Group #18019659 — subgroups

Subgroups — C₁₉×#13700 #18019659

Order 4864.

188 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 70 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×C₁₉×D₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#2258	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2221	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1425	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1419	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×D₁₆	C₂²	1	2	yes	—
7	C₁₉×#504	C₂²	1	1	yes	characteristic
8	C₁₉×#472	—	2	2	no	—
9	C₁₉×D₃₂	—	2	1	no	—
10	C₁₉×D₃₂	—	2	1	no	—
11	C₁₉×M₆₄	—	2	1	no	—
12	C₁₉×M₆₄	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×C₃₂	—	2	1	no	—
14	C₂×C₁₉×C₃₂	C₂²	1	1	yes	characteristic
15	C₁₉×#368	C₂²	1	2	yes	—
16	C₁₉×#361	—	2	2	no	—
17	C₁₉×AGL(1,ℤ/8)	—	4	2	no	—
18	C₂×D₈×C₁₉	—	2	2	no	—
19	C₁₉×#131	D₄	1	1	yes	characteristic
20	C₁₉×D₁₆	—	2	2	no	—
21	C₁₉×D₁₆	—	2	2	no	—
22	C₁₉×M₃₂	—	2	1	no	—
23	C₁₉×M₃₂	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₁₉	—	2	1	no	—
25	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
26	C₁₉×#108	—	2	2	no	—
27	C₁₉×#100	—	4	2	no	—
28	C₁₉×C₃₂	—	2	1	no	—
29	C₁₉×C₃₂	D₄	1	1	yes	characteristic
30	C₁₉×C₃₂	D₄	1	1	yes	characteristic
31	#41×C₁₉	D₈	1	1	yes	characteristic
32	C₂×D₄×C₁₉	—	4	2	no	—
33	SD₁₆×C₁₉	—	8	2	no	—
34	D₈×C₁₉	—	4	2	no	—
35	D₈×C₁₉	—	4	2	no	—
36	D₈×C₁₉	—	8	2	no	—
37	M₁₆×C₁₉	—	2	1	no	—
38	M₁₆×C₁₉	—	4	2	no	—
39	M₁₆×C₁₉	D₈	1	1	yes	characteristic
40	C₂×C₈×C₁₉	—	2	1	no	—
41	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₁₉	—	2	1	no	—
43	C₁₆×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₁₉	#41	1	1	yes	characteristic
45	#13700	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₁₉	—	8	2	no	—
47	Q₈×C₁₉	D₁₆	1	1	yes	characteristic
48	D₄×C₁₉	—	2	1	no	—
49	D₄×C₁₉	—	8	2	no	—
50	D₄×C₁₉	—	8	2	no	—
51	D₄×C₁₉	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
54	C₈×C₁₉	—	2	1	no	—
55	C₈×C₁₉	—	8	2	no	—
56	C₈×C₁₉	#121	1	1	yes	characteristic
57	C₈×C₁₉	#135	1	1	yes	characteristic
58	C₂×D₃₂	C₂×C₁₉	1	1	yes	characteristic, complemented
59	#2258	C₂×C₁₉	1	1	yes	characteristic, complemented
60	#2221	C₂×C₁₉	1	2	yes	complemented
61	#1425	C₂×C₁₉	1	1	yes	characteristic, complemented
62	#1419	C₂×C₁₉	1	2	yes	complemented
63	C₂²×C₁₉	—	2	1	no	—
64	C₂²×C₁₉	—	16	2	no	—
65	C₂²×C₁₉	—	32	2	no	—
66	C₂²×C₁₉	#508	1	1	yes	characteristic
67	C₄×C₁₉	—	2	1	no	—
68	C₄×C₁₉	#466	1	1	yes	characteristic
69	C₄×C₁₉	C₂×D₁₆	1	1	yes	characteristic
70	C₂×D₁₆	C₂²×C₁₉	1	2	yes	—
71	#504	C₂²×C₁₉	1	1	yes	characteristic
72	#472	—	2	2	no	—
73	D₃₂	—	2	1	no	—
74	D₃₂	—	2	1	no	—
75	M₆₄	—	2	1	no	—
76	M₆₄	C₂²×C₁₉	1	1	yes	characteristic
77	C₂×C₃₂	—	2	1	no	—
78	C₂×C₃₂	C₂²×C₁₉	1	1	yes	characteristic
79	#368	C₂²×C₁₉	1	2	yes	—
80	#361	—	2	2	no	—
81	C₂×C₁₉	—	2	1	no	—
82	C₂×C₁₉	—	4	1	no	—
83	C₂×C₁₉	—	32	2	no	—
84	C₂×C₁₉	#2215	1	1	yes	characteristic
85	AGL(1,ℤ/8)	—	4	2	no	—
86	C₂×D₈	—	2	2	no	—
87	#131	D₄×C₁₉	1	1	yes	characteristic
88	D₁₆	—	2	2	no	—
89	D₁₆	—	2	2	no	—
90	M₃₂	—	2	1	no	—
91	M₃₂	D₄×C₁₉	1	1	yes	characteristic
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
94	#108	—	2	2	no	—
95	#100	—	4	2	no	—
96	C₃₂	—	2	1	no	—
97	C₃₂	D₄×C₁₉	1	1	yes	characteristic
98	C₃₂	D₄×C₁₉	1	1	yes	characteristic
99	C₁₉	#13700	1	1	yes	characteristic, direct product factor, Sylow
100	#41	D₈×C₁₉	1	1	yes	characteristic
101	C₂×D₄	—	4	2	no	—
102	SD₁₆	—	8	2	no	—
103	D₈	—	4	2	no	—
104	D₈	—	4	2	no	—
105	D₈	—	8	2	no	—
106	M₁₆	—	2	1	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	D₈×C₁₉	1	1	yes	characteristic
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	C₂×D₄×C₁₉	1	1	yes	characteristic
113	C₁₆	#41×C₁₉	1	1	yes	characteristic
114	C₂³	—	8	2	no	—
115	Q₈	C₁₉×D₁₆	1	1	yes	characteristic
116	D₄	—	2	1	no	—
117	D₄	—	8	2	no	—
118	D₄	—	8	2	no	—
119	D₄	C₁₉×D₁₆	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	8	2	no	—
124	C₈	C₁₉×#121	1	1	yes	characteristic
125	C₈	C₁₉×#135	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	16	2	no	—
128	C₂²	—	32	2	no	—
129	C₂²	C₁₉×#508	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	C₁₉×#466	1	1	yes	characteristic
132	C₄	C₂×C₁₉×D₁₆	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	4	1	no	—
135	C₂	—	32	2	no	—
136	C₂	C₁₉×#2215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₉×D₃₂

C₂

yes

maximal, characteristic, complemented

C₁₉×#2258

C₂

yes

maximal, characteristic, complemented

C₁₉×#2221

C₂

yes

maximal, complemented

C₁₉×#1425

C₂

yes

maximal, characteristic, complemented

C₁₉×#1419

C₂

yes

maximal, complemented

C₂×C₁₉×D₁₆

C₂²

yes

—

C₁₉×#504

C₂²

yes

characteristic

C₁₉×#472

—

C₁₉×D₃₂

—

C₁₉×D₃₂

—

C₁₉×M₆₄

—

C₁₉×M₆₄

C₂²

yes

characteristic

C₂×C₁₉×C₃₂

—

C₂×C₁₉×C₃₂

C₂²

yes

characteristic

C₁₉×#368

C₂²

yes

—

C₁₉×#361

—

C₁₉×AGL(1,ℤ/8)

—

C₂×D₈×C₁₉

—

C₁₉×#131

D₄

yes

characteristic

C₁₉×D₁₆

—

C₁₉×D₁₆

—

C₁₉×M₃₂

—

C₁₉×M₃₂

D₄

yes

characteristic

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

C₂³

yes

characteristic

C₁₉×#108

—

C₁₉×#100

—

C₁₉×C₃₂

—

C₁₉×C₃₂

D₄

yes

characteristic

C₁₉×C₃₂

D₄

yes

characteristic

#41×C₁₉

D₈

yes

characteristic

C₂×D₄×C₁₉

—

SD₁₆×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

D₈

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

C₂×D₄

yes

C₁₆×C₁₉

yes

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

Q₈×C₁₉

D₁₆

yes

characteristic

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

D₁₆

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

C₂×D₃₂

C₂×C₁₉

yes

characteristic, complemented

#2258

C₂×C₁₉

yes

characteristic, complemented

#2221

C₂×C₁₉

yes

complemented

#1425

C₂×C₁₉

yes

characteristic, complemented

#1419

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#508

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

#466

yes

characteristic

C₄×C₁₉

C₂×D₁₆

yes

characteristic

C₂×D₁₆

C₂²×C₁₉

yes

—

#504

C₂²×C₁₉

yes

characteristic

#472

—

D₃₂

—

D₃₂

—

M₆₄

—

M₆₄

C₂²×C₁₉

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

C₂²×C₁₉

yes

characteristic

#368

C₂²×C₁₉

yes

—

#361

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#2215

yes

characteristic

AGL(1,ℤ/8)

—

C₂×D₈

—

#131

D₄×C₁₉

yes

characteristic

D₁₆

—

D₁₆

—

M₃₂

—

M₃₂

D₄×C₁₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂³×C₁₉

yes

characteristic

#108

—

#100

—

C₃₂

—

C₃₂

D₄×C₁₉

yes

characteristic

C₃₂

D₄×C₁₉

yes

characteristic

C₁₉

#13700

yes

characteristic, direct product factor, Sylow

100

#41

D₈×C₁₉

yes

characteristic

101

C₂×D₄

—

102

SD₁₆

—

103

D₈

—

104

D₈

—

105

D₈

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

D₈×C₁₉

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

111

C₁₆

—

112

C₁₆

C₂×D₄×C₁₉

yes

characteristic

113

C₁₆

#41×C₁₉

yes

characteristic

114

C₂³

—

115

Q₈

C₁₉×D₁₆

yes

characteristic

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

C₁₉×D₁₆

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

C₁₉×#121

yes

characteristic

125

C₈

C₁₉×#135

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₁₉×#508

yes

characteristic

130

C₄

—

131

C₄

C₁₉×#466

yes

characteristic

132

C₄

C₂×C₁₉×D₁₆

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

—

136

C₂

C₁₉×#2215

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₉
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₁₉×#131 ◁ C₁₉×#504 ◁ G

subgroup

G/H

center, socle

C₂×C₁₉

#2215

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₁₉

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₁₉×#131 ◁ C₁₉×#504 ◁ G

N	H ≅ G/N	type
1. C₂×C₁₉×D₃₂	C₂	semidirect
2. C₁₉×#2258	C₂	semidirect
3. C₁₉×#2221	C₂	semidirect
4. C₁₉×#1425	C₂	semidirect
5. C₁₉×#1419	C₂	semidirect
45. #13700	C₁₉	direct (N × H)
58. C₂×D₃₂	C₂×C₁₉	semidirect
59. #2258	C₂×C₁₉	semidirect
60. #2221	C₂×C₁₉	semidirect
61. #1425	C₂×C₁₉	semidirect
62. #1419	C₂×C₁₉	semidirect
99. C₁₉	#13700	direct (N × H)

H ≅ G/N

type

1. C₂×C₁₉×D₃₂

C₂

semidirect

2. C₁₉×#2258

C₂

semidirect

3. C₁₉×#2221

C₂

semidirect

4. C₁₉×#1425

C₂

semidirect

5. C₁₉×#1419

C₂

semidirect

45. #13700

C₁₉

direct (N × H)

58. C₂×D₃₂

C₂×C₁₉

semidirect

59. #2258

C₂×C₁₉

semidirect

60. #2221

C₂×C₁₉

semidirect

61. #1425

C₂×C₁₉

semidirect

62. #1419

C₂×C₁₉

semidirect

99. C₁₉

#13700

direct (N × H)

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

subgroup	index
C₂×C₁₉×D₃₂	2
C₁₉×#2258	2
C₁₉×#2221	2
C₁₉×#1425	2
C₁₉×#1419	2
#13700	19

subgroup

index

C₂×C₁₉×D₃₂

C₁₉×#2258

C₁₉×#2221

C₁₉×#1425

C₁₉×#1419

#13700

Subgroups — C₁₉×#13700 #18019659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#13700 #18019659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13700 #18019659