Group #20272540 — subgroups

Subgroups — #20272540

Order 6784.

166 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16810529	C₂	1	1	yes	maximal, characteristic
2	#16810378	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#336	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₀₆	C₄	1	1	yes	characteristic
5	C₂²×#88322	—	2	1	no	—
6	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
7	#13133124	—	2	1	no	—
8	#13133124	—	2	1	no	—
9	C₂²×C₈×C₅₃	—	2	1	no	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂²×Dic₅₃	—	2	1	no	—
15	C₂²×Dic₅₃	—	2	1	no	—
16	C₂×Dic₁₀₆	—	2	1	no	—
17	C₂×Dic₁₀₆	—	4	1	no	—
18	C₂×Dic₁₀₆	—	4	1	no	—
19	C₂×Dic₁₀₆	C₈	1	2	yes	complemented
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	—	4	1	no	—
22	C₂×C₈×C₅₃	—	4	1	no	—
23	C₄²×C₅₃	—	4	1	no	—
24	C₄⋊Dic₅₃	—	4	1	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	—	4	1	no	—
27	C₂×#88322	—	4	1	no	—
28	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
38	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
41	C₂×Dic₅₃	—	2	2	no	—
42	C₂×Dic₅₃	—	4	1	no	—
43	C₂×Dic₅₃	—	4	1	no	—
44	C₂×Dic₅₃	—	4	1	no	—
45	C₂×Dic₅₃	—	4	1	no	—
46	C₂×Dic₅₃	—	4	1	no	—
47	Dic₁₀₆	—	4	2	no	—
48	Dic₁₀₆	—	4	2	no	—
49	C₈×C₅₃	—	4	2	no	—
50	#88322	—	4	2	no	—
51	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂²×C₅₃	#99	1	1	yes	characteristic
53	C₂²×C₅₃	#100	1	1	yes	characteristic
54	C₂²×C₅₃	#103	1	1	yes	characteristic
55	C₂²×C₅₃	#103	1	1	yes	characteristic
56	C₂²×C₅₃	#104	1	1	yes	characteristic
57	C₂²×C₅₃	#104	1	1	yes	characteristic
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	2	1	no	—
64	Dic₅₃	—	4	2	no	—
65	Dic₅₃	—	8	1	no	—
66	#1272	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	#323	1	1	yes	characteristic
68	C₂×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
69	C₂×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
70	C₂×C₅₃	#328	1	1	yes	characteristic
71	C₂×C₅₃	#328	1	1	yes	characteristic
72	C₂×C₅₃	#329	1	1	yes	characteristic
73	C₂×C₅₃	#333	1	1	yes	characteristic
74	#391	—	53	1	no	—
75	#336	—	53	1	no	—
76	#336	D₅₃	1	1	yes	characteristic
77	C₅₃	#1272	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₈	—	53	1	no	—
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	—	106	1	no	—
81	C₂×C₄⋊C₄	—	53	1	no	—
82	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
83	#95	—	106	1	no	—
84	#95	—	106	1	no	—
85	C₂×Q₈	—	53	2	no	—
86	C₂×Q₈	—	106	1	no	—
87	C₂×Q₈	—	212	1	no	—
88	C₂×Q₈	—	212	1	no	—
89	C₂²×C₄	—	106	1	no	—
90	C₂²×C₄	—	106	1	no	—
91	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
92	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	106	2	no	—
98	C₂×C₈	—	212	1	no	—
99	C₂×C₈	—	212	1	no	—
100	C₄⋊C₄	—	212	1	no	—
101	C₄²	—	4	1	no	—
102	C₂³	#11719921	1	1	yes	characteristic
103	Q₈	—	212	2	no	—
104	Q₈	—	212	2	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	106	2	no	—
112	C₂×C₄	—	212	1	no	—
113	C₂×C₄	—	212	1	no	—
114	C₂×C₄	—	212	1	no	—
115	C₂×C₄	—	212	1	no	—
116	C₂×C₄	—	212	1	no	—
117	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
118	C₂×C₄	#11719912	1	1	yes	characteristic
119	C₂×C₄	#11719913	1	1	yes	characteristic
120	C₂×C₄	Dic₂₁₂	1	1	yes	characteristic
121	C₂×C₄	#11719923	1	1	yes	characteristic
122	C₂×C₄	#11719925	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	212	2	no	—
125	C₂²	#13133129	1	1	yes	characteristic
126	C₂²	#13133131	1	1	yes	characteristic
127	C₂²	#13133133	1	1	yes	characteristic
128	C₂²	#13133140	1	1	yes	characteristic
129	C₂²	#13133142	1	1	yes	characteristic
130	C₂²	#13133143	1	1	yes	characteristic
131	C₂²	#13133145	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	424	1	no	—
139	C₄	—	212	2	no	—
140	C₂	#16810346	1	1	yes	characteristic
141	C₂	#16810353	1	1	yes	characteristic
142	C₂	#16810355	1	1	yes	characteristic
143	C₂	#16810359	1	1	yes	characteristic
144	C₂	#16810361	1	1	yes	characteristic
145	C₂	#16810362	1	1	yes	characteristic
146	C₂	#16810371	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810529

C₂

yes

maximal, characteristic

#16810378

C₂

yes

maximal, characteristic

C₅₃×#336

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₄

yes

characteristic

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

#13133124

—

#13133124

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₈

yes

complemented

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂²⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

Q₈⋊C₈

yes

characteristic

C₂×C₅₃

Q₈⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

C₅₃

#1272

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

—

106

—

#95

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

100

C₄⋊C₄

—

212

—

101

C₄²

—

102

C₂³

#11719921

yes

characteristic

103

Q₈

—

212

—

104

Q₈

—

212

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

212

—

113

C₂×C₄

—

212

—

114

C₂×C₄

—

212

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

C₈×D₅₃

yes

characteristic

118

C₂×C₄

#11719912

yes

characteristic

119

C₂×C₄

#11719913

yes

characteristic

120

C₂×C₄

Dic₂₁₂

yes

characteristic

121

C₂×C₄

#11719923

yes

characteristic

122

C₂×C₄

#11719925

yes

characteristic

123

C₈

—

124

C₈

—

212

—

125

C₂²

#13133129

yes

characteristic

126

C₂²

#13133131

yes

characteristic

127

C₂²

#13133133

yes

characteristic

128

C₂²

#13133140

yes

characteristic

129

C₂²

#13133142

yes

characteristic

130

C₂²

#13133143

yes

characteristic

131

C₂²

#13133145

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

424

—

139

C₄

—

212

—

140

C₂

#16810346

yes

characteristic

141

C₂

#16810353

yes

characteristic

142

C₂

#16810355

yes

characteristic

143

C₂

#16810359

yes

characteristic

144

C₂

#16810361

yes

characteristic

145

C₂

#16810362

yes

characteristic

146

C₂

#16810371

yes

characteristic

	subgroup	G/H
center	C₂³	#11719921
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂²⋊C₄
hypercenter	#336	D₅₃
hypocenter	C₅₃	#1272
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

#11719921

commutator

C₂×C₄×C₅₃

C₂×C₈

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₅₃×#336

C₂

socle

C₂³×C₅₃

C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
19. C₂×Dic₁₀₆	C₈	semidirect
77. C₅₃	#1272	semidirect

H ≅ G/N

type

19. C₂×Dic₁₀₆

C₈

semidirect

77. C₅₃

#1272

semidirect

Subgroups — #20272540

Conjugacy classes of subgroups

Complements