Group #18019654 — subgroups

Subgroups — C₁₉×#13695 #18019654

Order 4864.

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₉×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2216	C₂	1	2	yes	maximal
4	C₁₉×#1423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1416	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×Q₃₂	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×#467	—	2	2	no	—
9	C₁₉×Q₆₄	—	2	2	no	—
10	C₂×C₁₉×C₃₂	—	2	2	no	—
11	C₂×C₁₉×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×C₃₂	C₂²	1	1	yes	characteristic
13	C₁₉×#366	C₂²	1	2	yes	—
14	C₁₉×#358	—	2	2	no	—
15	C₂×Q₁₆×C₁₉	—	2	2	no	—
16	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
17	C₁₉×C₂²⋊Q₈	—	4	2	no	—
18	C₁₉×Q₃₂	—	2	2	no	—
19	C₁₉×Q₃₂	—	2	2	no	—
20	C₂×C₁₆×C₁₉	—	2	1	no	—
21	C₂×C₁₆×C₁₉	—	2	1	no	—
22	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
24	C₁₉×#107	—	2	2	no	—
25	C₁₉×#103	—	4	2	no	—
26	C₁₉×C₃₂	—	2	1	no	—
27	C₁₉×C₃₂	D₄	1	2	yes	—
28	C₂×Q₈×C₁₉	—	4	2	no	—
29	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
30	Q₁₆×C₁₉	—	4	2	no	—
31	Q₁₆×C₁₉	—	4	2	no	—
32	C₂×C₈×C₁₉	—	2	1	no	—
33	C₂×C₈×C₁₉	—	2	1	no	—
34	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₉	—	4	2	no	—
37	C₄⋊C₄×C₁₉	—	8	2	no	—
38	C₂²⋊C₄×C₁₉	—	8	2	no	—
39	C₁₆×C₁₉	—	2	1	no	—
40	C₁₆×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₁₉	#41	1	1	yes	characteristic
42	#13695	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂³×C₁₉	D₁₆	1	1	yes	characteristic
44	Q₈×C₁₉	—	8	2	no	—
45	Q₈×C₁₉	—	8	2	no	—
46	C₂×C₄×C₁₉	—	2	1	no	—
47	C₂×C₄×C₁₉	—	2	1	no	—
48	C₂×C₄×C₁₉	—	8	2	no	—
49	C₂×C₄×C₁₉	—	8	2	no	—
50	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
51	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
52	C₈×C₁₉	—	2	1	no	—
53	C₈×C₁₉	#121	1	1	yes	characteristic
54	C₈×C₁₉	#135	1	1	yes	characteristic
55	C₂×Q₆₄	C₂×C₁₉	1	1	yes	characteristic, complemented
56	C₂²×C₃₂	C₂×C₁₉	1	1	yes	characteristic
57	#2216	C₂×C₁₉	1	2	yes	—
58	#1423	C₂×C₁₉	1	1	yes	characteristic, complemented
59	#1416	C₂×C₁₉	1	2	yes	complemented
60	C₂²×C₁₉	—	2	1	no	—
61	C₂²×C₁₉	—	2	1	no	—
62	C₂²×C₁₉	Q₆₄	1	2	yes	complemented
63	C₂²×C₁₉	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₁₉	—	2	1	no	—
65	C₄×C₁₉	—	16	2	no	—
66	C₄×C₁₉	—	16	2	no	—
67	C₄×C₁₉	#466	1	1	yes	characteristic
68	C₄×C₁₉	#508	1	1	yes	characteristic
69	C₂×Q₃₂	C₂²×C₁₉	1	2	yes	—
70	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
71	#467	—	2	2	no	—
72	Q₆₄	—	2	2	no	—
73	C₂×C₃₂	—	2	2	no	—
74	C₂×C₃₂	C₂²×C₁₉	1	1	yes	characteristic
75	C₂×C₃₂	C₂²×C₁₉	1	1	yes	characteristic
76	#366	C₂²×C₁₉	1	2	yes	—
77	#358	—	2	2	no	—
78	C₂×C₁₉	—	2	2	no	—
79	C₂×C₁₉	#2215	1	1	yes	characteristic
80	C₂×C₁₉	C₂×Q₆₄	1	1	yes	characteristic
81	C₂×C₁₉	#2262	1	1	yes	characteristic
82	C₂×Q₁₆	—	2	2	no	—
83	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
84	C₂²⋊Q₈	—	4	2	no	—
85	Q₃₂	—	2	2	no	—
86	Q₃₂	—	2	2	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
90	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
91	#107	—	2	2	no	—
92	#103	—	4	2	no	—
93	C₃₂	—	2	1	no	—
94	C₃₂	D₄×C₁₉	1	2	yes	—
95	C₁₉	#13695	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×Q₈	—	4	2	no	—
97	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
98	Q₁₆	—	4	2	no	—
99	Q₁₆	—	4	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	8	2	no	—
106	C₂²⋊C₄	—	8	2	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	C₂×D₄×C₁₉	1	1	yes	characteristic
109	C₁₆	#41×C₁₉	1	1	yes	characteristic
110	C₂³	C₁₉×D₁₆	1	1	yes	characteristic
111	Q₈	—	8	2	no	—
112	Q₈	—	8	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	C₁₉×D₁₆	1	1	yes	characteristic
118	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	C₁₉×#121	1	1	yes	characteristic
121	C₈	C₁₉×#135	1	1	yes	characteristic
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₁₉×Q₆₄	1	2	yes	complemented
125	C₂²	C₂×C₁₉×D₁₆	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	16	2	no	—
128	C₄	—	16	2	no	—
129	C₄	C₁₉×#466	1	1	yes	characteristic
130	C₄	C₁₉×#508	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	C₁₉×#2215	1	1	yes	characteristic
133	C₂	C₂×C₁₉×Q₆₄	1	1	yes	characteristic
134	C₂	C₁₉×#2262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₉×Q₆₄

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₉×C₃₂

C₂

yes

maximal, characteristic

C₁₉×#2216

C₂

yes

maximal

C₁₉×#1423

C₂

yes

maximal, characteristic, complemented

C₁₉×#1416

C₂

yes

maximal, complemented

C₂×C₁₉×Q₃₂

C₂²

yes

—

C₂²×C₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#467

—

C₁₉×Q₆₄

—

C₂×C₁₉×C₃₂

—

C₂×C₁₉×C₃₂

C₂²

yes

characteristic

C₂×C₁₉×C₃₂

C₂²

yes

characteristic

C₁₉×#366

C₂²

yes

—

C₁₉×#358

—

C₂×Q₁₆×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₁₉×C₂²⋊Q₈

—

C₁₉×Q₃₂

—

C₁₉×Q₃₂

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

C₂³

yes

characteristic

C₁₉×#107

—

C₁₉×#103

—

C₁₉×C₃₂

—

C₁₉×C₃₂

D₄

yes

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

D₈

yes

characteristic

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

characteristic

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

C₂×D₄

yes

C₁₆×C₁₉

yes

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

D₁₆

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

D₁₆

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

C₂×Q₆₄

C₂×C₁₉

yes

characteristic, complemented

C₂²×C₃₂

C₂×C₁₉

yes

characteristic

#2216

C₂×C₁₉

yes

—

#1423

C₂×C₁₉

yes

characteristic, complemented

#1416

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

Q₆₄

yes

complemented

C₂²×C₁₉

C₂×D₁₆

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×Q₃₂

C₂²×C₁₉

yes

—

C₂²×C₁₆

C₂²×C₁₉

yes

characteristic

#467

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂²×C₁₉

yes

characteristic

C₂×C₃₂

C₂²×C₁₉

yes

characteristic

#366

C₂²×C₁₉

yes

—

#358

—

C₂×C₁₉

—

C₂×C₁₉

#2215

yes

characteristic

C₂×C₁₉

C₂×Q₆₄

yes

characteristic

C₂×C₁₉

#2262

yes

characteristic

C₂×Q₁₆

—

C₂²×C₈

D₄×C₁₉

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₉

yes

characteristic

C₂×C₁₆

C₂³×C₁₉

yes

characteristic

#107

—

#103

—

C₃₂

—

C₃₂

D₄×C₁₉

yes

—

C₁₉

#13695

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂²×C₄

D₈×C₁₉

yes

characteristic

Q₁₆

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

D₈×C₁₉

yes

characteristic

103

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₁₆

—

108

C₁₆

C₂×D₄×C₁₉

yes

characteristic

109

C₁₆

#41×C₁₉

yes

characteristic

110

C₂³

C₁₉×D₁₆

yes

characteristic

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₁₉×D₁₆

yes

characteristic

118

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

119

C₈

—

120

C₈

C₁₉×#121

yes

characteristic

121

C₈

C₁₉×#135

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

C₁₉×Q₆₄

yes

complemented

125

C₂²

C₂×C₁₉×D₁₆

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₁₉×#466

yes

characteristic

130

C₄

C₁₉×#508

yes

characteristic

131

C₂

—

132

C₂

C₁₉×#2215

yes

characteristic

133

C₂

C₂×C₁₉×Q₆₄

yes

characteristic

134

C₂

C₁₉×#2262

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₉
prime radical	C₂³×C₁₉	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂²×C₈×C₁₉ ◁ C₂²×C₁₆×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₁₉

prime radical

C₂³×C₁₉

D₁₆

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂²×C₈×C₁₉ ◁ C₂²×C₁₆×C₁₉ ◁ G

N	H ≅ G/N	type
1. C₂×C₁₉×Q₆₄	C₂	semidirect
4. C₁₉×#1423	C₂	semidirect
5. C₁₉×#1416	C₂	semidirect
42. #13695	C₁₉	direct (N × H)
55. C₂×Q₆₄	C₂×C₁₉	semidirect
58. #1423	C₂×C₁₉	semidirect
59. #1416	C₂×C₁₉	semidirect
62. C₂²×C₁₉	Q₆₄	semidirect
95. C₁₉	#13695	direct (N × H)
124. C₂²	C₁₉×Q₆₄	semidirect

H ≅ G/N

type

1. C₂×C₁₉×Q₆₄

C₂

semidirect

4. C₁₉×#1423

C₂

semidirect

5. C₁₉×#1416

C₂

semidirect

42. #13695

C₁₉

direct (N × H)

55. C₂×Q₆₄

C₂×C₁₉

semidirect

58. #1423

C₂×C₁₉

semidirect

59. #1416

C₂×C₁₉

semidirect

62. C₂²×C₁₉

Q₆₄

semidirect

95. C₁₉

#13695

direct (N × H)

124. C₂²

C₁₉×Q₆₄

semidirect

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

subgroup	index
C₂×C₁₉×Q₆₄	2
C₂²×C₁₉×C₃₂	2
C₁₉×#2216	2
C₁₉×#1423	2
C₁₉×#1416	2
#13695	19

subgroup

index

C₂×C₁₉×Q₆₄

C₂²×C₁₉×C₃₂

C₁₉×#2216

C₁₉×#1423

C₁₉×#1416

#13695

Subgroups — C₁₉×#13695 #18019654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#13695 #18019654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13695 #18019654