Group #20274730 — subgroups

Subgroups — #20274730

Order 6784.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133141	C₂	1	2	yes	maximal
3	#16810489	C₂	1	1	yes	maximal, characteristic
4	#16810385	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	C₄⋊#88322	—	2	2	no	—
9	#13133141	—	2	4	no	—
10	#13133124	—	2	1	no	—
11	#13133124	—	2	1	no	—
12	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₅₃	—	2	2	no	—
14	C₄×Q₈×C₅₃	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	—	2	1	no	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	—	2	1	no	—
20	C₂×Q₈×C₅₃	—	2	1	no	—
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×Q₈×C₅₃	D₄	1	2	yes	—
24	C₂×Q₈×C₅₃	Q₈	1	2	yes	—
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	C₂²×C₄×C₅₃	—	2	1	no	—
27	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	—	2	4	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₄²×C₅₃	—	2	4	no	—
35	C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₄²×C₅₃	D₄	1	2	yes	—
37	C₄⋊Dic₅₃	—	2	2	no	—
38	C₄⋊Dic₅₃	—	2	2	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂×#88322	—	2	2	no	—
42	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₅₃	—	2	2	no	—
44	Q₈×C₅₃	—	2	2	no	—
45	Q₈×C₅₃	—	2	2	no	—
46	Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	4	2	no	—
54	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂×Dic₅₃	—	4	2	no	—
63	C₂×Dic₅₃	—	4	2	no	—
64	C₂×Dic₅₃	—	4	2	no	—
65	#88322	—	4	2	no	—
66	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₅₃	#121	1	1	yes	characteristic
68	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₅₃	#135	1	2	yes	—
70	C₂²×C₅₃	#137	1	2	yes	—
71	C₄×C₅₃	—	2	4	no	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	—	4	2	no	—
75	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₅₃	#117	1	1	yes	characteristic
77	C₄×C₅₃	#121	1	1	yes	characteristic
78	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
79	Dic₅₃	—	8	2	no	—
80	#1805	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	#380	1	1	yes	characteristic
82	C₂×C₅₃	#416	1	1	yes	characteristic
83	C₂×C₅₃	#419	1	1	yes	characteristic
84	C₂×C₅₃	#464	1	2	yes	—
85	C₂×C₅₃	#479	1	2	yes	—
86	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
87	C₂×C₄⋊C₈	—	53	1	no	—
88	C₂×#103	—	53	2	no	—
89	#382	—	53	1	no	—
90	#340	—	53	2	no	—
91	C₅₃	#1805	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
93	C₂²×C₈	—	53	2	no	—
94	C₄×Q₈	—	2	2	no	—
95	C₄×Q₈	Dic₅₃	1	4	yes	complemented
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	—	53	2	no	—
98	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
99	C₂×C₄²	—	2	1	no	—
100	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
101	C₄⋊C₈	—	106	2	no	—
102	#103	—	106	4	no	—
103	#95	—	106	1	no	—
104	#95	—	106	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	Dic₁₀₆	1	2	yes	—
109	C₂×Q₈	D₂₁₂	1	2	yes	—
110	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	106	2	no	—
114	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	—	106	2	no	—
119	C₂×C₈	—	106	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	106	2	no	—
123	C₄⋊C₄	—	106	2	no	—
124	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
125	C₄²	—	2	4	no	—
126	C₄²	C₂×Dic₅₃	1	2	yes	—
127	C₄²	C₂²⋊D₅₃	1	2	yes	—
128	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
129	Q₈	—	2	2	no	—
130	Q₈	—	2	2	no	—
131	Q₈	—	2	2	no	—
132	Q₈	C₄⋊Dic₅₃	1	4	yes	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	212	2	no	—
141	C₂×C₄	—	212	2	no	—
142	C₂×C₄	—	212	2	no	—
143	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
144	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
145	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
146	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
147	C₂×C₄	#11719945	1	1	yes	characteristic
148	C₂×C₄	#11719945	1	1	yes	characteristic
149	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
151	C₈	—	212	2	no	—
152	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
153	C₂²	#13133191	1	1	yes	characteristic
154	C₂²	#13133196	1	1	yes	characteristic
155	C₂²	#13133237	1	2	yes	—
156	C₂²	#13133248	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	4	no	—
160	C₄	—	4	2	no	—
161	C₄	—	424	2	no	—
162	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
163	C₄	#13133176	1	1	yes	characteristic
164	C₄	#13133191	1	1	yes	characteristic
165	C₄	#13133196	1	1	yes	characteristic
166	C₂	#16810483	1	1	yes	characteristic
167	C₂	#16810619	1	1	yes	characteristic
168	C₂	#16810632	1	1	yes	characteristic
169	C₂	#16810845	1	2	yes	—
170	C₂	#16810930	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₂×#13133141

C₂

yes

maximal

#16810489

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#479

yes

—

C₂×C₄×Q₈

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#103

—

#382

—

#340

—

C₅₃

#1805

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₂×D₅₃

yes

characteristic

101

C₄⋊C₈

—

106

—

102

#103

—

106

—

103

#95

—

106

—

104

#95

—

106

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

Dic₁₀₆

yes

—

109

C₂×Q₈

D₂₁₂

yes

—

110

C₂×Q₈

C₂×Dic₅₃

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

106

—

114

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

115

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

116

C₂²×C₄

C₂²×D₅₃

yes

characteristic

117

C₂×C₈

—

106

—

118

C₂×C₈

—

106

—

119

C₂×C₈

—

106

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

106

—

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

C₂×Dic₅₃

yes

—

125

C₄²

—

126

C₄²

C₂×Dic₅₃

yes

—

127

C₄²

C₂²⋊D₅₃

yes

—

128

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

C₄⋊Dic₅₃

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

212

—

141

C₂×C₄

—

212

—

142

C₂×C₄

—

212

—

143

C₂×C₄

C₂²⋊Dic₅₃

yes

—

144

C₂×C₄

C₂²⋊Dic₅₃

yes

—

145

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

146

C₂×C₄

C₂×D₂₁₂

yes

characteristic

147

C₂×C₄

#11719945

yes

characteristic

148

C₂×C₄

#11719945

yes

characteristic

149

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

151

C₈

—

212

—

152

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

153

C₂²

#13133191

yes

characteristic

154

C₂²

#13133196

yes

characteristic

155

C₂²

#13133237

yes

—

156

C₂²

#13133248

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

424

—

162

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

163

C₄

#13133176

yes

characteristic

164

C₄

#13133191

yes

characteristic

165

C₄

#13133196

yes

characteristic

166

C₂

#16810483

yes

characteristic

167

C₂

#16810619

yes

characteristic

168

C₂

#16810632

yes

characteristic

169

C₂

#16810845

yes

—

170

C₂

#16810930

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#1805
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄×Q₈

D₅₃

hypocenter

C₅₃

#1805

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
14. C₄×Q₈×C₅₃	C₄	semidirect
91. C₅₃	#1805	semidirect
95. C₄×Q₈	Dic₅₃	semidirect

H ≅ G/N

type

14. C₄×Q₈×C₅₃

C₄

semidirect

91. C₅₃

#1805

semidirect

95. C₄×Q₈

Dic₅₃

semidirect

Subgroups — #20274730

Conjugacy classes of subgroups

Complements