Group #20291059 — subgroups

Subgroups — C₂×#16811547 #20291059

Order 6784.

152 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 66 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×#13133287	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133285	C₂	1	1	yes	maximal, characteristic
3	#16811547	C₂	1	4	yes	maximal, direct product factor
4	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719936	—	2	1	no	—
6	C₂×C₄⋊#6157	—	2	1	no	—
7	#13133287	—	2	1	no	—
8	#13133287	C₂²	1	2	yes	—
9	#13133285	—	2	1	no	—
10	#13133285	C₂²	1	2	yes	—
11	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#13133156	C₄	1	2	yes	complemented
13	#13133156	C₂²	1	2	yes	—
14	C₂×#11719908	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#6157	—	4	1	no	—
17	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
18	C₂×#88324	—	4	1	no	—
19	C₄⋊#6157	—	2	2	no	—
20	C₄⋊#6157	—	4	1	no	—
21	#11719936	—	2	2	no	—
22	#11719936	—	4	1	no	—
23	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#88322	Q₈	1	1	yes	characteristic
25	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₈×D₅₃	D₄	1	1	yes	characteristic
27	C₈×D₅₃	Q₈	1	1	yes	characteristic
28	C₈×D₅₃	C₂³	1	1	yes	characteristic
29	C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
30	#11719908	C₂×C₄	1	2	yes	complemented
31	C₂²×D₅₃	SD₁₆	1	1	yes	characteristic
32	C₂×#6157	—	4	2	no	—
33	C₂×#6157	—	8	2	no	—
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
36	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
38	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
39	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
40	#88324	—	4	2	no	—
41	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	#88322	C₄⋊C₄	1	1	yes	characteristic
44	#88322	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₅₃	#106	1	1	yes	characteristic
46	C₂×D₅₃	—	2	2	no	—
47	C₂×D₅₃	#106	1	1	yes	characteristic
48	C₂×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
49	#6157	—	8	2	no	—
50	C₄×C₅₃	#106	1	1	yes	characteristic
51	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	Dic₅₃	#106	1	1	yes	characteristic
53	Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×#365	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	#365	1	2	yes	complemented
56	C₂×C₅₃	C₂×#106	1	1	yes	characteristic
57	D₅₃	—	2	1	no	—
58	D₅₃	—	2	1	no	—
59	C₂×M₃₂	—	53	1	no	—
60	C₂×#108	—	53	1	no	—
61	C₂×#106	—	53	1	no	—
62	#365	—	53	4	no	—
63	C₅₃	C₂×#365	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₁₆	—	106	1	no	—
65	C₂²×C₈	—	53	1	no	—
66	C₂×C₄⋊C₄	—	106	1	no	—
67	M₃₂	—	53	2	no	—
68	M₃₂	—	53	2	no	—
69	C₂×C₁₆	—	53	1	no	—
70	C₂×C₁₆	#6157	1	1	yes	characteristic, complemented
71	#108	—	53	2	no	—
72	#108	—	106	1	no	—
73	#106	—	53	2	no	—
74	#106	—	106	1	no	—
75	C₂²×C₄	—	53	1	no	—
76	C₂²×C₄	—	212	1	no	—
77	M₁₆	—	106	2	no	—
78	M₁₆	—	212	1	no	—
79	C₂×C₈	—	53	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	—	212	1	no	—
85	C₂×C₈	C₂×#6157	1	1	yes	characteristic
86	C₄⋊C₄	—	106	2	no	—
87	C₄⋊C₄	—	212	1	no	—
88	C₁₆	—	53	2	no	—
89	C₁₆	C₂×#6157	1	2	yes	complemented
90	C₂³	—	53	1	no	—
91	C₂×C₄	—	53	1	no	—
92	C₂×C₄	—	53	1	no	—
93	C₂×C₄	—	53	1	no	—
94	C₂×C₄	—	53	1	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	424	2	no	—
97	C₂×C₄	—	212	2	no	—
98	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
99	C₈	—	53	1	no	—
100	C₈	—	53	1	no	—
101	C₈	—	212	2	no	—
102	C₈	C₄⋊#6157	1	1	yes	characteristic
103	C₈	C₂²×#6157	1	1	yes	characteristic
104	C₂²	—	53	1	no	—
105	C₂²	—	53	1	no	—
106	C₂²	—	106	2	no	—
107	C₂²	#13133285	1	1	yes	characteristic
108	C₄	—	53	1	no	—
109	C₄	—	53	1	no	—
110	C₄	—	424	2	no	—
111	C₄	#13133285	1	1	yes	characteristic
112	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
113	C₂	—	106	1	no	—
114	C₂	—	106	1	no	—
115	C₂	#16811547	1	2	yes	direct product factor
116	C₂	C₂×#13133285	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133287

C₂

yes

maximal, characteristic

C₂×#13133285

C₂

yes

maximal, characteristic

#16811547

C₂

yes

maximal, direct product factor

C₂×#13133156

C₂

yes

maximal, characteristic

C₂×#11719936

—

C₂×C₄⋊#6157

—

#13133287

—

#13133287

C₂²

yes

—

#13133285

—

#13133285

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

D₄

yes

—

—

—

—

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×#88322

Q₈

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

C₈×D₅₃

C₂³

yes

characteristic

C₁₆×C₅₃

C₂×C₄

yes

complemented

#11719908

C₂×C₄

yes

complemented

C₂²×D₅₃

SD₁₆

yes

characteristic

C₂×#6157

—

C₂×#6157

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

#88324

—

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

#88322

C₂×Q₈

yes

characteristic

C₂²×C₅₃

#106

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

#106

yes

characteristic

C₂×D₅₃

C₂×SD₁₆

yes

characteristic

#6157

—

C₄×C₅₃

#106

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

Dic₅₃

#106

yes

characteristic

Dic₅₃

C₂×SD₁₆

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

C₂×M₃₂

—

C₂×#108

—

C₂×#106

—

#365

—

C₅₃

C₂×#365

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

106

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

#6157

yes

characteristic, complemented

#108

—

#108

—

106

—

#106

—

#106

—

106

—

C₂²×C₄

—

C₂²×C₄

—

212

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₁₆

—

C₁₆

C₂×#6157

yes

complemented

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂×C₄

C₄⋊#6157

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

212

—

102

C₈

C₄⋊#6157

yes

characteristic

103

C₈

C₂²×#6157

yes

characteristic

104

C₂²

—

105

C₂²

—

106

C₂²

—

106

—

107

C₂²

#13133285

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

424

—

111

C₄

#13133285

yes

characteristic

112

C₄

C₂×C₄⋊#6157

yes

characteristic

113

C₂

—

106

—

114

C₂

—

106

—

115

C₂

#16811547

yes

direct product factor

116

C₂

C₂×#13133285

yes

characteristic

	subgroup	G/H
center	C₂²	#13133285
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₈	C₂²×#6157
Fitting	C₂×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#106
hypercenter	C₂×C₁₆	#6157
hypocenter	C₅₃	C₂×#365
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₂²

#13133285

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₈

C₂²×#6157

Fitting

C₂×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂×C₁₆

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
3. #16811547	C₂	direct (N × H)
12. #13133156	C₄	semidirect
14. C₂×#11719908	C₄	semidirect
15. C₂×C₁₆×C₅₃	C₄	semidirect
29. C₁₆×C₅₃	C₂×C₄	semidirect
30. #11719908	C₂×C₄	semidirect
55. C₂×C₅₃	#365	semidirect
63. C₅₃	C₂×#365	semidirect
70. C₂×C₁₆	#6157	semidirect
89. C₁₆	C₂×#6157	semidirect
115. C₂	#16811547	direct (N × H)

H ≅ G/N

type

3. #16811547

C₂

direct (N × H)

12. #13133156

C₄

semidirect

14. C₂×#11719908

C₄

semidirect

15. C₂×C₁₆×C₅₃

C₄

semidirect

29. C₁₆×C₅₃

C₂×C₄

semidirect

30. #11719908

C₂×C₄

semidirect

55. C₂×C₅₃

#365

semidirect

63. C₅₃

C₂×#365

semidirect

70. C₂×C₁₆

#6157

semidirect

89. C₁₆

C₂×#6157

semidirect

115. C₂

#16811547

direct (N × H)

Subgroups — C₂×#16811547 #20291059

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811547 #20291059

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811547 #20291059