Group #20287812 — subgroups

Subgroups — #504×D₅₃ #20287812

Order 6784.

368 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 211 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	#131×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16811041	C₂	1	3	yes	maximal, complemented
3	#16811040	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811039	C₂	1	3	yes	maximal, complemented
5	M₃₂×D₅₃	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₆×D₅₃	C₂	1	3	yes	maximal, complemented
7	C₅₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×D₅₃	C₄	1	1	yes	characteristic
9	#13133225	C₄	1	3	yes	—
10	#13133224	C₄	1	1	yes	characteristic
11	#13133223	C₂²	1	3	yes	—
12	M₁₆×D₅₃	C₄	1	3	yes	—
13	C₂×C₈×D₅₃	C₂²	1	3	yes	—
14	#13133157	C₂²	1	3	yes	complemented
15	#13133156	C₂²	1	6	yes	complemented
16	C₂×#11719908	C₂²	1	3	yes	complemented
17	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
18	C₁₆×D₅₃	C₂²	1	3	yes	—
19	C₁₆×D₅₃	C₂²	1	3	yes	complemented
20	C₁₆×D₅₃	C₂²	1	3	yes	complemented
21	#131×C₅₃	C₂²	1	1	yes	characteristic
22	M₃₂×C₅₃	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₅₃	C₂²	1	3	yes	complemented
24	#41×C₅₃	C₂×C₄	1	1	yes	characteristic
25	#11719949	C₈	1	1	yes	characteristic
26	Q₈×D₅₃	C₈	1	1	yes	characteristic
27	#11719947	C₈	1	3	yes	—
28	D₄×D₅₃	C₈	1	3	yes	—
29	#11719945	C₂×C₄	1	3	yes	—
30	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
31	M₁₆×C₅₃	C₂×C₄	1	3	yes	—
32	C₂×C₈×C₅₃	C₂³	1	3	yes	—
33	#11719917	C₂×C₄	1	3	yes	—
34	C₂×#88322	C₂×C₄	1	3	yes	—
35	#11719912	C₂×C₄	1	6	yes	—
36	C₈×D₅₃	C₂×C₄	1	3	yes	—
37	C₈×D₅₃	C₂×C₄	1	3	yes	—
38	C₈×D₅₃	C₂³	1	1	yes	characteristic
39	C₈×D₅₃	C₂³	1	3	yes	—
40	C₁₆×C₅₃	C₂³	1	3	yes	—
41	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
42	#11719908	C₂³	1	1	yes	characteristic
43	#11719908	C₂³	1	3	yes	—
44	C₂²×D₅₃	C₂×C₈	1	3	yes	—
45	Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
46	D₄×C₅₃	C₂×C₈	1	3	yes	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
48	C₂²⋊D₅₃	C₂×C₈	1	6	yes	—
49	C₂×Dic₅₃	C₂×C₈	1	3	yes	—
50	D₂₁₂	C₂×C₈	1	3	yes	—
51	C₄×D₅₃	C₂×C₈	1	3	yes	—
52	C₄×D₅₃	C₂×C₈	1	3	yes	—
53	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₄×D₅₃	C₂²×C₄	1	3	yes	—
55	Dic₁₀₆	C₂×C₈	1	3	yes	—
56	C₈×C₅₃	C₂²×C₄	1	3	yes	—
57	C₈×C₅₃	C₂⁴	1	1	yes	characteristic
58	#88322	C₂²×C₄	1	1	yes	characteristic
59	#88322	C₂²×C₄	1	3	yes	—
60	C₂²×C₅₃	C₂²×C₈	1	3	yes	—
61	C₂×D₅₃	—	2	6	no	—
62	C₂×D₅₃	C₂²×C₈	1	3	yes	—
63	C₂×D₅₃	C₂²×C₈	1	1	yes	characteristic
64	C₄×C₅₃	C₂²×C₈	1	3	yes	—
65	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	Dic₅₃	C₂²×C₈	1	3	yes	—
67	Dic₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×#504	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	3	no	—
70	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
71	D₅₃	—	2	3	no	—
72	D₅₃	#504	1	2	yes	direct product factor
73	C₂×#131	—	53	1	no	—
74	#504	—	53	1	no	—
75	#504	—	53	3	no	—
76	#504	—	53	3	no	—
77	#504	D₅₃	1	1	yes	characteristic, direct product factor
78	C₂×M₃₂	—	53	3	no	—
79	C₂²×C₁₆	—	53	3	no	—
80	C₅₃	C₂×#504	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	—	53	1	no	—
82	#131	—	53	1	no	—
83	#131	—	53	3	no	—
84	#131	—	53	3	no	—
85	#131	C₂×D₅₃	1	1	yes	characteristic
86	C₂×M₁₆	—	53	3	no	—
87	C₂²×C₈	—	53	3	no	—
88	M₃₂	—	53	3	no	—
89	M₃₂	—	53	6	no	—
90	M₃₂	C₂×D₅₃	1	3	yes	complemented
91	C₂×C₁₆	—	53	3	no	—
92	C₂×C₁₆	—	53	3	no	—
93	C₂×C₁₆	—	53	3	no	—
94	C₂×C₁₆	—	53	1	no	—
95	C₂×C₁₆	—	53	3	no	—
96	C₂×C₁₆	C₂×D₅₃	1	3	yes	complemented
97	#41	—	53	1	no	—
98	#41	—	53	3	no	—
99	#41	—	53	3	no	—
100	#41	C₄×D₅₃	1	1	yes	characteristic
101	C₂×Q₈	—	53	1	no	—
102	C₂×D₄	—	53	3	no	—
103	C₂²×C₄	—	53	3	no	—
104	M₁₆	—	53	3	no	—
105	M₁₆	—	53	6	no	—
106	M₁₆	C₄×D₅₃	1	3	yes	—
107	C₂×C₈	—	53	3	no	—
108	C₂×C₈	—	53	3	no	—
109	C₂×C₈	—	53	3	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	3	no	—
112	C₂×C₈	C₂²×D₅₃	1	3	yes	—
113	C₁₆	—	53	1	no	—
114	C₁₆	—	53	3	no	—
115	C₁₆	C₂²×D₅₃	1	3	yes	—
116	C₁₆	C₂²×D₅₃	1	1	yes	characteristic
117	C₂³	—	53	3	no	—
118	Q₈	—	53	3	no	—
119	Q₈	C₈×D₅₃	1	1	yes	characteristic
120	D₄	—	53	6	no	—
121	D₄	—	53	3	no	—
122	D₄	C₈×D₅₃	1	3	yes	—
123	C₂×C₄	—	53	3	no	—
124	C₂×C₄	—	53	3	no	—
125	C₂×C₄	—	53	3	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	53	3	no	—
128	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
129	C₈	—	53	1	no	—
130	C₈	—	53	3	no	—
131	C₈	C₂×C₄×D₅₃	1	3	yes	—
132	C₈	C₂³×D₅₃	1	1	yes	characteristic
133	C₂²	—	53	1	no	—
134	C₂²	—	53	3	no	—
135	C₂²	—	106	6	no	—
136	C₂²	C₂×C₈×D₅₃	1	3	yes	—
137	C₄	—	53	1	no	—
138	C₄	—	53	3	no	—
139	C₄	C₂×C₈×D₅₃	1	3	yes	—
140	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
141	C₂	—	2	3	no	—
142	C₂	—	53	2	no	—
143	C₂	—	106	3	no	—
144	C₂	C₂²×C₈×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#131×D₅₃

C₂

yes

maximal, characteristic

#16811041

C₂

yes

maximal, complemented

#16811040

C₂

yes

maximal, characteristic, complemented

#16811039

C₂

yes

maximal, complemented

M₃₂×D₅₃

C₂

yes

maximal, complemented

C₂×C₁₆×D₅₃

C₂

yes

maximal, complemented

C₅₃×#504

C₂

yes

maximal, characteristic, complemented

#41×D₅₃

C₄

yes

characteristic

#13133225

C₄

yes

—

#13133224

C₄

yes

characteristic

#13133223

C₂²

yes

—

M₁₆×D₅₃

C₄

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

#13133157

C₂²

yes

complemented

#13133156

C₂²

yes

complemented

C₂×#11719908

C₂²

yes

complemented

C₁₆×D₅₃

C₂²

yes

characteristic

C₁₆×D₅₃

C₂²

yes

—

C₁₆×D₅₃

C₂²

yes

complemented

C₁₆×D₅₃

C₂²

yes

complemented

#131×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

complemented

C₂×C₁₆×C₅₃

C₂²

yes

complemented

#41×C₅₃

C₂×C₄

yes

characteristic

#11719949

C₈

yes

characteristic

Q₈×D₅₃

C₈

yes

characteristic

#11719947

C₈

yes

—

D₄×D₅₃

C₈

yes

—

#11719945

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

#11719917

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

#11719912

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

C₂³

yes

characteristic

C₈×D₅₃

C₂³

yes

—

C₁₆×C₅₃

C₂³

yes

—

C₁₆×C₅₃

C₂³

yes

characteristic

#11719908

C₂³

yes

characteristic

#11719908

C₂³

yes

—

C₂²×D₅₃

C₂×C₈

yes

—

Q₈×C₅₃

C₂×C₈

yes

characteristic

D₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂²⋊D₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

D₂₁₂

C₂×C₈

yes

—

C₄×D₅₃

C₂×C₈

yes

—

C₄×D₅₃

C₂×C₈

yes

—

C₄×D₅₃

C₂²×C₄

yes

characteristic

C₄×D₅₃

C₂²×C₄

yes

—

Dic₁₀₆

C₂×C₈

yes

—

C₈×C₅₃

C₂²×C₄

yes

—

C₈×C₅₃

C₂⁴

yes

characteristic

#88322

C₂²×C₄

yes

characteristic

#88322

C₂²×C₄

yes

—

C₂²×C₅₃

C₂²×C₈

yes

—

C₂×D₅₃

—

C₂×D₅₃

C₂²×C₈

yes

—

C₂×D₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

—

C₄×C₅₃

C₂³×C₄

yes

characteristic

Dic₅₃

C₂²×C₈

yes

—

Dic₅₃

C₂²×C₈

yes

characteristic

C₂×#504

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂³×C₈

yes

characteristic

D₅₃

—

D₅₃

#504

yes

direct product factor

—

—

—

—

D₅₃

yes

characteristic, direct product factor

C₂×M₃₂

—

C₂²×C₁₆

—

C₅₃

C₂×#504

yes

characteristic, complemented, Sylow

C₂×#41

—

—

—

—

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

M₃₂

C₂×D₅₃

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

—

—

—

100

#41

C₄×D₅₃

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

C₄×D₅₃

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₅₃

yes

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

C₂²×D₅₃

yes

—

116

C₁₆

C₂²×D₅₃

yes

characteristic

117

C₂³

—

118

Q₈

—

119

Q₈

C₈×D₅₃

yes

characteristic

120

D₄

—

121

D₄

—

122

D₄

C₈×D₅₃

yes

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₄×D₅₃

yes

—

129

C₈

—

130

C₈

—

131

C₈

C₂×C₄×D₅₃

yes

—

132

C₈

C₂³×D₅₃

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

106

—

136

C₂²

C₂×C₈×D₅₃

yes

—

137

C₄

—

138

C₄

—

139

C₄

C₂×C₈×D₅₃

yes

—

140

C₄

C₂²×C₄×D₅₃

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

106

—

144

C₂

C₂²×C₈×D₅₃

yes

characteristic

	subgroup	G/H
center	C₁₆	C₂²×D₅₃
commutator, socle	C₂×C₅₃	C₂³×C₈
Frattini	C₈	C₂³×D₅₃
Fitting	C₅₃×#504	C₂
hypercenter	#504	D₅₃
hypocenter	C₅₃	C₂×#504
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ #504

subgroup

G/H

center

C₁₆

C₂²×D₅₃

commutator, socle

C₂×C₅₃

C₂³×C₈

Frattini

C₈

C₂³×D₅₃

Fitting

C₅₃×#504

C₂

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₁₆ ◁ #504

N	H ≅ G/N	type
2. #16811041	C₂	semidirect
3. #16811040	C₂	semidirect
4. #16811039	C₂	semidirect
5. M₃₂×D₅₃	C₂	semidirect
6. C₂×C₁₆×D₅₃	C₂	semidirect
7. C₅₃×#504	C₂	semidirect
14. #13133157	C₂²	semidirect
15. #13133156	C₂²	semidirect
16. C₂×#11719908	C₂²	semidirect
19. C₁₆×D₅₃	C₂²	semidirect
20. C₁₆×D₅₃	C₂²	semidirect
22. M₃₂×C₅₃	C₂²	semidirect
23. C₂×C₁₆×C₅₃	C₂²	semidirect
72. D₅₃	#504	direct (N × H)
77. #504	D₅₃	direct (N × H)
80. C₅₃	C₂×#504	semidirect
90. M₃₂	C₂×D₅₃	semidirect
96. C₂×C₁₆	C₂×D₅₃	semidirect

H ≅ G/N

type

2. #16811041

C₂

semidirect

3. #16811040

C₂

semidirect

4. #16811039

C₂

semidirect

5. M₃₂×D₅₃

C₂

semidirect

6. C₂×C₁₆×D₅₃

C₂

semidirect

7. C₅₃×#504

C₂

semidirect

14. #13133157

C₂²

semidirect

15. #13133156

C₂²

semidirect

16. C₂×#11719908

C₂²

semidirect

19. C₁₆×D₅₃

C₂²

semidirect

20. C₁₆×D₅₃

C₂²

semidirect

22. M₃₂×C₅₃

C₂²

semidirect

23. C₂×C₁₆×C₅₃

C₂²

semidirect

72. D₅₃

#504

direct (N × H)

77. #504

D₅₃

direct (N × H)

80. C₅₃

C₂×#504

semidirect

90. M₃₂

C₂×D₅₃

semidirect

96. C₂×C₁₆

C₂×D₅₃

semidirect

Subgroups — #504×D₅₃ #20287812

Conjugacy classes of subgroups

Complements

Subgroups — #504×D53 #20287812

Conjugacy classes of subgroups

Complements

Subgroups — #504×D₅₃ #20287812