Group #20290934 — subgroups

Subgroups — #20290934

Order 6784.

160 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 78 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	D₅₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic
2	#16811586	C₂	1	1	yes	maximal, characteristic
3	#16811514	C₂	1	2	yes	maximal, complemented
4	#16811513	C₂	1	2	yes	maximal, complemented
5	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
6	D₅₃⋊M₁₆	C₄	1	1	yes	characteristic
7	#13133296	C₄	1	1	yes	characteristic
8	#13133292	C₂²	1	2	yes	—
9	#13133291	C₄	1	2	yes	complemented
10	D₅₃⋊C₁₆	C₄	1	2	yes	complemented
11	#13133271	C₂²	1	2	yes	—
12	C₈×#6157	C₄	1	2	yes	—
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133156	C₄	1	1	yes	characteristic, complemented
15	#13133156	C₄	1	1	yes	characteristic, complemented
16	#13133156	C₂²	1	1	yes	characteristic
17	#13133156	C₂²	1	1	yes	characteristic
18	C₂×#11719908	C₄	1	1	yes	characteristic, complemented
19	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²⋊#6157	—	2	1	no	—
22	#11719940	—	2	1	no	—
23	C₄⋊#6157	—	2	1	no	—
24	C₄×#6157	C₂×C₄	1	2	yes	—
25	#11719936	—	2	1	no	—
26	D₅₃⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#88322	C₂×C₄	1	1	yes	characteristic
29	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
30	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₈×D₅₃	C₂³	1	1	yes	characteristic
33	#11719910	C₂×C₄	1	4	yes	—
34	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
36	#11719908	C₂×C₄	1	1	yes	characteristic
37	#11719908	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₅₃	M₁₆	1	1	yes	characteristic
39	C₂×#6157	—	2	2	no	—
40	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×Dic₅₃	M₁₆	1	1	yes	characteristic
42	C₄×D₅₃	C₄²	1	1	yes	characteristic
43	C₄×D₅₃	M₁₆	1	1	yes	characteristic
44	C₄×D₅₃	M₁₆	1	1	yes	characteristic
45	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
46	#88324	—	2	2	no	—
47	C₈×C₅₃	C₄²	1	1	yes	characteristic
48	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	#88322	C₄²	1	1	yes	characteristic
50	#88322	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₅₃	#97	1	1	yes	characteristic
52	C₂×D₅₃	—	4	1	no	—
53	C₂×D₅₃	#97	1	1	yes	characteristic
54	C₂×D₅₃	C₂×M₁₆	1	1	yes	characteristic
55	#6157	—	4	2	no	—
56	C₄×C₅₃	#97	1	1	yes	characteristic
57	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
58	Dic₅₃	#97	1	1	yes	characteristic
59	Dic₅₃	C₂×M₁₆	1	1	yes	characteristic
60	#2110	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	1	no	—
62	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
63	D₅₃	—	2	1	no	—
64	D₅₃	—	2	1	no	—
65	C₂×M₃₂	—	53	1	no	—
66	C₂×M₃₂	—	53	1	no	—
67	#405	—	53	1	no	—
68	#347	—	53	2	no	—
69	#347	—	53	2	no	—
70	C₅₃	#2110	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	53	1	no	—
72	C₂²×C₈	—	53	1	no	—
73	#117	—	53	1	no	—
74	M₃₂	—	53	2	no	—
75	M₃₂	—	53	2	no	—
76	M₃₂	—	53	1	no	—
77	M₃₂	—	53	1	no	—
78	M₃₂	—	53	1	no	—
79	M₃₂	—	53	1	no	—
80	C₂×C₁₆	—	53	2	no	—
81	C₂×C₁₆	—	53	1	no	—
82	C₂×C₁₆	#6157	1	1	yes	characteristic, complemented
83	#97	—	53	2	no	—
84	C₄×C₈	—	53	2	no	—
85	C₂²×C₄	—	53	1	no	—
86	M₁₆	—	106	1	no	—
87	M₁₆	—	106	1	no	—
88	C₂×C₈	—	53	1	no	—
89	C₂×C₈	—	53	1	no	—
90	C₂×C₈	—	53	1	no	—
91	C₂×C₈	—	53	1	no	—
92	C₂×C₈	—	53	1	no	—
93	C₂×C₈	—	53	2	no	—
94	C₂×C₈	C₂×#6157	1	1	yes	characteristic
95	C₄⋊C₄	—	106	1	no	—
96	C₂²⋊C₄	—	106	1	no	—
97	C₄²	—	53	2	no	—
98	C₁₆	—	53	4	no	—
99	C₁₆	—	53	1	no	—
100	C₁₆	—	53	1	no	—
101	C₁₆	C₂×#6157	1	1	yes	characteristic
102	C₁₆	C₂×#6157	1	1	yes	characteristic
103	C₂³	—	53	1	no	—
104	C₂×C₄	—	53	1	no	—
105	C₂×C₄	—	53	1	no	—
106	C₂×C₄	—	53	1	no	—
107	C₂×C₄	—	53	1	no	—
108	C₂×C₄	—	53	1	no	—
109	C₂×C₄	—	106	2	no	—
110	C₂×C₄	C₄×#6157	1	1	yes	characteristic
111	C₈	—	53	1	no	—
112	C₈	—	53	1	no	—
113	C₈	—	106	2	no	—
114	C₈	C₄×#6157	1	1	yes	characteristic
115	C₈	C₂²×#6157	1	1	yes	characteristic
116	C₂²	—	53	1	no	—
117	C₂²	—	53	1	no	—
118	C₂²	—	212	1	no	—
119	C₂²	#13133271	1	1	yes	characteristic
120	C₄	—	53	1	no	—
121	C₄	—	53	1	no	—
122	C₄	—	212	2	no	—
123	C₄	#13133271	1	1	yes	characteristic
124	C₄	C₂×C₄×#6157	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	106	1	no	—
127	C₂	—	106	1	no	—
128	C₂	C₂×#13133271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊M₃₂

C₂

yes

maximal, characteristic

#16811586

C₂

yes

maximal, characteristic

#16811514

C₂

yes

maximal, complemented

#16811513

C₂

yes

maximal, complemented

C₂×#13133156

C₂

yes

maximal, characteristic

D₅₃⋊M₁₆

C₄

yes

characteristic

#13133296

C₄

yes

characteristic

#13133292

C₂²

yes

—

#13133291

C₄

yes

complemented

D₅₃⋊C₁₆

C₄

yes

complemented

#13133271

C₂²

yes

—

C₈×#6157

C₄

yes

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133156

C₄

yes

characteristic, complemented

#13133156

C₄

yes

characteristic, complemented

#13133156

C₂²

yes

characteristic

#13133156

C₂²

yes

characteristic

C₂×#11719908

C₄

yes

characteristic, complemented

C₂×C₁₆×C₅₃

C₄

yes

characteristic, complemented

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂²⋊#6157

—

#11719940

—

C₄⋊#6157

—

C₄×#6157

C₂×C₄

yes

—

#11719936

—

D₅₃⋊C₈

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂³

yes

characteristic

#11719910

C₂×C₄

yes

—

C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₁₆×C₅₃

C₂×C₄

yes

characteristic

#11719908

C₂×C₄

yes

characteristic

#11719908

C₂×C₄

yes

characteristic

C₂²×D₅₃

M₁₆

yes

characteristic

C₂×#6157

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

M₁₆

yes

characteristic

C₄×D₅₃

C₄²

yes

characteristic

C₄×D₅₃

M₁₆

yes

characteristic

C₄×D₅₃

M₁₆

yes

characteristic

C₄×D₅₃

C₂²×C₄

yes

characteristic

#88324

—

C₈×C₅₃

C₄²

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

#88322

C₄²

yes

characteristic

#88322

C₂²×C₄

yes

characteristic

C₂²×C₅₃

#97

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

#97

yes

characteristic

C₂×D₅₃

C₂×M₁₆

yes

characteristic

#6157

—

C₄×C₅₃

#97

yes

characteristic

C₄×C₅₃

C₂×C₄²

yes

characteristic

Dic₅₃

#97

yes

characteristic

Dic₅₃

C₂×M₁₆

yes

characteristic

#2110

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#97

yes

characteristic

D₅₃

—

D₅₃

—

C₂×M₃₂

—

C₂×M₃₂

—

#405

—

#347

—

#347

—

C₅₃

#2110

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#6157

yes

characteristic, complemented

#97

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₁₆

—

C₁₆

—

100

C₁₆

—

101

C₁₆

C₂×#6157

yes

characteristic

102

C₁₆

C₂×#6157

yes

characteristic

103

C₂³

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

106

—

110

C₂×C₄

C₄×#6157

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

106

—

114

C₈

C₄×#6157

yes

characteristic

115

C₈

C₂²×#6157

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

—

212

—

119

C₂²

#13133271

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

212

—

123

C₄

#13133271

yes

characteristic

124

C₄

C₂×C₄×#6157

yes

characteristic

125

C₂

—

126

C₂

—

106

—

127

C₂

—

106

—

128

C₂

C₂×#13133271

yes

characteristic

	subgroup	G/H
center	C₈	C₄×#6157
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₈	C₂²×#6157
Fitting	C₂×C₁₆×C₅₃	C₄
socle	C₂×C₅₃	C₂×#97
hypercenter	C₂×C₁₆	#6157
hypocenter	C₅₃	#2110
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₈

C₄×#6157

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₈

C₂²×#6157

Fitting

C₂×C₁₆×C₅₃

C₄

socle

C₂×C₅₃

C₂×#97

C₂×C₁₆

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
3. #16811514	C₂	semidirect
4. #16811513	C₂	semidirect
9. #13133291	C₄	semidirect
10. D₅₃⋊C₁₆	C₄	semidirect
14. #13133156	C₄	semidirect
15. #13133156	C₄	semidirect
18. C₂×#11719908	C₄	semidirect
19. C₂×C₁₆×C₅₃	C₄	semidirect
70. C₅₃	#2110	semidirect
82. C₂×C₁₆	#6157	semidirect

H ≅ G/N

type

3. #16811514

C₂

semidirect

4. #16811513

C₂

semidirect

9. #13133291

C₄

semidirect

10. D₅₃⋊C₁₆

C₄

semidirect

14. #13133156

C₄

semidirect

15. #13133156

C₄

semidirect

18. C₂×#11719908

C₄

semidirect

19. C₂×C₁₆×C₅₃

C₄

semidirect

70. C₅₃

#2110

semidirect

82. C₂×C₁₆

#6157

semidirect

Subgroups — #20290934

Conjugacy classes of subgroups

Complements