Group #20277907 — subgroups

Subgroups — #20277907

Order 6784.

456 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 213 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	#16810522	C₂	1	3	yes	maximal, complemented
2	#16810516	C₂	1	3	yes	maximal
3	#16810509	C₂	1	3	yes	maximal, complemented
4	#16810498	C₂	1	3	yes	maximal
5	#16810493	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#126×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
12	C₂×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
13	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
14	#13133124	C₂²	1	3	yes	—
15	#13133124	C₂²	1	3	yes	—
16	#13133124	C₂²	1	3	yes	—
17	#13133124	C₂²	1	3	yes	—
18	#126×C₅₃	C₄	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
20	C₂×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
25	C₂²×Dic₅₃	—	4	3	no	—
26	C₂²×Dic₅₃	C₂³	1	3	yes	—
27	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	C₂³	1	3	yes	—
30	C₄⋊C₄×C₅₃	C₂×C₄	1	12	yes	complemented
31	C₂²⋊C₄×C₅₃	C₂×C₄	1	12	yes	—
32	C₄²×C₅₃	C₂×C₄	1	4	yes	complemented
33	C₂²⋊Dic₅₃	—	4	3	no	—
34	C₄⋊Dic₅₃	—	4	3	no	—
35	C₄⋊Dic₅₃	—	4	3	no	—
36	C₄⋊Dic₅₃	—	4	3	no	—
37	C₄×Dic₅₃	—	4	3	no	—
38	C₂³×C₅₃	—	2	3	no	—
39	C₂³×C₅₃	—	2	3	no	—
40	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	3	no	—
43	C₂×C₄×C₅₃	—	2	3	no	—
44	C₂×C₄×C₅₃	—	2	3	no	—
45	C₂×C₄×C₅₃	—	2	3	no	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
48	C₂×Dic₅₃	—	2	3	no	—
49	C₂×Dic₅₃	—	2	3	no	—
50	C₂×Dic₅₃	—	2	3	no	—
51	C₂×Dic₅₃	—	2	3	no	—
52	C₂×Dic₅₃	—	2	3	no	—
53	C₂×Dic₅₃	—	2	3	no	—
54	C₂×Dic₅₃	—	2	6	no	—
55	C₂×Dic₅₃	—	4	4	no	—
56	C₂²×C₅₃	—	2	6	no	—
57	C₂²×C₅₃	—	4	1	no	—
58	C₂²×C₅₃	—	4	3	no	—
59	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₅₃	H(5,𝔽₂)	1	3	yes	—
61	C₂²×C₅₃	2₋¹⁺⁴	1	3	yes	—
62	C₄×C₅₃	—	2	6	no	—
63	C₄×C₅₃	—	2	6	no	—
64	Dic₅₃	—	4	3	no	—
65	Dic₅₃	—	4	1	no	—
66	Dic₅₃	—	4	1	no	—
67	Dic₅₃	—	4	3	no	—
68	#2383	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	4	2	no	—
70	C₂×C₅₃	#520	1	3	yes	—
71	C₂×C₅₃	#560	1	1	yes	characteristic
72	C₂×C₅₃	#563	1	3	yes	—
73	C₂×#126	D₅₃	1	1	yes	characteristic
74	#389	—	53	3	no	—
75	#388	—	53	3	no	—
76	#387	—	53	3	no	—
77	#385	—	53	3	no	—
78	#384	—	53	1	no	—
79	#383	—	53	1	no	—
80	C₅₃	#2383	1	1	yes	characteristic, complemented, Sylow
81	C₂³×C₄	—	53	1	no	—
82	#126	Dic₅₃	1	8	yes	complemented
83	C₂×C₄⋊C₄	—	53	3	no	—
84	C₂×C₄⋊C₄	—	53	3	no	—
85	C₂×C₄⋊C₄	—	53	3	no	—
86	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
87	C₂×C₂²⋊C₄	—	53	3	no	—
88	C₂×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
89	C₂×C₄²	—	53	3	no	—
90	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
91	#95	—	53	3	no	—
92	#95	—	53	3	no	—
93	#95	—	53	3	no	—
94	#95	—	53	3	no	—
95	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
96	C₂²×C₄	—	53	3	no	—
97	C₂²×C₄	—	53	1	no	—
98	C₂²×C₄	—	53	1	no	—
99	C₂²×C₄	—	53	3	no	—
100	C₂²×C₄	—	212	3	no	—
101	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
102	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
103	C₄⋊C₄	—	212	3	no	—
104	C₄⋊C₄	—	212	3	no	—
105	C₄⋊C₄	—	212	3	no	—
106	C₄⋊C₄	C₂×Dic₅₃	1	12	yes	complemented
107	C₂²⋊C₄	—	212	3	no	—
108	C₂²⋊C₄	C₂×Dic₅₃	1	12	yes	—
109	C₄²	—	212	3	no	—
110	C₄²	C₂×Dic₅₃	1	4	yes	complemented
111	C₂³	—	2	3	no	—
112	C₂³	—	2	3	no	—
113	C₂³	C₂²×Dic₅₃	1	2	yes	—
114	C₂³	C₂³×D₅₃	1	1	yes	characteristic
115	C₂×C₄	—	2	3	no	—
116	C₂×C₄	—	2	3	no	—
117	C₂×C₄	—	2	3	no	—
118	C₂×C₄	—	2	3	no	—
119	C₂×C₄	—	106	3	no	—
120	C₂×C₄	—	106	3	no	—
121	C₂×C₄	—	106	3	no	—
122	C₂×C₄	—	106	3	no	—
123	C₂×C₄	—	106	3	no	—
124	C₂×C₄	—	106	3	no	—
125	C₂×C₄	—	106	6	no	—
126	C₂×C₄	—	212	4	no	—
127	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
128	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
129	C₂²	—	2	6	no	—
130	C₂²	—	4	1	no	—
131	C₂²	—	4	3	no	—
132	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
133	C₂²	#13133263	1	3	yes	—
134	C₂²	#13133264	1	3	yes	—
135	C₄	—	2	6	no	—
136	C₄	—	2	6	no	—
137	C₄	—	212	1	no	—
138	C₄	—	212	1	no	—
139	C₄	—	212	3	no	—
140	C₄	—	212	3	no	—
141	C₂	—	4	2	no	—
142	C₂	#16811106	1	3	yes	—
143	C₂	#16811364	1	1	yes	characteristic
144	C₂	#16811375	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810522

C₂

yes

maximal, complemented

#16810516

C₂

yes

maximal

#16810509

C₂

yes

maximal, complemented

#16810498

C₂

yes

maximal

#16810493

C₂

yes

maximal, characteristic, complemented

#16810491

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₅₃

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#126×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

complemented

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

complemented

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

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×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2383

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#520

yes

—

C₂×C₅₃

#560

yes

characteristic

C₂×C₅₃

#563

yes

—

C₂×#126

D₅₃

yes

—

—

—

—

—

—

C₅₃

#2383

yes

characteristic, complemented, Sylow

C₂³×C₄

—

#126

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

—

—

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

212

—

101

C₂²×C₄

C₂²×D₅₃

yes

—

102

C₂²×C₄

C₂²×D₅₃

yes

—

103

C₄⋊C₄

—

212

—

104

C₄⋊C₄

—

212

—

105

C₄⋊C₄

—

212

—

106

C₄⋊C₄

C₂×Dic₅₃

yes

complemented

107

C₂²⋊C₄

—

212

—

108

C₂²⋊C₄

C₂×Dic₅₃

yes

—

109

C₄²

—

212

—

110

C₄²

C₂×Dic₅₃

yes

complemented

111

C₂³

—

112

C₂³

—

113

C₂³

C₂²×Dic₅₃

yes

—

114

C₂³

C₂³×D₅₃

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

212

—

127

C₂×C₄

C₂²×Dic₅₃

yes

—

128

C₂×C₄

C₂²×Dic₅₃

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×Dic₅₃

yes

characteristic

133

C₂²

#13133263

yes

—

134

C₂²

#13133264

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

212

—

138

C₄

—

212

—

139

C₄

—

212

—

140

C₄

—

212

—

141

C₂

—

142

C₂

#16811106

yes

—

143

C₂

#16811364

yes

characteristic

144

C₂

#16811375

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×#126×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×#126	D₅₃
hypocenter	C₅₃	#2383
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Fitting

C₂×#126×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. #16810522	C₂	semidirect
3. #16810509	C₂	semidirect
5. #16810493	C₂	semidirect
6. #16810491	C₂	semidirect
18. #126×C₅₃	C₄	semidirect
30. C₄⋊C₄×C₅₃	C₂×C₄	semidirect
32. C₄²×C₅₃	C₂×C₄	semidirect
80. C₅₃	#2383	semidirect
82. #126	Dic₅₃	semidirect
106. C₄⋊C₄	C₂×Dic₅₃	semidirect
110. C₄²	C₂×Dic₅₃	semidirect

H ≅ G/N

type

1. #16810522

C₂

semidirect

3. #16810509

C₂

semidirect

5. #16810493

C₂

semidirect

6. #16810491

C₂

semidirect

18. #126×C₅₃

C₄

semidirect

30. C₄⋊C₄×C₅₃

C₂×C₄

semidirect

32. C₄²×C₅₃

C₂×C₄

semidirect

80. C₅₃

#2383

semidirect

82. #126

Dic₅₃

semidirect

106. C₄⋊C₄

C₂×Dic₅₃

semidirect

110. C₄²

C₂×Dic₅₃

semidirect

Subgroups — #20277907

Conjugacy classes of subgroups

Complements