristic
179	C₂	#16810767	1	1	yes	characteristic
180	C₂	#16810794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811352

C₂

yes

maximal, characteristic, complemented

#16810969

C₂

yes

maximal, characteristic

#16810820

C₂

yes

maximal, characteristic

#16810368

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, characteristic, complemented

#16810353

C₂

yes

maximal, characteristic, complemented

C₅₃×#460

C₂

yes

maximal, characteristic

C₂×#11719923

—

#13133217

—

#13133217

C₂²

yes

characteristic

#13133208

—

#13133208

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×SD₁₆×C₅₃

—

#127×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

SD₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

SD₁₆

yes

complemented

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1626

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

D₅₃

yes

characteristic

#332

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₅₃

#1626

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

100

C₂×SD₁₆

—

106

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

106

—

103

#127

C₂×D₅₃

yes

characteristic

104

#125

—

106

—

105

#125

—

106

—

106

C₄×Q₈

—

107

C₄×Q₈

—

106

—

108

C₄×Q₈

—

106

—

109

C₄×Q₈

C₂×D₅₃

yes

characteristic

110

C₄⋊C₈

—

111

C₄⋊C₈

C₂×D₅₃

yes

characteristic

112

#102

—

113

#102

—

106

—

114

#102

—

106

—

115

C₄×C₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

106

—

118

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

SD₁₆

—

123

SD₁₆

—

212

—

124

C₂×C₈

—

125

C₂×C₈

—

106

—

126

C₂×C₈

—

106

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

106

—

131

C₄⋊C₄

—

106

—

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

—

106

—

135

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

136

C₄²

—

137

C₄²

—

106

—

138

C₄²

—

106

—

139

C₄²

C₂²×D₅₃

yes

characteristic

140

C₂³

—

141

Q₈

—

142

Q₈

—

106

—

143

Q₈

—

106

—

144

Q₈

—

212

—

145

Q₈

—

212

—

146

Q₈

#11719923

yes

complemented

147

D₄

—

148

D₄

—

149

D₄

—

150

D₄

—

151

D₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

106

—

155

C₂×C₄

—

106

—

156

C₂×C₄

—

106

—

157

C₂×C₄

D₄×D₅₃

yes

characteristic

158

C₂×C₄

D₄×D₅₃

yes

characteristic

159

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

160

C₈

—

161

C₈

—

212

—

162

C₈

—

212

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

#13133185

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

212

—

171

C₄

—

212

—

172

C₄

—

212

—

173

C₄

C₂×#11719923

yes

characteristic

174

C₄

#13133238

yes

characteristic

175

C₄

#13133244

yes

characteristic

176

C₄

#13133247

yes

characteristic

177

C₂

—

178

C₂

#16810761

yes

characteristic

179

C₂

#16810767

yes

characteristic

180

C₂

#16810794

yes

characteristic

	subgroup	G/H
center	C₂²	#13133185
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#460	C₂
socle	C₂²×C₅₃	#120
hypercenter	#460	D₅₃
hypocenter	C₅₃	#1626
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

subgroup

G/H

center

C₂²

#13133185

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#460

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #460

N	H ≅ G/N	type
1. #16811352	C₂	semidirect
4. #16810368	C₂	semidirect
5. Q₈⋊#88322	C₂	semidirect
6. #16810353	C₂	semidirect
51. Q₈×C₅₃	SD₁₆	semidirect
98. C₅₃	#1626	semidirect
146. Q₈	#11719923	semidirect

H ≅ G/N

type

1. #16811352

C₂

semidirect

4. #16810368

C₂

semidirect

5. Q₈⋊#88322

C₂

semidirect

6. #16810353

C₂

semidirect

51. Q₈×C₅₃

SD₁₆

semidirect

98. C₅₃

#1626

semidirect

146. Q₈

#11719923

semidirect

Subgroups — #20273810

Conjugacy classes of subgroups

Complements