Group #19105961 — subgroups

Subgroups — C₄₁×AGL(1,ℤ/16) #19105961

Order 5248.

154 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 72 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	C₄₁×#509	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#442	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#360	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#347	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×D₈×C₄₁	C₂²	1	1	yes	characteristic
10	#131×C₄₁	—	2	1	no	—
11	C₄×D₄×C₄₁	—	2	1	no	—
12	SD₃₂×C₄₁	C₄	1	2	yes	complemented
13	D₁₆×C₄₁	C₄	1	2	yes	complemented
14	M₃₂×C₄₁	C₂²	1	1	yes	characteristic
15	M₃₂×C₄₁	C₂²	1	1	yes	characteristic
16	#108×C₄₁	C₂²	1	1	yes	characteristic
17	#106×C₄₁	C₂²	1	1	yes	characteristic
18	#104×C₄₁	—	2	1	no	—
19	#102×C₄₁	—	2	1	no	—
20	#97×C₄₁	C₂²	1	1	yes	characteristic
21	#41×C₄₁	—	2	1	no	—
22	C₂×D₄×C₄₁	—	2	1	no	—
23	C₂²×C₄×C₄₁	—	4	1	no	—
24	Q₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
25	SD₁₆×C₄₁	—	2	1	no	—
26	D₈×C₄₁	—	2	1	no	—
27	D₈×C₄₁	C₂×C₄	1	2	yes	—
28	D₈×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
29	M₁₆×C₄₁	—	2	1	no	—
30	M₁₆×C₄₁	—	4	1	no	—
31	C₂×C₈×C₄₁	—	4	1	no	—
32	C₂×C₈×C₄₁	D₄	1	1	yes	characteristic
33	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₄₁	—	2	1	no	—
35	C₂²⋊C₄×C₄₁	—	4	1	no	—
36	C₄²×C₄₁	D₄	1	1	yes	characteristic
37	C₁₆×C₄₁	—	2	1	no	—
38	C₁₆×C₄₁	C₂×C₄	1	2	yes	complemented
39	C₂³×C₄₁	—	4	1	no	—
40	Q₈×C₄₁	—	2	1	no	—
41	D₄×C₄₁	—	2	2	no	—
42	D₄×C₄₁	—	2	1	no	—
43	D₄×C₄₁	—	4	1	no	—
44	D₄×C₄₁	—	4	1	no	—
45	C₂×C₄×C₄₁	—	2	1	no	—
46	C₂×C₄×C₄₁	—	4	1	no	—
47	C₂×C₄×C₄₁	—	4	1	no	—
48	C₂×C₄×C₄₁	—	8	2	no	—
49	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
50	C₈×C₄₁	—	2	1	no	—
51	C₈×C₄₁	—	4	1	no	—
52	C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₄₁	#41	1	1	yes	characteristic
54	C₂²×C₄₁	—	4	1	no	—
55	C₂²×C₄₁	—	4	2	no	—
56	C₂²×C₄₁	—	8	1	no	—
57	C₂²×C₄₁	—	8	1	no	—
58	C₂²×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₄×C₄₁	—	4	1	no	—
60	C₄×C₄₁	—	4	1	no	—
61	C₄×C₄₁	—	8	1	no	—
62	C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
63	C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	AGL(1,ℤ/16)	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₄₁	—	2	1	no	—
66	C₂×C₄₁	—	8	1	no	—
67	C₂×C₄₁	—	8	2	no	—
68	C₂×C₄₁	#442	1	1	yes	characteristic
69	#509	C₂×C₄₁	1	1	yes	characteristic
70	#444	C₂×C₄₁	1	1	yes	characteristic, complemented
71	#442	C₂×C₄₁	1	1	yes	characteristic, complemented
72	#365	C₂×C₄₁	1	1	yes	characteristic, complemented
73	#361	C₂×C₄₁	1	1	yes	characteristic, complemented
74	#360	C₂×C₄₁	1	1	yes	characteristic, complemented
75	#347	C₂×C₄₁	1	1	yes	characteristic, complemented
76	C₄₁	AGL(1,ℤ/16)	1	1	yes	characteristic, direct product factor, Sylow
77	#135	C₂²×C₄₁	1	1	yes	characteristic
78	C₂×D₈	C₂²×C₄₁	1	1	yes	characteristic
79	#131	—	2	1	no	—
80	C₄×D₄	—	2	1	no	—
81	SD₃₂	C₄×C₄₁	1	2	yes	complemented
82	D₁₆	C₄×C₄₁	1	2	yes	complemented
83	M₃₂	C₂²×C₄₁	1	1	yes	characteristic
84	M₃₂	C₂²×C₄₁	1	1	yes	characteristic
85	#108	C₂²×C₄₁	1	1	yes	characteristic
86	#106	C₂²×C₄₁	1	1	yes	characteristic
87	#104	—	2	1	no	—
88	#102	—	2	1	no	—
89	#97	C₂²×C₄₁	1	1	yes	characteristic
90	#41	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂²×C₄	—	4	1	no	—
93	Q₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
94	SD₁₆	—	2	1	no	—
95	D₈	—	2	1	no	—
96	D₈	C₂×C₄×C₄₁	1	2	yes	—
97	D₈	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
98	M₁₆	—	2	1	no	—
99	M₁₆	—	4	1	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	D₄×C₄₁	1	1	yes	characteristic
102	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	—	4	1	no	—
105	C₄²	D₄×C₄₁	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	C₂×C₄×C₄₁	1	2	yes	complemented
108	C₂³	—	4	1	no	—
109	Q₈	—	2	1	no	—
110	D₄	—	2	2	no	—
111	D₄	—	2	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	4	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	4	1	no	—
121	C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
122	C₈	#41×C₄₁	1	1	yes	characteristic
123	C₂²	—	4	1	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	8	1	no	—
131	C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
132	C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	8	2	no	—
135	C₂	—	8	1	no	—
136	C₂	C₄₁×#442	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₁×#509

C₂

yes

maximal, characteristic

C₄₁×#444

C₂

yes

maximal, characteristic, complemented

C₄₁×#442

C₂

yes

maximal, characteristic, complemented

C₄₁×#365

C₂

yes

maximal, characteristic, complemented

C₄₁×#361

C₂

yes

maximal, characteristic, complemented

C₄₁×#360

C₂

yes

maximal, characteristic, complemented

C₄₁×#347

C₂

yes

maximal, characteristic, complemented

#135×C₄₁

C₂²

yes

characteristic

C₂×D₈×C₄₁

C₂²

yes

characteristic

#131×C₄₁

—

C₄×D₄×C₄₁

—

SD₃₂×C₄₁

C₄

yes

complemented

D₁₆×C₄₁

C₄

yes

complemented

M₃₂×C₄₁

C₂²

yes

characteristic

M₃₂×C₄₁

C₂²

yes

characteristic

#108×C₄₁

C₂²

yes

characteristic

#106×C₄₁

C₂²

yes

characteristic

#104×C₄₁

—

#102×C₄₁

—

#97×C₄₁

C₂²

yes

characteristic

#41×C₄₁

—

C₂×D₄×C₄₁

—

C₂²×C₄×C₄₁

—

Q₁₆×C₄₁

C₂×C₄

yes

characteristic, complemented

SD₁₆×C₄₁

—

D₈×C₄₁

—

D₈×C₄₁

C₂×C₄

yes

—

D₈×C₄₁

C₂×C₄

yes

characteristic, complemented

M₁₆×C₄₁

—

M₁₆×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

D₄

yes

characteristic

C₂×C₈×C₄₁

C₂³

yes

characteristic

C₄⋊C₄×C₄₁

—

C₂²⋊C₄×C₄₁

—

C₄²×C₄₁

D₄

yes

characteristic

C₁₆×C₄₁

—

C₁₆×C₄₁

C₂×C₄

yes

complemented

C₂³×C₄₁

—

Q₈×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₂×D₄

yes

characteristic

C₈×C₄₁

—

C₈×C₄₁

—

C₈×C₄₁

C₂²×C₄

yes

characteristic

C₈×C₄₁

#41

yes

characteristic

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

AGL(1,ℤ/8)

yes

characteristic

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

C₄×D₄

yes

C₄×C₄₁

yes

C₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

#442

yes

characteristic

#509

C₂×C₄₁

yes

characteristic

#444

C₂×C₄₁

yes

characteristic, complemented

#442

C₂×C₄₁

yes

characteristic, complemented

#365

C₂×C₄₁

yes

characteristic, complemented

#361

C₂×C₄₁

yes

characteristic, complemented

#360

C₂×C₄₁

yes

characteristic, complemented

#347

C₂×C₄₁

yes

characteristic, complemented

C₄₁

AGL(1,ℤ/16)

yes

characteristic, direct product factor, Sylow

#135

C₂²×C₄₁

yes

characteristic

C₂×D₈

C₂²×C₄₁

yes

characteristic

#131

—

C₄×D₄

—

SD₃₂

C₄×C₄₁

yes

complemented

D₁₆

C₄×C₄₁

yes

complemented

M₃₂

C₂²×C₄₁

yes

characteristic

M₃₂

C₂²×C₄₁

yes

characteristic

#108

C₂²×C₄₁

yes

characteristic

#106

C₂²×C₄₁

yes

—

—

C₂²×C₄₁

yes

characteristic

#41

—

C₂×D₄

—

C₂²×C₄

—

Q₁₆

C₂×C₄×C₄₁

yes

characteristic, complemented

SD₁₆

—

D₈

—

D₈

C₂×C₄×C₄₁

yes

—

D₈

C₂×C₄×C₄₁

yes

characteristic, complemented

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

D₄×C₄₁

yes

characteristic

102

C₂×C₈

C₂³×C₄₁

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

D₄×C₄₁

yes

characteristic

106

C₁₆

—

107

C₁₆

C₂×C₄×C₄₁

yes

complemented

108

C₂³

—

109

Q₈

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂×D₄×C₄₁

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

C₂²×C₄×C₄₁

yes

characteristic

122

C₈

#41×C₄₁

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

AGL(1,ℤ/8)×C₄₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₄×D₄×C₄₁

yes

characteristic

132

C₄

AGL(1,ℤ/8)×C₄₁

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

—

136

C₂

C₄₁×#442

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	#442
commutator	C₈	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₈	C₂³×C₄₁
prime radical	C₄₁×#509	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂×C₄×C₄₁ ◁ #97×C₄₁ ◁ G

subgroup

G/H

center, socle

C₂×C₄₁

#442

commutator

C₈

C₂²×C₄×C₄₁

Frattini, cosocle

C₂×C₈

C₂³×C₄₁

prime radical

C₄₁×#509

C₂

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄₁ ◁ C₂×C₄×C₄₁ ◁ #97×C₄₁ ◁ G

N	H ≅ G/N	type
2. C₄₁×#444	C₂	semidirect
3. C₄₁×#442	C₂	semidirect
4. C₄₁×#365	C₂	semidirect
5. C₄₁×#361	C₂	semidirect
6. C₄₁×#360	C₂	semidirect
7. C₄₁×#347	C₂	semidirect
12. SD₃₂×C₄₁	C₄	semidirect
13. D₁₆×C₄₁	C₄	semidirect
24. Q₁₆×C₄₁	C₂×C₄	semidirect
28. D₈×C₄₁	C₂×C₄	semidirect
38. C₁₆×C₄₁	C₂×C₄	semidirect
64. AGL(1,ℤ/16)	C₄₁	direct (N × H)
70. #444	C₂×C₄₁	semidirect
71. #442	C₂×C₄₁	semidirect
72. #365	C₂×C₄₁	semidirect
73. #361	C₂×C₄₁	semidirect
74. #360	C₂×C₄₁	semidirect
75. #347	C₂×C₄₁	semidirect
76. C₄₁	AGL(1,ℤ/16)	direct (N × H)
81. SD₃₂	C₄×C₄₁	semidirect
82. D₁₆	C₄×C₄₁	semidirect
93. Q₁₆	C₂×C₄×C₄₁	semidirect
97. D₈	C₂×C₄×C₄₁	semidirect
107. C₁₆	C₂×C₄×C₄₁	semidirect

H ≅ G/N

type

2. C₄₁×#444

C₂

semidirect

3. C₄₁×#442

C₂

semidirect

4. C₄₁×#365

C₂

semidirect

5. C₄₁×#361

C₂

semidirect

6. C₄₁×#360

C₂

semidirect

7. C₄₁×#347

C₂

semidirect

12. SD₃₂×C₄₁

C₄

semidirect

13. D₁₆×C₄₁

C₄

semidirect

24. Q₁₆×C₄₁

C₂×C₄

semidirect

28. D₈×C₄₁

C₂×C₄

semidirect

38. C₁₆×C₄₁

C₂×C₄

semidirect

64. AGL(1,ℤ/16)

C₄₁

direct (N × H)

70. #444

C₂×C₄₁

semidirect

71. #442

C₂×C₄₁

semidirect

72. #365

C₂×C₄₁

semidirect

73. #361

C₂×C₄₁

semidirect

74. #360

C₂×C₄₁

semidirect

75. #347

C₂×C₄₁

semidirect

76. C₄₁

AGL(1,ℤ/16)

direct (N × H)

81. SD₃₂

C₄×C₄₁

semidirect

82. D₁₆

C₄×C₄₁

semidirect

93. Q₁₆

C₂×C₄×C₄₁

semidirect

97. D₈

C₂×C₄×C₄₁

semidirect

107. C₁₆

C₂×C₄×C₄₁

semidirect

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

subgroup	index
C₄₁×#509	2
C₄₁×#444	2
C₄₁×#442	2
C₄₁×#365	2
C₄₁×#361	2
C₄₁×#360	2
C₄₁×#347	2
AGL(1,ℤ/16)	41

subgroup

index

C₄₁×#509

C₄₁×#444

C₄₁×#442

C₄₁×#365

C₄₁×#361

C₄₁×#360

C₄₁×#347

AGL(1,ℤ/16)

Subgroups — C₄₁×AGL(1,ℤ/16) #19105961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C41×AGL(1,ℤ/16) #19105961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×AGL(1,ℤ/16) #19105961