Group #20276043 — subgroups

Subgroups — #20276043

Order 6784.

282 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 97 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133150	C₂	1	2	yes	maximal
3	#16810385	C₂	1	2	yes	maximal
4	#16810378	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#389	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	—	2	1	no	—
7	C₂²×#88322	C₂²	1	2	yes	—
8	#13133217	—	2	4	no	—
9	C₂×C₄⋊Dic₅₃	—	2	1	no	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	#13133150	—	2	4	no	—
12	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#95×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	—	2	1	no	—
18	C₂²×C₄×C₅₃	—	2	1	no	—
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	2	no	—
23	C₂×Dic₁₀₆	—	2	4	no	—
24	C₂×Dic₁₀₆	—	4	2	no	—
25	C₂×Dic₁₀₆	—	4	2	no	—
26	C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₄⋊C₄×C₅₃	—	4	2	no	—
29	C₄⋊C₄×C₅₃	Q₈	1	2	yes	complemented
30	C₄²×C₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	4	no	—
32	C₄⋊Dic₅₃	—	2	2	no	—
33	C₄⋊Dic₅₃	D₄	1	4	yes	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	Q₈	1	4	yes	complemented
37	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	D₈	1	2	yes	—
49	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	4	no	—
55	C₂×Dic₅₃	—	4	2	no	—
56	C₂×Dic₅₃	—	4	2	no	—
57	Dic₁₀₆	—	4	8	no	—
58	#88322	—	2	4	no	—
59	C₂²×C₅₃	#120	1	1	yes	characteristic
60	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₅₃	#137	1	1	yes	characteristic
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	—	8	1	no	—
70	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
71	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
72	Dic₅₃	—	4	4	no	—
73	#2057	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	#395	1	1	yes	characteristic
75	C₂×C₅₃	#447	1	1	yes	characteristic
76	C₂×C₅₃	#451	1	1	yes	characteristic
77	C₂×C₅₃	#474	1	1	yes	characteristic
78	C₂×C₅₃	#477	1	1	yes	characteristic
79	C₂×C₅₃	#500	1	1	yes	characteristic
80	C₂×C₅₃	#501	1	1	yes	characteristic
81	C₂×C₄⋊Q₈	—	53	1	no	—
82	C₂×#107	—	53	2	no	—
83	#389	D₅₃	1	1	yes	characteristic
84	#340	—	53	2	no	—
85	#336	—	53	1	no	—
86	C₅₃	#2057	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	—	106	1	no	—
88	C₂²×C₈	—	53	2	no	—
89	C₄⋊Q₈	—	106	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	53	2	no	—
92	C₂×C₄⋊C₄	—	106	1	no	—
93	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
96	#107	—	106	4	no	—
97	#95	—	2	1	no	—
98	C₂×Q₈	—	106	4	no	—
99	C₂×Q₈	—	212	2	no	—
100	C₂×Q₈	—	212	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	106	2	no	—
104	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
107	C₂×C₈	—	53	4	no	—
108	C₂×C₈	—	106	2	no	—
109	C₂×C₈	—	106	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	53	4	no	—
114	C₄⋊C₄	—	106	4	no	—
115	C₄⋊C₄	—	106	2	no	—
116	C₄⋊C₄	Dic₁₀₆	1	2	yes	complemented
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
119	Q₈	—	212	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	106	4	no	—
131	C₂×C₄	—	212	2	no	—
132	C₂×C₄	—	212	2	no	—
133	C₂×C₄	#11719922	1	2	yes	—
134	C₂×C₄	#11719925	1	2	yes	—
135	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
136	C₂×C₄	#11719945	1	1	yes	characteristic
137	C₂×C₄	D₄×D₅₃	1	2	yes	—
138	C₂×C₄	Q₈×D₅₃	1	2	yes	—
139	C₈	—	106	4	no	—
140	C₂²	#13133185	1	1	yes	characteristic
141	C₂²	#13133196	1	1	yes	characteristic
142	C₂²	#13133215	1	1	yes	characteristic
143	C₂²	C₂×#11719922	1	1	yes	characteristic
144	C₂²	C₂×#11719925	1	1	yes	characteristic
145	C₂²	#13133242	1	1	yes	characteristic
146	C₂²	#13133247	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	8	1	no	—
151	C₄	—	212	4	no	—
152	C₄	#13133195	1	2	yes	—
153	C₄	#13133216	1	2	yes	—
154	C₂	#16810543	1	1	yes	characteristic
155	C₂	#16810756	1	1	yes	characteristic
156	C₂	#16810778	1	1	yes	characteristic
157	C₂	#16810903	1	1	yes	characteristic
158	C₂	#16810918	1	1	yes	characteristic
159	C₂	#16811026	1	1	yes	characteristic
160	C₂	#16811033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

C₂×#13133150

C₂

yes

maximal

#16810385

C₂

yes

maximal

#16810378

C₂

yes

maximal, characteristic

C₅₃×#389

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

C₂²×#88322

C₂²

yes

—

#13133217

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133150

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

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

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

Q₈

yes

complemented

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

Q₈

yes

complemented

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

C₄⋊Q₈

yes

—

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

—

C₂×#107

—

#389

D₅₃

yes

characteristic

#340

—

#336

—

C₅₃

#2057

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂²×C₈

—

C₄⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

#107

—

106

—

#95

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

100

C₂×Q₈

—

212

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

106

—

104

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

106

C₂²×C₄

C₂²×D₅₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

—

106

—

116

C₄⋊C₄

Dic₁₀₆

yes

complemented

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

119

Q₈

—

212

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

106

—

131

C₂×C₄

—

212

—

132

C₂×C₄

—

212

—

133

C₂×C₄

#11719922

yes

—

134

C₂×C₄

#11719925

yes

—

135

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

136

C₂×C₄

#11719945

yes

characteristic

137

C₂×C₄

D₄×D₅₃

yes

—

138

C₂×C₄

Q₈×D₅₃

yes

—

139

C₈

—

106

—

140

C₂²

#13133185

yes

characteristic

141

C₂²

#13133196

yes

characteristic

142

C₂²

#13133215

yes

characteristic

143

C₂²

C₂×#11719922

yes

characteristic

144

C₂²

C₂×#11719925

yes

characteristic

145

C₂²

#13133242

yes

characteristic

146

C₂²

#13133247

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

212

—

152

C₄

#13133195

yes

—

153

C₄

#13133216

yes

—

154

C₂

#16810543

yes

characteristic

155

C₂

#16810756

yes

characteristic

156

C₂

#16810778

yes

characteristic

157

C₂

#16810903

yes

characteristic

158

C₂

#16810918

yes

characteristic

159

C₂

#16811026

yes

characteristic

160

C₂

#16811033

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#389	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#389	D₅₃
hypocenter	C₅₃	#2057
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#389

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
29. C₄⋊C₄×C₅₃	Q₈	semidirect
36. C₂×#88322	Q₈	semidirect
86. C₅₃	#2057	semidirect
116. C₄⋊C₄	Dic₁₀₆	semidirect

H ≅ G/N

type

29. C₄⋊C₄×C₅₃

Q₈

semidirect

36. C₂×#88322

Q₈

semidirect

86. C₅₃

#2057

semidirect

116. C₄⋊C₄

Dic₁₀₆

semidirect

Subgroups — #20276043

Conjugacy classes of subgroups

Complements