Group #20290929 — subgroups

Subgroups — C₂×#16811514 #20290929

Order 6784.

168 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 82 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×#13133292	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133271	C₂	1	1	yes	maximal, characteristic
3	#16811514	C₂	1	4	yes	maximal, direct product factor
4	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×#6157	C₄	1	1	yes	characteristic
7	#13133292	C₄	1	2	yes	complemented
8	#13133292	C₂²	1	2	yes	—
9	C₂×#11719910	C₄	1	2	yes	complemented
10	#13133271	C₄	1	2	yes	—
11	#13133271	C₂²	1	2	yes	—
12	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#13133156	C₄	1	2	yes	complemented
14	#13133156	C₂²	1	2	yes	—
15	C₂×#11719908	C₄	1	1	yes	characteristic, complemented
16	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×#6157	—	2	1	no	—
18	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×#88324	—	2	1	no	—
20	C₄×#6157	—	2	1	no	—
21	C₄×#6157	C₂×C₄	1	2	yes	—
22	D₅₃⋊C₈	—	2	1	no	—
23	D₅₃⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×#88322	C₂×C₄	1	1	yes	characteristic
26	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₈×D₅₃	C₂³	1	1	yes	characteristic
30	#11719910	C₂×C₄	1	4	yes	complemented
31	C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
32	#11719908	C₂×C₄	1	2	yes	complemented
33	C₂²×D₅₃	M₁₆	1	1	yes	characteristic
34	C₂×#6157	—	2	2	no	—
35	C₂×#6157	—	4	2	no	—
36	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×Dic₅₃	M₁₆	1	1	yes	characteristic
38	C₄×D₅₃	C₄²	1	1	yes	characteristic
39	C₄×D₅₃	M₁₆	1	1	yes	characteristic
40	C₄×D₅₃	M₁₆	1	1	yes	characteristic
41	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
42	#88324	—	2	2	no	—
43	C₈×C₅₃	C₄²	1	1	yes	characteristic
44	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	#88322	C₄²	1	1	yes	characteristic
46	#88322	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₅₃	#97	1	1	yes	characteristic
48	C₂×D₅₃	—	2	2	no	—
49	C₂×D₅₃	#97	1	1	yes	characteristic
50	C₂×D₅₃	C₂×M₁₆	1	1	yes	characteristic
51	#6157	—	4	2	no	—
52	C₄×C₅₃	#97	1	1	yes	characteristic
53	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	Dic₅₃	#97	1	1	yes	characteristic
55	Dic₅₃	C₂×M₁₆	1	1	yes	characteristic
56	C₂×#347	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	#347	1	2	yes	complemented
58	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
59	D₅₃	—	2	1	no	—
60	D₅₃	—	2	1	no	—
61	C₂×M₃₂	—	53	1	no	—
62	C₂×M₃₂	—	53	1	no	—
63	C₂×#97	—	53	1	no	—
64	#347	—	53	4	no	—
65	C₅₃	C₂×#347	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₈	—	53	1	no	—
67	C₂²×C₈	—	53	1	no	—
68	C₂×C₄²	—	53	1	no	—
69	M₃₂	—	53	2	no	—
70	M₃₂	—	53	2	no	—
71	M₃₂	—	53	2	no	—
72	M₃₂	—	53	2	no	—
73	C₂×C₁₆	—	53	2	no	—
74	C₂×C₁₆	—	53	1	no	—
75	C₂×C₁₆	#6157	1	1	yes	characteristic, complemented
76	#97	—	53	2	no	—
77	#97	—	53	2	no	—
78	C₂²×C₄	—	53	1	no	—
79	C₂²×C₄	—	106	1	no	—
80	C₂×C₈	—	53	1	no	—
81	C₂×C₈	—	53	1	no	—
82	C₂×C₈	—	53	1	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	—	53	1	no	—
85	C₂×C₈	—	53	2	no	—
86	C₂×C₈	—	106	1	no	—
87	C₂×C₈	—	106	1	no	—
88	C₂×C₈	C₂×#6157	1	1	yes	characteristic
89	C₄²	—	53	2	no	—
90	C₄²	—	106	1	no	—
91	C₁₆	—	53	4	no	—
92	C₁₆	—	53	2	no	—
93	C₁₆	C₂×#6157	1	2	yes	complemented
94	C₂³	—	53	1	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	53	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	106	2	no	—
101	C₂×C₄	—	212	2	no	—
102	C₂×C₄	C₄×#6157	1	1	yes	characteristic
103	C₈	—	53	1	no	—
104	C₈	—	53	1	no	—
105	C₈	—	106	2	no	—
106	C₈	C₄×#6157	1	1	yes	characteristic
107	C₈	C₂²×#6157	1	1	yes	characteristic
108	C₂²	—	53	1	no	—
109	C₂²	—	53	1	no	—
110	C₂²	—	106	2	no	—
111	C₂²	#13133271	1	1	yes	characteristic
112	C₄	—	53	1	no	—
113	C₄	—	53	1	no	—
114	C₄	—	212	2	no	—
115	C₄	#13133271	1	1	yes	characteristic
116	C₄	C₂×C₄×#6157	1	1	yes	characteristic
117	C₂	—	106	1	no	—
118	C₂	—	106	1	no	—
119	C₂	#16811514	1	2	yes	direct product factor
120	C₂	C₂×#13133271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133292

C₂

yes

maximal, characteristic

C₂×#13133271

C₂

yes

maximal, characteristic

#16811514

C₂

yes

maximal, direct product factor

C₂×#13133156

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₄

yes

characteristic

C₂×C₄×#6157

C₄

yes

characteristic

#13133292

C₄

yes

complemented

#13133292

C₂²

yes

—

C₂×#11719910

C₄

yes

complemented

#13133271

C₄

yes

—

#13133271

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133156

C₄

yes

complemented

#13133156

C₂²

yes

—

C₂×#11719908

C₄

yes

characteristic, complemented

C₂×C₁₆×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

—

C₂×C₄

yes

—

D₅₃⋊C₈

—

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

complemented

C₁₆×C₅₃

C₂×C₄

yes

complemented

#11719908

C₂×C₄

yes

complemented

C₂²×D₅₃

M₁₆

yes

characteristic

C₂×#6157

—

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

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×#97

yes

characteristic

D₅₃

—

D₅₃

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×#97

—

#347

—

C₅₃

C₂×#347

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#6157

yes

characteristic, complemented

#97

—

#97

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄²

—

C₄²

—

106

—

C₁₆

—

C₁₆

—

C₁₆

C₂×#6157

yes

complemented

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

106

—

101

C₂×C₄

—

212

—

102

C₂×C₄

C₄×#6157

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

106

—

106

C₈

C₄×#6157

yes

characteristic

107

C₈

C₂²×#6157

yes

characteristic

108

C₂²

—

109

C₂²

—

110

C₂²

—

106

—

111

C₂²

#13133271

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

212

—

115

C₄

#13133271

yes

characteristic

116

C₄

C₂×C₄×#6157

yes

characteristic

117

C₂

—

106

—

118

C₂

—

106

—

119

C₂

#16811514

yes

direct product factor

120

C₂

C₂×#13133271

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#6157
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₈	C₂²×#6157
Fitting	C₂×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#97
hypercenter	C₂×C₁₆	#6157
hypocenter	C₅₃	C₂×#347
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

subgroup

G/H

center

C₂×C₄

C₄×#6157

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₈

C₂²×#6157

Fitting

C₂×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂×C₁₆

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

N	H ≅ G/N	type
3. #16811514	C₂	direct (N × H)
7. #13133292	C₄	semidirect
9. C₂×#11719910	C₄	semidirect
13. #13133156	C₄	semidirect
15. C₂×#11719908	C₄	semidirect
16. C₂×C₁₆×C₅₃	C₄	semidirect
30. #11719910	C₂×C₄	semidirect
31. C₁₆×C₅₃	C₂×C₄	semidirect
32. #11719908	C₂×C₄	semidirect
57. C₂×C₅₃	#347	semidirect
65. C₅₃	C₂×#347	semidirect
75. C₂×C₁₆	#6157	semidirect
93. C₁₆	C₂×#6157	semidirect
119. C₂	#16811514	direct (N × H)

H ≅ G/N

type

3. #16811514

C₂

direct (N × H)

7. #13133292

C₄

semidirect

9. C₂×#11719910

C₄

semidirect

13. #13133156

C₄

semidirect

15. C₂×#11719908

C₄

semidirect

16. C₂×C₁₆×C₅₃

C₄

semidirect

30. #11719910

C₂×C₄

semidirect

31. C₁₆×C₅₃

C₂×C₄

semidirect

32. #11719908

C₂×C₄

semidirect

57. C₂×C₅₃

#347

semidirect

65. C₅₃

C₂×#347

semidirect

75. C₂×C₁₆

#6157

semidirect

93. C₁₆

C₂×#6157

semidirect

119. C₂

#16811514

direct (N × H)

Subgroups — C₂×#16811514 #20290929

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811514 #20290929

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811514 #20290929