Group #20277267 — subgroups

Subgroups — #126×Dic₅₃ #20277267

Order 6784.

516 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 229 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	#16810516	C₂	1	3	yes	maximal
2	#16810509	C₂	1	3	yes	maximal, complemented
3	#16810491	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊C₄×Dic₅₃	C₂	1	3	yes	maximal, complemented
5	C₂²⋊C₄×Dic₅₃	C₂	1	3	yes	maximal
6	C₄²×Dic₅₃	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×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
12	C₂×C₄×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	—	2	6	no	—
26	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₃	C₂³	1	3	yes	—
28	C₂²×Dic₅₃	C₂³	1	3	yes	—
29	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
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	—	2	6	no	—
34	C₄⋊Dic₅₃	—	2	6	no	—
35	C₄×Dic₅₃	—	2	6	no	—
36	C₄×Dic₅₃	—	2	6	no	—
37	C₄×Dic₅₃	—	2	6	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	—	4	4	no	—
55	C₂×Dic₅₃	#41	1	6	yes	—
56	C₂×Dic₅₃	#41	1	6	yes	—
57	C₂²×C₅₃	—	2	6	no	—
58	C₂²×C₅₃	—	4	1	no	—
59	C₂²×C₅₃	—	4	3	no	—
60	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
61	C₂²×C₅₃	C₂×#41	1	3	yes	—
62	C₂²×C₅₃	C₂×#41	1	3	yes	—
63	C₄×C₅₃	—	2	6	no	—
64	C₄×C₅₃	—	2	6	no	—
65	Dic₅₃	—	2	6	no	—
66	Dic₅₃	—	4	1	no	—
67	Dic₅₃	—	4	3	no	—
68	Dic₅₃	#126	1	4	yes	direct product factor
69	C₄×#126	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	4	2	no	—
71	C₂×C₅₃	C₄×#41	1	3	yes	—
72	C₂×C₅₃	C₂×#126	1	1	yes	characteristic
73	C₂×C₅₃	#529	1	3	yes	—
74	C₂×#126	D₅₃	1	1	yes	characteristic
75	#388	—	53	3	no	—
76	#387	—	53	3	no	—
77	#383	—	53	1	no	—
78	C₄×C₄⋊C₄	—	53	3	no	—
79	C₄×C₂²⋊C₄	—	53	3	no	—
80	C₄³	—	53	1	no	—
81	C₅₃	C₄×#126	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₄	—	53	1	no	—
83	#126	Dic₅₃	1	8	yes	direct product factor
84	C₂×C₄⋊C₄	—	53	3	no	—
85	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
86	C₂×C₂²⋊C₄	—	53	3	no	—
87	C₂×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
88	C₂×C₄²	—	53	3	no	—
89	C₂×C₄²	—	53	3	no	—
90	C₂×C₄²	—	53	3	no	—
91	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
92	#95	—	53	3	no	—
93	#95	—	53	3	no	—
94	#95	—	53	3	no	—
95	#95	—	53	3	no	—
96	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
97	C₂²×C₄	—	53	1	no	—
98	C₂²×C₄	—	53	3	no	—
99	C₂²×C₄	—	53	1	no	—
100	C₂²×C₄	—	53	3	no	—
101	C₂²×C₄	—	106	6	no	—
102	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
103	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
104	C₄⋊C₄	—	106	6	no	—
105	C₄⋊C₄	C₂×Dic₅₃	1	12	yes	complemented
106	C₂²⋊C₄	—	106	6	no	—
107	C₂²⋊C₄	C₂×Dic₅₃	1	12	yes	—
108	C₄²	—	106	6	no	—
109	C₄²	—	106	6	no	—
110	C₄²	—	106	6	no	—
111	C₄²	C₂×Dic₅₃	1	4	yes	complemented
112	C₂³	—	2	3	no	—
113	C₂³	—	2	3	no	—
114	C₂³	C₂²×Dic₅₃	1	2	yes	—
115	C₂³	C₂³×D₅₃	1	1	yes	characteristic
116	C₂×C₄	—	2	3	no	—
117	C₂×C₄	—	2	3	no	—
118	C₂×C₄	—	2	3	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	53	6	no	—
121	C₂×C₄	—	53	6	no	—
122	C₂×C₄	—	106	3	no	—
123	C₂×C₄	—	106	3	no	—
124	C₂×C₄	—	106	3	no	—
125	C₂×C₄	—	106	3	no	—
126	C₂×C₄	—	106	3	no	—
127	C₂×C₄	—	106	3	no	—
128	C₂×C₄	—	212	4	no	—
129	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
130	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
131	C₂²	—	2	6	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	4	3	no	—
134	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
135	C₂²	#41×D₅₃	1	3	yes	—
136	C₂²	#41×D₅₃	1	3	yes	—
137	C₄	—	2	6	no	—
138	C₄	—	2	6	no	—
139	C₄	—	53	4	no	—
140	C₄	—	106	6	no	—
141	C₄	—	212	1	no	—
142	C₄	—	212	3	no	—
143	C₂	—	4	2	no	—
144	C₂	#41×Dic₅₃	1	3	yes	—
145	C₂	#126×D₅₃	1	1	yes	characteristic
146	C₂	#16811156	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810516

C₂

yes

maximal

#16810509

C₂

yes

maximal, complemented

#16810491

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×Dic₅₃

C₂

yes

maximal, complemented

C₂²⋊C₄×Dic₅₃

C₂

yes

maximal

C₄²×Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₅₃

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×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

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

characteristic

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

#41

yes

—

C₂×Dic₅₃

#41

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

C₂×#41

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#126

yes

direct product factor

C₄×#126

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₄×#41

yes

—

C₂×C₅₃

C₂×#126

yes

characteristic

C₂×C₅₃

#529

yes

—

C₂×#126

D₅₃

yes

—

—

—

C₄×C₄⋊C₄

—

C₄×C₂²⋊C₄

—

C₄³

—

C₅₃

C₄×#126

yes

characteristic, complemented, Sylow

C₂³×C₄

—

#126

Dic₅₃

yes

direct product factor

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

—

—

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

106

—

102

C₂²×C₄

C₂²×D₅₃

yes

—

103

C₂²×C₄

C₂²×D₅₃

yes

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

C₂×Dic₅₃

yes

complemented

106

C₂²⋊C₄

—

106

—

107

C₂²⋊C₄

C₂×Dic₅₃

yes

—

108

C₄²

—

106

—

109

C₄²

—

106

—

110

C₄²

—

106

—

111

C₄²

C₂×Dic₅₃

yes

complemented

112

C₂³

—

113

C₂³

—

114

C₂³

C₂²×Dic₅₃

yes

—

115

C₂³

C₂³×D₅₃

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

212

—

129

C₂×C₄

C₂²×Dic₅₃

yes

—

130

C₂×C₄

C₂²×Dic₅₃

yes

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂³×Dic₅₃

yes

characteristic

135

C₂²

#41×D₅₃

yes

—

136

C₂²

#41×D₅₃

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

106

—

141

C₄

—

212

—

142

C₄

—

212

—

143

C₂

—

144

C₂

#41×Dic₅₃

yes

—

145

C₂

#126×D₅₃

yes

characteristic

146

C₂

#16811156

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	C₄×#126
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
2. #16810509	C₂	semidirect
3. #16810491	C₂	semidirect
4. C₄⋊C₄×Dic₅₃	C₂	semidirect
6. C₄²×Dic₅₃	C₂	semidirect
18. #126×C₅₃	C₄	semidirect
30. C₄⋊C₄×C₅₃	C₂×C₄	semidirect
32. C₄²×C₅₃	C₂×C₄	semidirect
68. Dic₅₃	#126	direct (N × H)
81. C₅₃	C₄×#126	semidirect
83. #126	Dic₅₃	direct (N × H)
105. C₄⋊C₄	C₂×Dic₅₃	semidirect
111. C₄²	C₂×Dic₅₃	semidirect

H ≅ G/N

type

2. #16810509

C₂

semidirect

3. #16810491

C₂

semidirect

4. C₄⋊C₄×Dic₅₃

C₂

semidirect

6. C₄²×Dic₅₃

C₂

semidirect

18. #126×C₅₃

C₄

semidirect

30. C₄⋊C₄×C₅₃

C₂×C₄

semidirect

32. C₄²×C₅₃

C₂×C₄

semidirect

68. Dic₅₃

#126

direct (N × H)

81. C₅₃

C₄×#126

semidirect

83. #126

Dic₅₃

direct (N × H)

105. C₄⋊C₄

C₂×Dic₅₃

semidirect

111. C₄²

C₂×Dic₅₃

semidirect

Subgroups — #126×Dic₅₃ #20277267

Conjugacy classes of subgroups

Complements

Subgroups — #126×Dic53 #20277267

Conjugacy classes of subgroups

Complements

Subgroups — #126×Dic₅₃ #20277267