Group #20275853 — subgroups

Subgroups — #20275853

Order 6784.

290 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13133140	C₂	1	2	yes	maximal
2	#16810529	C₂	1	1	yes	maximal, characteristic
3	#16810385	C₂	1	2	yes	maximal
4	#16810378	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133140	—	2	4	no	—
10	#13133124	—	2	1	no	—
11	#13133124	—	2	1	no	—
12	C₂²×Q₈×C₅₃	—	2	1	no	—
13	C₄⋊Q₈×C₅₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	4	no	—
18	C₂×Q₈×C₅₃	—	4	2	no	—
19	C₂×Q₈×C₅₃	—	4	2	no	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₃	—	2	1	no	—
25	C₂²×Dic₅₃	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	—	2	2	no	—
28	C₂×Dic₁₀₆	—	2	1	no	—
29	C₂×Dic₁₀₆	Q₈	1	2	yes	complemented
30	C₄⋊C₄×C₅₃	—	2	4	no	—
31	C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₄⋊C₄×C₅₃	D₄	1	4	yes	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	—	2	2	no	—
37	C₂×#88322	—	2	2	no	—
38	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₅₃	—	4	8	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#41	1	2	yes	—
52	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂×Dic₅₃	—	2	2	no	—
54	C₂×Dic₅₃	—	4	1	no	—
55	C₂×Dic₅₃	—	4	1	no	—
56	C₂×Dic₅₃	—	4	1	no	—
57	C₂×Dic₅₃	—	4	1	no	—
58	C₂×Dic₅₃	—	4	1	no	—
59	Dic₁₀₆	—	2	4	no	—
60	Dic₁₀₆	—	4	2	no	—
61	#88322	—	4	2	no	—
62	C₂²×C₅₃	#120	1	1	yes	characteristic
63	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₅₃	#124	1	1	yes	characteristic
65	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₅₃	#137	1	1	yes	characteristic
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	—	4	4	no	—
72	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₅₃	#124	1	2	yes	—
74	Dic₅₃	—	4	2	no	—
75	Dic₅₃	—	8	1	no	—
76	#2024	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	#393	1	1	yes	characteristic
78	C₂×C₅₃	#450	1	1	yes	characteristic
79	C₂×C₅₃	#451	1	1	yes	characteristic
80	C₂×C₅₃	#475	1	1	yes	characteristic
81	C₂×C₅₃	#477	1	1	yes	characteristic
82	C₂×C₅₃	#487	1	1	yes	characteristic
83	C₂×C₅₃	#489	1	1	yes	characteristic
84	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
85	C₂×#103	—	53	2	no	—
86	#391	—	53	1	no	—
87	#340	—	53	2	no	—
88	#336	—	53	1	no	—
89	C₅₃	#2024	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	—	53	1	no	—
92	C₂²×C₈	—	53	2	no	—
93	C₄⋊Q₈	—	2	4	no	—
94	C₂×C₄⋊C₄	—	2	1	no	—
95	C₂×C₄⋊C₄	—	53	1	no	—
96	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
97	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
98	#103	—	106	4	no	—
99	#95	—	106	1	no	—
100	#95	—	106	1	no	—
101	C₂×Q₈	—	2	4	no	—
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	4	2	no	—
104	C₂×Q₈	—	53	2	no	—
105	C₂×Q₈	—	106	1	no	—
106	C₂×Q₈	—	106	2	no	—
107	C₂×Q₈	—	106	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	106	1	no	—
110	C₂²×C₄	—	106	1	no	—
111	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
114	C₂×C₈	—	106	2	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂×C₈	—	106	2	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	106	2	no	—
120	C₄⋊C₄	C₂²⋊D₅₃	1	4	yes	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
123	Q₈	—	4	8	no	—
124	Q₈	—	106	4	no	—
125	Q₈	—	212	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	106	2	no	—
134	C₂×C₄	—	212	1	no	—
135	C₂×C₄	—	212	1	no	—
136	C₂×C₄	—	212	1	no	—
137	C₂×C₄	—	212	1	no	—
138	C₂×C₄	—	212	1	no	—
139	C₂×C₄	#11719923	1	2	yes	—
140	C₂×C₄	#11719925	1	2	yes	—
141	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
142	C₂×C₄	#11719949	1	2	yes	—
143	C₂×C₄	#11719949	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
145	C₈	—	212	2	no	—
146	C₂²	#13133186	1	1	yes	characteristic
147	C₂²	#13133193	1	1	yes	characteristic
148	C₂²	#13133206	1	1	yes	characteristic
149	C₂²	C₂×#11719923	1	1	yes	characteristic
150	C₂²	C₂×#11719925	1	1	yes	characteristic
151	C₂²	#13133243	1	1	yes	characteristic
152	C₂²	#13133248	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	4	no	—
156	C₄	—	424	1	no	—
157	C₄	—	212	2	no	—
158	C₄	#13133193	1	2	yes	—
159	C₄	#13133206	1	2	yes	—
160	C₂	#16810534	1	1	yes	characteristic
161	C₂	#16810774	1	1	yes	characteristic
162	C₂	#16810779	1	1	yes	characteristic
163	C₂	#16810905	1	1	yes	characteristic
164	C₂	#16810915	1	1	yes	characteristic
165	C₂	#16810975	1	1	yes	characteristic
166	C₂	#16810985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133140

C₂

yes

maximal

#16810529

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

#16810378

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

Q₈

yes

complemented

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

#124

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#124

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

D₅₃

yes

—

—

—

—

C₅₃

#2024

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

106

—

#95

—

106

—

100

#95

—

106

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

—

106

C₂×Q₈

—

106

—

107

C₂×Q₈

—

106

—

108

C₂²×C₄

—

109

C₂²×C₄

—

106

—

110

C₂²×C₄

—

106

—

111

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

112

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

113

C₂²×C₄

C₂²×D₅₃

yes

characteristic

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₂×C₈

—

106

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

106

—

120

C₄⋊C₄

C₂²⋊D₅₃

yes

—

121

C₄²

—

122

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

123

Q₈

—

124

Q₈

—

106

—

125

Q₈

—

212

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

212

—

135

C₂×C₄

—

212

—

136

C₂×C₄

—

212

—

137

C₂×C₄

—

212

—

138

C₂×C₄

—

212

—

139

C₂×C₄

#11719923

yes

—

140

C₂×C₄

#11719925

yes

—

141

C₂×C₄

Q₈×D₅₃

yes

characteristic

142

C₂×C₄

#11719949

yes

—

143

C₂×C₄

#11719949

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

145

C₈

—

212

—

146

C₂²

#13133186

yes

characteristic

147

C₂²

#13133193

yes

characteristic

148

C₂²

#13133206

yes

characteristic

149

C₂²

C₂×#11719923

yes

characteristic

150

C₂²

C₂×#11719925

yes

characteristic

151

C₂²

#13133243

yes

characteristic

152

C₂²

#13133248

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

424

—

157

C₄

—

212

—

158

C₄

#13133193

yes

—

159

C₄

#13133206

yes

—

160

C₂

#16810534

yes

characteristic

161

C₂

#16810774

yes

characteristic

162

C₂

#16810779

yes

characteristic

163

C₂

#16810905

yes

characteristic

164

C₂

#16810915

yes

characteristic

165

C₂

#16810975

yes

characteristic

166

C₂

#16810985

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#2024
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#2024

LCS

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
29. C₂×Dic₁₀₆	Q₈	semidirect
89. C₅₃	#2024	semidirect

H ≅ G/N

type

29. C₂×Dic₁₀₆

Q₈

semidirect

89. C₅₃

#2024

semidirect

Subgroups — #20275853

Conjugacy classes of subgroups

Complements