Group #18018422 — subgroups

Subgroups — C₁₉×#12463 #18018422

Order 4864.

172 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2253	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#2231	C₂	1	2	yes	maximal
3	C₁₉×#2170	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1373	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1366	C₂	1	2	yes	maximal, complemented
6	C₁₉×#504	—	2	1	no	—
7	C₁₉×#499	C₂²	1	2	yes	—
8	C₁₉×#443	C₂²	1	1	yes	characteristic
9	C₁₉×#367	—	2	1	no	—
10	C₁₉×#366	—	2	1	no	—
11	C₁₉×#365	—	2	2	no	—
12	C₁₉×#364	C₂²	1	1	yes	characteristic
13	C₁₉×#364	C₂²	1	2	yes	—
14	C₁₉×#360	—	2	2	no	—
15	C₁₉×#350	—	2	1	no	—
16	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#135	D₄	1	1	yes	characteristic
18	C₁₉×#131	—	2	1	no	—
19	C₁₉×#125	—	2	2	no	—
20	C₁₉×M₃₂	—	2	1	no	—
21	C₁₉×M₃₂	—	2	2	no	—
22	C₁₉×M₃₂	—	4	1	no	—
23	C₂×C₁₆×C₁₉	—	4	1	no	—
24	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
26	C₁₉×#108	D₄	1	1	yes	characteristic
27	C₁₉×#107	—	2	2	no	—
28	C₁₉×#106	—	2	2	no	—
29	C₁₉×#104	—	2	1	no	—
30	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
31	#41×C₁₉	—	2	1	no	—
32	Q₁₆×C₁₉	Q₁₆	1	1	yes	characteristic
33	SD₁₆×C₁₉	—	2	1	no	—
34	D₈×C₁₉	Q₁₆	1	1	yes	characteristic
35	M₁₆×C₁₉	—	2	1	no	—
36	M₁₆×C₁₉	—	4	1	no	—
37	C₂×C₈×C₁₉	—	4	1	no	—
38	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₁₉	—	4	2	no	—
41	C₄⋊C₄×C₁₉	—	8	2	no	—
42	C₄²×C₁₉	#41	1	1	yes	characteristic
43	C₁₆×C₁₉	—	2	1	no	—
44	C₁₆×C₁₉	—	4	1	no	—
45	C₁₆×C₁₉	—	4	2	no	—
46	C₁₆×C₁₉	SD₁₆	1	1	yes	characteristic
47	C₁₆×C₁₉	SD₁₆	1	1	yes	characteristic
48	#12463	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	Q₈×C₁₉	—	2	1	no	—
50	D₄×C₁₉	—	2	1	no	—
51	D₄×C₁₉	—	4	1	no	—
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	4	1	no	—
54	C₂×C₄×C₁₉	—	8	2	no	—
55	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
56	C₈×C₁₉	—	4	1	no	—
57	C₈×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₈×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₁₉	#135	1	1	yes	characteristic
60	C₈×C₁₉	#137	1	1	yes	characteristic
61	#2253	C₂×C₁₉	1	1	yes	characteristic, complemented
62	#2231	C₂×C₁₉	1	2	yes	—
63	#2170	C₂×C₁₉	1	1	yes	characteristic
64	#1373	C₂×C₁₉	1	1	yes	characteristic, complemented
65	#1366	C₂×C₁₉	1	2	yes	complemented
66	C₂²×C₁₉	—	4	1	no	—
67	C₂²×C₁₉	#472	1	1	yes	characteristic
68	C₄×C₁₉	—	4	1	no	—
69	C₄×C₁₉	—	4	1	no	—
70	C₄×C₁₉	—	16	2	no	—
71	C₄×C₁₉	#461	1	1	yes	characteristic
72	C₄×C₁₉	#467	1	1	yes	characteristic
73	#504	—	2	1	no	—
74	#499	C₂²×C₁₉	1	2	yes	—
75	#443	C₂²×C₁₉	1	1	yes	characteristic
76	#367	—	2	1	no	—
77	#366	—	2	1	no	—
78	#365	—	2	2	no	—
79	#364	C₂²×C₁₉	1	1	yes	characteristic
80	#364	C₂²×C₁₉	1	2	yes	—
81	#360	—	2	2	no	—
82	#350	—	2	1	no	—
83	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
84	C₂×C₁₉	—	2	1	no	—
85	C₂×C₁₉	—	8	1	no	—
86	C₂×C₁₉	#1675	1	1	yes	characteristic
87	#135	D₄×C₁₉	1	1	yes	characteristic
88	#131	—	2	1	no	—
89	#125	—	2	2	no	—
90	M₃₂	—	2	1	no	—
91	M₃₂	—	2	2	no	—
92	M₃₂	—	4	1	no	—
93	C₂×C₁₆	—	4	1	no	—
94	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
95	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
96	#108	D₄×C₁₉	1	1	yes	characteristic
97	#107	—	2	2	no	—
98	#106	—	2	2	no	—
99	#104	—	2	1	no	—
100	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
101	C₁₉	#12463	1	1	yes	characteristic, direct product factor, Sylow
102	#41	—	2	1	no	—
103	Q₁₆	Q₁₆×C₁₉	1	1	yes	characteristic
104	SD₁₆	—	2	1	no	—
105	D₈	Q₁₆×C₁₉	1	1	yes	characteristic
106	M₁₆	—	2	1	no	—
107	M₁₆	—	4	1	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	8	2	no	—
113	C₄²	#41×C₁₉	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	1	no	—
116	C₁₆	—	4	2	no	—
117	C₁₆	SD₁₆×C₁₉	1	1	yes	characteristic
118	C₁₆	SD₁₆×C₁₉	1	1	yes	characteristic
119	Q₈	—	2	1	no	—
120	D₄	—	2	1	no	—
121	D₄	—	4	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
126	C₈	—	4	1	no	—
127	C₈	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
128	C₈	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
129	C₈	C₁₉×#135	1	1	yes	characteristic
130	C₈	C₁₉×#137	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	C₁₉×#472	1	1	yes	characteristic
133	C₄	—	4	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	16	2	no	—
136	C₄	C₁₉×#461	1	1	yes	characteristic
137	C₄	C₁₉×#467	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	8	1	no	—
140	C₂	C₁₉×#1675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2253

C₂

yes

maximal, characteristic, complemented

C₁₉×#2231

C₂

yes

maximal

C₁₉×#2170

C₂

yes

maximal, characteristic

C₁₉×#1373

C₂

yes

maximal, characteristic, complemented

C₁₉×#1366

C₂

yes

maximal, complemented

C₁₉×#504

—

C₁₉×#499

C₂²

yes

—

C₁₉×#443

C₂²

yes

characteristic

C₁₉×#367

—

C₁₉×#366

—

C₁₉×#365

—

C₁₉×#364

C₂²

yes

characteristic

C₁₉×#364

C₂²

yes

—

C₁₉×#360

—

C₁₉×#350

—

C₄×C₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#135

D₄

yes

characteristic

C₁₉×#131

—

C₁₉×#125

—

C₁₉×M₃₂

—

C₁₉×M₃₂

—

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₁₉×#108

D₄

yes

characteristic

C₁₉×#107

—

C₁₉×#106

—

C₁₉×#104

—

C₄×C₈×C₁₉

C₂³

yes

characteristic

#41×C₁₉

—

Q₁₆×C₁₉

Q₁₆

yes

characteristic

SD₁₆×C₁₉

—

D₈×C₁₉

Q₁₆

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

#41

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

SD₁₆

yes

characteristic

C₁₆×C₁₉

SD₁₆

yes

characteristic

#12463

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#121

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

C₂×SD₁₆

yes

characteristic

C₈×C₁₉

C₂×Q₁₆

yes

C₈×C₁₉

yes

C₈×C₁₉

yes

C₂×C₁₉

yes

characteristic, complemented

#2231

C₂×C₁₉

yes

—

#2170

C₂×C₁₉

yes

characteristic

#1373

C₂×C₁₉

yes

characteristic, complemented

#1366

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

#472

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₂²×C₁₉

yes

—

#443

C₂²×C₁₉

yes

—

—

—

C₂²×C₁₉

yes

characteristic

#364

C₂²×C₁₉

yes

—

#360

—

#350

—

C₄×C₁₆

C₂²×C₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1675

yes

characteristic

#135

D₄×C₁₉

yes

characteristic

#131

—

#125

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₉

yes

characteristic

C₂×C₁₆

D₄×C₁₉

yes

characteristic

#108

D₄×C₁₉

yes

—

—

—

100

C₄×C₈

C₂³×C₁₉

yes

characteristic

101

C₁₉

#12463

yes

characteristic, direct product factor, Sylow

102

#41

—

103

Q₁₆

Q₁₆×C₁₉

yes

characteristic

104

SD₁₆

—

105

D₈

Q₁₆×C₁₉

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

110

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

#41×C₁₉

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

SD₁₆×C₁₉

yes

characteristic

118

C₁₆

SD₁₆×C₁₉

yes

characteristic

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₁₉×#121

yes

characteristic

126

C₈

—

127

C₈

C₂×SD₁₆×C₁₉

yes

characteristic

128

C₈

C₂×Q₁₆×C₁₉

yes

characteristic

129

C₈

C₁₉×#135

yes

characteristic

130

C₈

C₁₉×#137

yes

characteristic

131

C₂²

—

132

C₂²

C₁₉×#472

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₁₉×#461

yes

characteristic

137

C₄

C₁₉×#467

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

C₁₉×#1675

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1675
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₁₉×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₄×C₈×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂×C₁₉

#1675

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₉

prime radical

C₁₉×#135

D₄

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₄×C₈×C₁₉ ◁ G

N	H ≅ G/N	type
1. C₁₉×#2253	C₂	semidirect
4. C₁₉×#1373	C₂	semidirect
5. C₁₉×#1366	C₂	semidirect
48. #12463	C₁₉	direct (N × H)
61. #2253	C₂×C₁₉	semidirect
64. #1373	C₂×C₁₉	semidirect
65. #1366	C₂×C₁₉	semidirect
101. C₁₉	#12463	direct (N × H)

H ≅ G/N

type

1. C₁₉×#2253

C₂

semidirect

4. C₁₉×#1373

C₂

semidirect

5. C₁₉×#1366

C₂

semidirect

48. #12463

C₁₉

direct (N × H)

61. #2253

C₂×C₁₉

semidirect

64. #1373

C₂×C₁₉

semidirect

65. #1366

C₂×C₁₉

semidirect

101. C₁₉

#12463

direct (N × H)

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

subgroup	index
C₁₉×#2253	2
C₁₉×#2231	2
C₁₉×#2170	2
C₁₉×#1373	2
C₁₉×#1366	2
#12463	19

subgroup

index

C₁₉×#2253

C₁₉×#2231

C₁₉×#2170

C₁₉×#1373

C₁₉×#1366

#12463

Subgroups — C₁₉×#12463 #18018422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12463 #18018422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12463 #18018422