Subgroups — C 2 × C 19 × AGL(1,ℤ/8) #11926249

maximal, direct product factor

C₂

yes

C₂×SD₁₆×C₁₉

C₂

yes

maximal, complemented

C₂×D₈×C₁₉

C₂

yes

maximal, complemented

C₂×M₁₆×C₁₉

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₁₉

—

#41×C₁₉

—

#41×C₁₉

C₂²

yes

C₂×Q₈×C₁₉

C₂²

yes

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₂²

yes

—

C₂×D₄×C₁₉

C₂²

yes

C₂×D₄×C₁₉

C₂²

yes

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²

yes

SD₁₆×C₁₉

C₂²

yes

D₈×C₁₉

C₂²

yes

M₁₆×C₁₉

C₂²

yes

C₂×C₈×C₁₉

C₂²

yes

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

D₄

yes

Q₈×C₁₉

—

Q₈×C₁₉

C₂³

yes

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

C₂³

yes

—

D₄×C₁₉

C₂³

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

D₄

yes

—

C₂×C₄×C₁₉

D₄

yes

C₂×C₄×C₁₉

C₂³

yes

—

C₂×C₄×C₁₉

C₂³

yes

C₈×C₁₉

C₂³

yes

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×D₄

yes

—

C₂²×C₁₉

C₂×D₄

yes

C₄×C₁₉

—

C₄×C₁₉

C₂×D₄

yes

—

C₄×C₁₉

C₂×D₄

yes

C₄×C₁₉

C₂⁴

yes

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

AGL(1,ℤ/8)

yes

direct product factor

C₂×C₁₉

C₂²×D₄

yes

C₂×#41

C₂×C₁₉

yes

C₂²×D₄

C₂×C₁₉

yes

AGL(1,ℤ/8)

C₂×C₁₉

yes

direct product factor

C₂×SD₁₆

C₂×C₁₉

yes

C₂×D₈

C₂×C₁₉

yes

C₂×M₁₆

C₂×C₁₉

yes

C₁₉

characteristic, direct product factor, Sylow

yes

C₂⁴

—

#41

—

#41

C₂²×C₁₉

yes

C₂×Q₈

C₂²×C₁₉

yes

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×C₁₉

yes

—

C₂×D₄

C₂²×C₁₉

yes

C₂×D₄

C₂²×C₁₉

yes

C₂²×C₄

—

C₂²×C₄

C₂²×C₁₉

yes

SD₁₆

C₂²×C₁₉

yes

D₈

C₂²×C₁₉

yes

M₁₆

C₂²×C₁₉

yes

C₂×C₈

C₂²×C₁₉

yes

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

D₄×C₁₉

yes

Q₈

—

Q₈

C₂³×C₁₉

yes

D₄

—

D₄

—

D₄

—

D₄

—

D₄

C₂³×C₁₉

yes

—

D₄

C₂³×C₁₉

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×C₁₉

yes

100

C₂×C₄

D₄×C₁₉

yes

—

101

C₂×C₄

C₂³×C₁₉

yes

102

C₂×C₄

C₂³×C₁₉

yes

—

103

C₈

C₂³×C₁₉

yes

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×D₄×C₁₉

yes

—

111

C₂²

C₂×D₄×C₁₉

yes

112

C₄

—

113

C₄

C₂×D₄×C₁₉

yes

—

114

C₄

C₂×D₄×C₁₉

yes

115

C₄

C₂⁴×C₁₉

yes

116

C₂

—

117

C₂

—

118

C₂

—

119

C₂

yes

direct product factor

120

C₂

C₂²×D₄×C₁₉

yes

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₁₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

C₂×D₄

commutator, Frattini, cosocle

C₄

C₂⁴×C₁₉

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ G

N	H ≅ G/N	type
1. C₂×#41×C₁₉	C₂	semidirect
2. C₂²×D₄×C₁₉	C₂	semidirect
3. C₁₉×AGL(1,ℤ/8)	C₂	direct (N × H)
4. C₂×SD₁₆×C₁₉	C₂	semidirect
5. C₂×D₈×C₁₉	C₂	semidirect
6. C₂×M₁₆×C₁₉	C₂	semidirect
9. #41×C₁₉	C₂²	semidirect
10. C₂×Q₈×C₁₉	C₂²	semidirect
15. C₂×D₄×C₁₉	C₂²	semidirect
16. C₂×D₄×C₁₉	C₂²	semidirect
19. SD₁₆×C₁₉	C₂²	semidirect
20. D₈×C₁₉	C₂²	semidirect
21. M₁₆×C₁₉	C₂²	semidirect
22. C₂×C₈×C₁₉	C₂²	semidirect
29. Q₈×C₁₉	C₂³	semidirect
35. D₄×C₁₉	C₂³	semidirect
42. C₈×C₁₉	C₂³	semidirect
55. C₂×AGL(1,ℤ/8)	C₁₉	direct (N × H)
59. C₂×C₁₉	AGL(1,ℤ/8)	direct (N × H)
61. C₂×#41	C₂×C₁₉	semidirect
62. C₂²×D₄	C₂×C₁₉	semidirect
63. AGL(1,ℤ/8)	C₂×C₁₉	direct (N × H)
64. C₂×SD₁₆	C₂×C₁₉	semidirect
65. C₂×D₈	C₂×C₁₉	semidirect
66. C₂×M₁₆	C₂×C₁₉	semidirect
67. C₁₉	C₂×AGL(1,ℤ/8)	direct (N × H)
70. #41	C₂²×C₁₉	semidirect
71. C₂×Q₈	C₂²×C₁₉	semidirect
76. C₂×D₄	C₂²×C₁₉	semidirect
77. C₂×D₄	C₂²×C₁₉	semidirect
80. SD₁₆	C₂²×C₁₉	semidirect
81. D₈	C₂²×C₁₉	semidirect
82. M₁₆	C₂²×C₁₉	semidirect
83. C₂×C₈	C₂²×C₁₉	semidirect
90. Q₈	C₂³×C₁₉	semidirect
96. D₄	C₂³×C₁₉	semidirect
103. C₈	C₂³×C₁₉	semidirect
119. C₂	C₁₉×AGL(1,ℤ/8)	direct (N × H)

H ≅ G/N

type

1. C₂×#41×C₁₉

C₂

semidirect

2. C₂²×D₄×C₁₉

C₂

semidirect

3. C₁₉×AGL(1,ℤ/8)

C₂

direct (N × H)

4. C₂×SD₁₆×C₁₉

C₂

semidirect

5. C₂×D₈×C₁₉

C₂

semidirect

6. C₂×M₁₆×C₁₉

C₂

semidirect

9. #41×C₁₉

C₂²

semidirect

10. C₂×Q₈×C₁₉

C₂²

semidirect

15. C₂×D₄×C₁₉

C₂²

semidirect

16. C₂×D₄×C₁₉

C₂²

semidirect

19. SD₁₆×C₁₉

C₂²

semidirect

20. D₈×C₁₉

C₂²

semidirect

21. M₁₆×C₁₉

C₂²

semidirect

22. C₂×C₈×C₁₉

C₂²

semidirect

29. Q₈×C₁₉

C₂³

semidirect

35. D₄×C₁₉

C₂³

semidirect

42. C₈×C₁₉

C₂³

semidirect

55. C₂×AGL(1,ℤ/8)

C₁₉

direct (N × H)

59. C₂×C₁₉

AGL(1,ℤ/8)

direct (N × H)

61. C₂×#41

C₂×C₁₉

semidirect

62. C₂²×D₄

C₂×C₁₉

semidirect

63. AGL(1,ℤ/8)

C₂×C₁₉

direct (N × H)

64. C₂×SD₁₆

C₂×C₁₉

semidirect

65. C₂×D₈

C₂×C₁₉

semidirect

66. C₂×M₁₆

C₂×C₁₉

semidirect

67. C₁₉

direct (N × H)

70. #41

C₂²×C₁₉

semidirect

71. C₂×Q₈

C₂²×C₁₉

semidirect

76. C₂×D₄

C₂²×C₁₉

semidirect

77. C₂×D₄

C₂²×C₁₉

semidirect

80. SD₁₆

C₂²×C₁₉

semidirect

81. D₈

C₂²×C₁₉

semidirect

82. M₁₆

C₂²×C₁₉

semidirect

83. C₂×C₈

C₂²×C₁₉

semidirect

90. Q₈

C₂³×C₁₉

semidirect

96. D₄

C₂³×C₁₉

semidirect

103. C₈

C₂³×C₁₉

semidirect

119. C₂

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#41×C₁₉	2
C₂²×D₄×C₁₉	2
C₁₉×AGL(1,ℤ/8)	2
C₂×SD₁₆×C₁₉	2
C₂×D₈×C₁₉	2
C₂×M₁₆×C₁₉	2
C₂×AGL(1,ℤ/8)	19

subgroup

index

C₂×#41×C₁₉

C₂²×D₄×C₁₉

C₂×SD₁₆×C₁₉

C₂×D₈×C₁₉

C₂×M₁₆×C₁₉