Group #20272840 — subgroups

Subgroups — #20272840

Order 6784.

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

normal subgroups only

This group has 232 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	C₂×#13133132	C₂	1	2	yes	maximal
2	C₅₃×#390	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
4	#13133132	—	2	2	no	—
5	#13133132	C₄	1	4	yes	complemented
6	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
9	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic, complemented
10	C₂⁴×C₅₃	D₄	1	2	yes	—
11	C₂×D₄×C₅₃	—	4	1	no	—
12	C₂×D₄×C₅₃	—	4	1	no	—
13	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
15	C₂×D₄×C₅₃	Q₈	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	—	2	1	no	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₅₃	—	4	2	no	—
20	C₄⋊C₄×C₅₃	—	4	1	no	—
21	C₂²⋊C₄×C₅₃	—	4	4	no	—
22	C₄²×C₅₃	—	4	1	no	—
23	C₂²⋊Dic₅₃	—	2	4	no	—
24	C₂²⋊Dic₅₃	—	4	2	no	—
25	C₂³×C₅₃	—	2	2	no	—
26	C₂³×C₅₃	—	2	2	no	—
27	C₂³×C₅₃	—	4	2	no	—
28	C₂³×C₅₃	—	4	2	no	—
29	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
30	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
32	D₄×C₅₃	—	4	2	no	—
33	D₄×C₅₃	—	4	1	no	—
34	D₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₅₃	—	4	4	no	—
45	C₂×Dic₅₃	—	8	2	no	—
46	C₂×Dic₅₃	—	8	2	no	—
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	—	2	2	no	—
50	C₂²×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	—	4	2	no	—
52	C₂²×C₅₃	—	4	2	no	—
53	C₂²×C₅₃	—	4	2	no	—
54	C₂²×C₅₃	—	4	2	no	—
55	C₂²×C₅₃	—	8	2	no	—
56	C₂²×C₅₃	#95	1	1	yes	characteristic
57	C₂²×C₅₃	#99	1	1	yes	characteristic
58	C₂²×C₅₃	#99	1	1	yes	characteristic
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	4	2	no	—
62	C₄×C₅₃	—	8	1	no	—
63	Dic₅₃	—	8	4	no	—
64	#1393	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	—	4	2	no	—
68	C₂×C₅₃	—	4	2	no	—
69	C₂×C₅₃	#342	1	1	yes	characteristic
70	C₂×C₅₃	#353	1	1	yes	characteristic
71	C₂×C₅₃	#354	1	1	yes	characteristic
72	C₂×#99	—	53	2	no	—
73	#390	D₅₃	1	1	yes	characteristic
74	C₅₃	#1393	1	1	yes	characteristic, complemented, Sylow
75	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
76	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic, complemented
77	C₂×C₂²⋊C₄	—	2	2	no	—
78	C₂×C₂²⋊C₄	—	106	2	no	—
79	C₂×C₄²	Dic₅₃	1	1	yes	characteristic, complemented
80	#99	—	53	4	no	—
81	#99	—	106	2	no	—
82	C₂⁴	C₂²⋊D₅₃	1	2	yes	—
83	C₂×D₄	—	4	1	no	—
84	C₂×D₄	—	4	1	no	—
85	C₂×D₄	Dic₁₀₆	1	1	yes	characteristic
86	C₂×D₄	C₄×D₅₃	1	2	yes	—
87	C₂×D₄	D₂₁₂	1	1	yes	characteristic
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	212	2	no	—
91	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
92	C₄⋊C₄	—	4	1	no	—
93	C₂²⋊C₄	—	4	4	no	—
94	C₂²⋊C₄	—	106	4	no	—
95	C₂²⋊C₄	—	212	2	no	—
96	C₄²	—	4	1	no	—
97	C₂³	—	2	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	2	no	—
101	C₂³	#11719919	1	2	yes	—
102	C₂³	#11719921	1	2	yes	—
103	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
104	D₄	—	4	1	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	424	2	no	—
115	C₂×C₄	—	424	2	no	—
116	C₂×C₄	—	212	4	no	—
117	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
118	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	4	2	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	4	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	#13133124	1	1	yes	characteristic
129	C₂²	#13133132	1	1	yes	characteristic
130	C₂²	#13133132	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	2	no	—
134	C₄	—	8	1	no	—
135	C₄	—	424	4	no	—
136	C₂	—	2	1	no	—
137	C₂	—	2	1	no	—
138	C₂	—	4	2	no	—
139	C₂	—	4	2	no	—
140	C₂	#16810390	1	1	yes	characteristic
141	C₂	#16810416	1	1	yes	characteristic
142	C₂	#16810418	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133132

C₂

yes

maximal

C₅₃×#390

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₅₃

—

#13133132

—

#13133132

C₄

yes

complemented

C₂²×D₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₄

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₄

yes

characteristic, complemented

C₂⁴×C₅₃

D₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₄⋊C₄

yes

—

D₄×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×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1393

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×#99

—

#390

D₅₃

yes

characteristic

C₅₃

#1393

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

Dic₅₃

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

106

—

C₂×C₄²

Dic₅₃

yes

characteristic, complemented

#99

—

#99

—

106

—

C₂⁴

C₂²⋊D₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₁₀₆

yes

characteristic

C₂×D₄

C₄×D₅₃

yes

—

C₂×D₄

D₂₁₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

212

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

#11719919

yes

—

102

C₂³

#11719921

yes

—

103

C₂³

C₂²⋊Dic₅₃

yes

characteristic

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

424

—

115

C₂×C₄

—

424

—

116

C₂×C₄

—

212

—

117

C₂×C₄

C₄×Dic₅₃

yes

characteristic

118

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#13133124

yes

characteristic

129

C₂²

#13133132

yes

characteristic

130

C₂²

#13133132

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

424

—

136

C₂

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

#16810390

yes

characteristic

141

C₂

#16810416

yes

characteristic

142

C₂

#16810418

yes

characteristic

	subgroup	G/H
center	C₂²	#13133132
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂²×D₄	C₂×D₅₃
Fitting	C₅₃×#390	C₂
socle	C₂²×C₅₃	#99
hypercenter	#390	D₅₃
hypocenter	C₅₃	#1393
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #390

subgroup

G/H

center

C₂²

#13133132

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂²×D₄

C₂×D₅₃

Fitting

C₅₃×#390

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #390

N	H ≅ G/N	type
5. #13133132	C₄	semidirect
7. C₂×C₄⋊C₄×C₅₃	C₄	semidirect
9. C₂×C₄²×C₅₃	C₄	semidirect
74. C₅₃	#1393	semidirect
76. C₂×C₄⋊C₄	Dic₅₃	semidirect
79. C₂×C₄²	Dic₅₃	semidirect

H ≅ G/N

type

5. #13133132

C₄

semidirect

7. C₂×C₄⋊C₄×C₅₃

C₄

semidirect

9. C₂×C₄²×C₅₃

C₄

semidirect

74. C₅₃

#1393

semidirect

76. C₂×C₄⋊C₄

Dic₅₃

semidirect

79. C₂×C₄²

Dic₅₃

semidirect

Subgroups — #20272840

Conjugacy classes of subgroups

Complements