Group #18017986 — subgroups

Subgroups — C₁₉×#12027 #18017986

Order 4864.

180 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₁₉×#2166	C₂	1	2	yes	maximal
2	C₁₉×#2108	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1373	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1369	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×M₃₂	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₁₉	C₄	1	1	yes	characteristic
7	C₁₉×#405	C₂²	1	1	yes	characteristic
8	C₁₉×#364	C₂²	1	4	yes	—
9	C₁₉×#349	—	2	2	no	—
10	C₁₉×#347	—	2	2	no	—
11	C₁₉×#346	C₄	1	1	yes	characteristic
12	C₁₉×#346	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₁₉	C₄	1	1	yes	characteristic
14	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₁₉×#117	Q₈	1	1	yes	characteristic
18	C₁₉×M₃₂	—	2	2	no	—
19	C₁₉×M₃₂	—	2	4	no	—
20	C₂×C₁₆×C₁₉	—	2	2	no	—
21	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₁₉×#97	D₄	1	1	yes	characteristic
26	C₁₉×#97	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
30	M₁₆×C₁₉	—	2	2	no	—
31	C₂×C₈×C₁₉	—	2	1	no	—
32	C₂×C₈×C₁₉	—	2	1	no	—
33	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂²⋊C₄×C₁₉	—	2	1	no	—
39	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
41	C₁₆×C₁₉	—	2	1	no	—
42	C₁₆×C₁₉	—	2	1	no	—
43	C₁₆×C₁₉	—	4	4	no	—
44	C₁₆×C₁₉	M₁₆	1	2	yes	—
45	C₁₆×C₁₉	M₁₆	1	2	yes	—
46	#12027	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₁₉	#108	1	1	yes	characteristic
48	C₂×C₄×C₁₉	—	2	1	no	—
49	C₂×C₄×C₁₉	—	2	1	no	—
50	C₂×C₄×C₁₉	—	2	1	no	—
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	#108	1	1	yes	characteristic
53	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₈×C₁₉	—	2	1	no	—
55	C₈×C₁₉	—	2	1	no	—
56	C₈×C₁₉	C₂×M₁₆	1	1	yes	characteristic
57	C₈×C₁₉	C₂×M₁₆	1	1	yes	characteristic
58	C₈×C₁₉	#135	1	2	yes	—
59	#2166	C₂×C₁₉	1	2	yes	—
60	#2108	C₂×C₁₉	1	1	yes	characteristic
61	#1373	C₂×C₁₉	1	2	yes	complemented
62	#1369	C₂×C₁₉	1	2	yes	complemented
63	C₂²×C₁₉	—	2	1	no	—
64	C₂²×C₁₉	—	4	1	no	—
65	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
66	C₄×C₁₉	—	2	1	no	—
67	C₄×C₁₉	—	4	1	no	—
68	C₄×C₁₉	—	4	1	no	—
69	C₄×C₁₉	#423	1	1	yes	characteristic
70	C₄×C₁₉	#427	1	1	yes	characteristic
71	C₂×M₃₂	C₄×C₁₉	1	1	yes	characteristic
72	C₂²×C₁₆	C₄×C₁₉	1	1	yes	characteristic
73	#405	C₂²×C₁₉	1	1	yes	characteristic
74	#364	C₂²×C₁₉	1	4	yes	—
75	#349	—	2	2	no	—
76	#347	—	2	2	no	—
77	#346	C₄×C₁₉	1	1	yes	characteristic
78	#346	C₂²×C₁₉	1	1	yes	characteristic
79	C₄×C₁₆	C₄×C₁₉	1	1	yes	characteristic
80	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
81	C₂×C₁₉	—	2	1	no	—
82	C₂×C₁₉	—	4	1	no	—
83	C₂×C₁₉	#1568	1	1	yes	characteristic
84	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
85	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
86	#117	Q₈×C₁₉	1	1	yes	characteristic
87	M₃₂	—	2	2	no	—
88	M₃₂	—	2	4	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
91	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
92	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
93	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
94	#97	D₄×C₁₉	1	1	yes	characteristic
95	#97	Q₈×C₁₉	1	1	yes	characteristic
96	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
97	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
98	C₁₉	#12027	1	1	yes	characteristic, direct product factor, Sylow
99	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
104	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
107	C₄⋊C₄	—	2	1	no	—
108	C₂²⋊C₄	—	2	1	no	—
109	C₄²	C₄⋊C₄×C₁₉	1	1	yes	characteristic
110	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	4	4	no	—
114	C₁₆	M₁₆×C₁₉	1	2	yes	—
115	C₁₆	M₁₆×C₁₉	1	2	yes	—
116	C₂³	C₁₉×#108	1	1	yes	characteristic
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	C₁₉×#108	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	C₂×M₁₆×C₁₉	1	1	yes	characteristic
126	C₈	C₂×M₁₆×C₁₉	1	1	yes	characteristic
127	C₈	C₁₉×#135	1	2	yes	—
128	C₂²	—	2	1	no	—
129	C₂²	—	4	1	no	—
130	C₂²	C₂×C₁₉×#108	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	C₁₉×#423	1	1	yes	characteristic
135	C₄	C₁₉×#427	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	4	1	no	—
138	C₂	C₁₉×#1568	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2166

C₂

yes

maximal

C₁₉×#2108

C₂

yes

maximal, characteristic

C₁₉×#1373

C₂

yes

maximal, complemented

C₁₉×#1369

C₂

yes

maximal, complemented

C₂×C₁₉×M₃₂

C₄

yes

characteristic

C₂²×C₁₆×C₁₉

C₄

yes

characteristic

C₁₉×#405

C₂²

yes

characteristic

C₁₉×#364

C₂²

yes

—

C₁₉×#349

—

C₁₉×#347

—

C₁₉×#346

C₄

yes

characteristic

C₁₉×#346

C₂²

yes

characteristic

C₄×C₁₆×C₁₉

C₄

yes

characteristic

C₄×C₁₆×C₁₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₁₉×#117

Q₈

yes

characteristic

C₁₉×M₃₂

—

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₉

C₂×C₄

yes

characteristic

C₁₉×#97

D₄

yes

characteristic

C₁₉×#97

Q₈

yes

characteristic

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₂×Q₈

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

M₁₆

yes

—

C₁₆×C₁₉

M₁₆

yes

—

#12027

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

#108

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#108

yes

characteristic

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

C₂×M₁₆

yes

characteristic

C₈×C₁₉

C₂×M₁₆

yes

characteristic

C₈×C₁₉

#135

yes

—

#2166

C₂×C₁₉

yes

—

#2108

C₂×C₁₉

yes

characteristic

#1373

C₂×C₁₉

yes

complemented

#1369

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×#108

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×M₃₂

C₄×C₁₉

yes

characteristic

C₂²×C₁₆

C₄×C₁₉

yes

characteristic

#405

C₂²×C₁₉

yes

characteristic

#364

C₂²×C₁₉

yes

—

#349

—

#347

—

#346

C₄×C₁₉

yes

characteristic

#346

C₂²×C₁₉

yes

characteristic

C₄×C₁₆

C₄×C₁₉

yes

characteristic

C₄×C₁₆

C₂²×C₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1568

yes

characteristic

C₂×M₁₆

D₄×C₁₉

yes

characteristic

C₂²×C₈

C₂×C₄×C₁₉

yes

characteristic

#117

Q₈×C₁₉

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₁₉

yes

characteristic

C₂×C₁₆

C₂×C₄×C₁₉

yes

characteristic

C₂×C₁₆

C₂×C₄×C₁₉

yes

characteristic

C₂×C₁₆

C₂×C₄×C₁₉

yes

characteristic

#97

D₄×C₁₉

yes

characteristic

#97

Q₈×C₁₉

yes

characteristic

C₄×C₈

C₂×C₄×C₁₉

yes

characteristic

C₄×C₈

C₂³×C₁₉

yes

characteristic

C₁₉

#12027

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄⋊C₄×C₁₉

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₄⋊C₄×C₁₉

yes

characteristic

104

C₂×C₈

C₄⋊C₄×C₁₉

yes

characteristic

105

C₂×C₈

C₂²×C₄×C₁₉

yes

characteristic

106

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

C₄⋊C₄×C₁₉

yes

characteristic

110

C₄²

C₂×Q₈×C₁₉

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

M₁₆×C₁₉

yes

—

115

C₁₆

M₁₆×C₁₉

yes

—

116

C₂³

C₁₉×#108

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₁₉×#108

yes

characteristic

122

C₂×C₄

C₂×C₄⋊C₄×C₁₉

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

C₂×M₁₆×C₁₉

yes

characteristic

126

C₈

C₂×M₁₆×C₁₉

yes

characteristic

127

C₈

C₁₉×#135

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₁₉×#108

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

C₁₉×#423

yes

characteristic

135

C₄

C₁₉×#427

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

C₁₉×#1568

yes

characteristic

	subgroup	G/H
center	C₄×C₁₉	#427
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
socle	C₂×C₁₉	#1568
prime radical	C₂³×C₁₉	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₁₉×#97 ◁ C₁₉×#405 ◁ G

subgroup

G/H

center

C₄×C₁₉

#427

commutator

C₂×C₈

C₂²×C₄×C₁₉

Frattini, cosocle

C₄×C₈

C₂³×C₁₉

C₂×C₁₉

C₂³×C₁₉

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₉ ◁ C₁₉×#97 ◁ C₁₉×#405 ◁ G

N	H ≅ G/N	type
3. C₁₉×#1373	C₂	semidirect
4. C₁₉×#1369	C₂	semidirect
46. #12027	C₁₉	direct (N × H)
61. #1373	C₂×C₁₉	semidirect
62. #1369	C₂×C₁₉	semidirect
98. C₁₉	#12027	direct (N × H)

H ≅ G/N

type

3. C₁₉×#1373

C₂

semidirect

4. C₁₉×#1369

C₂

semidirect

46. #12027

C₁₉

direct (N × H)

61. #1373

C₂×C₁₉

semidirect

62. #1369

C₂×C₁₉

semidirect

98. C₁₉

#12027

direct (N × H)

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

subgroup	index
C₁₉×#2166	2
C₁₉×#2108	2
C₁₉×#1373	2
C₁₉×#1369	2
#12027	19

subgroup

index

C₁₉×#2166

C₁₉×#2108

C₁₉×#1373

C₁₉×#1369

#12027

Subgroups — C₁₉×#12027 #18017986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12027 #18017986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12027 #18017986