Group #20273971 — subgroups

Subgroups — #20273971

Order 6784.

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

normal subgroups only

This group has 200 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	#16811026	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810908	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810828	C₂	1	1	yes	maximal, characteristic
4	#16810369	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#556	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719923	—	2	1	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133150	—	2	2	no	—
11	#13133148	—	2	1	no	—
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	#13133141	—	2	1	no	—
15	#13133140	—	2	1	no	—
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	#126×C₅₃	—	2	2	no	—
19	#124×C₅₃	—	2	1	no	—
20	C₂²⋊Q₈×C₅₃	—	2	1	no	—
21	C₄×Q₈×C₅₃	—	2	1	no	—
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
24	#117×C₅₃	—	2	1	no	—
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	—	2	1	no	—
29	C₂×Dic₁₀₆	—	2	1	no	—
30	C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
36	C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₂²⋊C₄×C₅₃	—	2	2	no	—
38	C₂²⋊C₄×C₅₃	—	2	1	no	—
39	C₄²×C₅₃	—	2	2	no	—
40	C₄²×C₅₃	—	2	1	no	—
41	C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#11719923	—	4	2	no	—
43	C₄⋊Dic₅₃	—	2	1	no	—
44	C₄⋊Dic₅₃	—	2	1	no	—
45	C₂×#88322	—	2	1	no	—
46	C₂×#88322	—	2	1	no	—
47	C₂×#88322	—	2	1	no	—
48	C₂³×C₅₃	—	2	1	no	—
49	Q₈×C₅₃	—	2	1	no	—
50	Q₈×C₅₃	—	2	1	no	—
51	Q₈×C₅₃	—	4	1	no	—
52	Q₈×C₅₃	—	4	1	no	—
53	D₄×C₅₃	—	2	1	no	—
54	D₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	—	4	1	no	—
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₅₃	—	4	1	no	—
65	Dic₁₀₆	—	4	1	no	—
66	Dic₁₀₆	—	4	1	no	—
67	#88322	—	2	2	no	—
68	#88322	—	4	1	no	—
69	#88322	—	4	1	no	—
70	C₂²×C₅₃	—	4	1	no	—
71	C₂²×C₅₃	—	4	1	no	—
72	C₂²×C₅₃	—	4	1	no	—
73	C₂²×C₅₃	#120	1	1	yes	characteristic
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	4	2	no	—
76	C₄×C₅₃	—	4	1	no	—
77	C₄×C₅₃	—	4	1	no	—
78	C₄×C₅₃	—	4	1	no	—
79	C₄×C₅₃	#135	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	—	8	1	no	—
84	#1650	—	53	1	no	maximal, Sylow
85	C₂×C₅₃	—	8	1	no	—
86	C₂×C₅₃	#449	1	1	yes	characteristic
87	C₂×C₅₃	#452	1	1	yes	characteristic
88	C₂×C₅₃	#456	1	1	yes	characteristic
89	#556	D₅₃	1	1	yes	characteristic
90	#500	—	53	1	no	—
91	#475	—	53	1	no	—
92	#461	—	53	1	no	—
93	#332	—	53	1	no	—
94	Q₈⋊C₈	—	53	1	no	—
95	D₄⋊C₈	—	53	1	no	—
96	C₅₃	#1650	1	1	yes	characteristic, complemented, Sylow
97	C₂×SD₁₆	—	106	1	no	—
98	C₄⋊Q₈	—	53	1	no	—
99	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
100	#126	—	2	2	no	—
101	#124	—	2	1	no	—
102	C₂²⋊Q₈	—	2	1	no	—
103	C₄×Q₈	—	2	1	no	—
104	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
105	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
106	#117	—	2	1	no	—
107	#107	—	106	2	no	—
108	#106	—	106	1	no	—
109	C₄⋊C₈	—	53	1	no	—
110	C₄⋊C₈	—	53	1	no	—
111	#103	—	106	1	no	—
112	#103	—	106	1	no	—
113	C₄×C₈	—	53	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	106	1	no	—
116	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
118	C₂²×C₄	—	2	1	no	—
119	SD₁₆	—	212	2	no	—
120	C₂×C₈	—	106	1	no	—
121	C₂×C₈	—	106	1	no	—
122	C₂×C₈	—	106	1	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	106	1	no	—
127	C₄⋊C₄	—	106	1	no	—
128	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
129	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
130	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	Q₈	—	2	1	no	—
139	Q₈	—	2	1	no	—
140	Q₈	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	212	1	no	—
143	Q₈	—	212	1	no	—
144	D₄	—	2	1	no	—
145	D₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	212	1	no	—
153	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
156	C₈	—	106	2	no	—
157	C₈	—	212	1	no	—
158	C₈	—	212	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	#13133186	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	4	1	no	—
168	C₄	—	424	1	no	—
169	C₄	#13133237	1	1	yes	characteristic
170	C₄	#13133237	1	1	yes	characteristic
171	C₄	#13133242	1	1	yes	characteristic
172	C₄	#13133247	1	1	yes	characteristic
173	C₂	—	8	1	no	—
174	C₂	#16810768	1	1	yes	characteristic
175	C₂	#16810785	1	1	yes	characteristic
176	C₂	#16810806	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811026

C₂

yes

maximal, characteristic, complemented

#16810908

C₂

yes

maximal, characteristic, complemented

#16810828

C₂

yes

maximal, characteristic

#16810369

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, characteristic, complemented

D₄⋊#88322

C₂

yes

maximal, characteristic

C₅₃×#556

C₂

yes

maximal, characteristic

C₂×#11719923

—

#13133217

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#126×C₅₃

—

#124×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

#117×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×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

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1650

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₅₃

#1650

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

100

#126

—

101

#124

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

C₄×Q₈

C₂×D₅₃

yes

characteristic

105

C₄×D₄

C₂×D₅₃

yes

characteristic

106

#117

—

107

#107

—

106

—

108

#106

—

106

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

—

106

—

112

#103

—

106

—

113

C₄×C₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

106

—

116

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

117

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

118

C₂²×C₄

—

119

SD₁₆

—

212

—

120

C₂×C₈

—

106

—

121

C₂×C₈

—

106

—

122

C₂×C₈

—

106

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

—

106

—

128

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

129

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

130

C₄⋊C₄

C₂²⋊D₅₃

yes

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₅₃

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

212

—

143

Q₈

—

212

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

212

—

153

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

154

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

155

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

156

C₈

—

106

—

157

C₈

—

212

—

158

C₈

—

212

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

#13133186

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

424

—

169

C₄

#13133237

yes

characteristic

170

C₄

#13133237

yes

characteristic

171

C₄

#13133242

yes

characteristic

172

C₄

#13133247

yes

characteristic

173

C₂

—

174

C₂

#16810768

yes

characteristic

175

C₂

#16810785

yes

characteristic

176

C₂

#16810806

yes

characteristic

	subgroup	G/H
center	C₂²	#13133186
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#556	C₂
socle	C₂²×C₅₃	#120
hypercenter	#556	D₅₃
hypocenter	C₅₃	#1650
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #556

subgroup

G/H

center

C₂²

#13133186

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#556

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #556

N	H ≅ G/N	type
1. #16811026	C₂	semidirect
2. #16810908	C₂	semidirect
4. #16810369	C₂	semidirect
5. Q₈⋊#88322	C₂	semidirect
96. C₅₃	#1650	semidirect

H ≅ G/N

type

1. #16811026

C₂

semidirect

2. #16810908

C₂

semidirect

4. #16810369

C₂

semidirect

5. Q₈⋊#88322

C₂

semidirect

96. C₅₃

#1650

semidirect

Subgroups — #20273971

Conjugacy classes of subgroups

Complements