Group #18019676 — subgroups

Subgroups — C₁₉×#13717 #18019676

Order 4864.

170 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 66 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×#2218	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2161	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1426	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1419	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1400	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₉×D₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
8	C₁₉×#466	—	2	1	no	—
9	C₂×C₁₉×#108	—	2	1	no	—
10	C₁₉×M₆₄	C₂²	1	2	yes	—
11	C₁₉×#368	—	2	1	no	—
12	C₁₉×#368	C₂²	1	2	yes	—
13	C₁₉×#367	C₂²	1	1	yes	characteristic
14	C₁₉×#357	—	2	1	no	—
15	C₂×D₈×C₁₉	—	2	1	no	—
16	C₂×M₁₆×C₁₉	—	4	1	no	—
17	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
18	C₁₉×#121	—	4	1	no	—
19	C₁₉×D₁₆	—	2	2	no	—
20	C₁₉×M₃₂	—	2	2	no	—
21	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
23	C₁₉×#108	—	2	2	no	—
24	C₁₉×#108	—	4	1	no	—
25	C₁₉×#107	—	2	1	no	—
26	C₁₉×#102	—	4	1	no	—
27	C₁₉×C₃₂	—	2	2	no	—
28	C₂×D₄×C₁₉	—	4	1	no	—
29	C₂×D₄×C₁₉	—	8	1	no	—
30	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
31	D₈×C₁₉	—	4	1	no	—
32	D₈×C₁₉	—	4	1	no	—
33	M₁₆×C₁₉	—	4	2	no	—
34	M₁₆×C₁₉	—	8	1	no	—
35	C₂×C₈×C₁₉	—	2	1	no	—
36	C₂×C₈×C₁₉	—	2	1	no	—
37	C₂×C₈×C₁₉	—	8	1	no	—
38	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
39	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₁₉	—	4	1	no	—
41	C₂²⋊C₄×C₁₉	—	8	1	no	—
42	C₁₆×C₁₉	—	2	1	no	—
43	C₁₆×C₁₉	#41	1	2	yes	—
44	#13717	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂³×C₁₉	—	8	1	no	—
46	C₂³×C₁₉	SD₃₂	1	1	yes	characteristic
47	D₄×C₁₉	—	8	1	no	—
48	D₄×C₁₉	—	8	1	no	—
49	D₄×C₁₉	—	16	2	no	—
50	C₂×C₄×C₁₉	—	2	1	no	—
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	—	8	1	no	—
53	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
54	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
55	C₈×C₁₉	—	2	1	no	—
56	C₈×C₁₉	—	8	2	no	—
57	C₈×C₁₉	#123	1	1	yes	characteristic
58	C₈×C₁₉	#137	1	1	yes	characteristic
59	#2218	C₂×C₁₉	1	1	yes	characteristic
60	#2161	C₂×C₁₉	1	1	yes	characteristic, complemented
61	#1426	C₂×C₁₉	1	2	yes	complemented
62	#1419	C₂×C₁₉	1	2	yes	complemented
63	#1400	C₂×C₁₉	1	1	yes	characteristic, complemented
64	C₂²×C₁₉	—	2	1	no	—
65	C₂²×C₁₉	—	4	1	no	—
66	C₂²×C₁₉	—	16	1	no	—
67	C₂²×C₁₉	—	16	1	no	—
68	C₂²×C₁₉	—	16	1	no	—
69	C₂²×C₁₉	C₂×SD₃₂	1	1	yes	characteristic
70	C₄×C₁₉	—	2	1	no	—
71	C₄×C₁₉	—	16	1	no	—
72	C₄×C₁₉	#480	1	1	yes	characteristic
73	C₄×C₁₉	#509	1	1	yes	characteristic
74	C₂×D₁₆	C₂²×C₁₉	1	1	yes	characteristic
75	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
76	#466	—	2	1	no	—
77	C₂×#108	—	2	1	no	—
78	M₆₄	C₂²×C₁₉	1	2	yes	—
79	#368	—	2	1	no	—
80	#368	C₂²×C₁₉	1	2	yes	—
81	#367	C₂²×C₁₉	1	1	yes	characteristic
82	#357	—	2	1	no	—
83	C₂×C₁₉	—	2	1	no	—
84	C₂×C₁₉	—	4	1	no	—
85	C₂×C₁₉	—	32	1	no	—
86	C₂×C₁₉	#2233	1	1	yes	characteristic
87	C₂×D₈	—	2	1	no	—
88	C₂×M₁₆	—	4	1	no	—
89	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
90	#121	—	4	1	no	—
91	D₁₆	—	2	2	no	—
92	M₃₂	—	2	2	no	—
93	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
94	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
95	#108	—	2	2	no	—
96	#108	—	4	1	no	—
97	#107	—	2	1	no	—
98	#102	—	4	1	no	—
99	C₃₂	—	2	2	no	—
100	C₁₉	#13717	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×D₄	—	4	1	no	—
102	C₂×D₄	—	8	1	no	—
103	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
104	D₈	—	4	1	no	—
105	D₈	—	4	1	no	—
106	M₁₆	—	4	2	no	—
107	M₁₆	—	8	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	8	1	no	—
111	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
113	C₄⋊C₄	—	4	1	no	—
114	C₂²⋊C₄	—	8	1	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	#41×C₁₉	1	2	yes	—
117	C₂³	—	8	1	no	—
118	C₂³	C₁₉×SD₃₂	1	1	yes	characteristic
119	D₄	—	8	1	no	—
120	D₄	—	8	1	no	—
121	D₄	—	16	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	C₁₉×SD₃₂	1	1	yes	characteristic
126	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
127	C₈	—	2	1	no	—
128	C₈	—	8	2	no	—
129	C₈	C₁₉×#123	1	1	yes	characteristic
130	C₈	C₁₉×#137	1	1	yes	characteristic
131	C₂²	—	2	1	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	16	1	no	—
134	C₂²	—	16	1	no	—
135	C₂²	—	16	1	no	—
136	C₂²	C₂×C₁₉×SD₃₂	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	16	1	no	—
139	C₄	C₁₉×#480	1	1	yes	characteristic
140	C₄	C₁₉×#509	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	4	1	no	—
143	C₂	—	32	1	no	—
144	C₂	C₁₉×#2233	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2218

C₂

yes

maximal, characteristic

C₁₉×#2161

C₂

yes

maximal, characteristic, complemented

C₁₉×#1426

C₂

yes

maximal, complemented

C₁₉×#1419

C₂

yes

maximal, complemented

C₁₉×#1400

C₂

yes

maximal, characteristic, complemented

C₂×C₁₉×D₁₆

C₂²

yes

characteristic

C₂×C₁₉×M₃₂

C₂²

yes

characteristic

C₁₉×#466

—

C₂×C₁₉×#108

—

C₁₉×M₆₄

C₂²

yes

—

C₁₉×#368

—

C₁₉×#368

C₂²

yes

—

C₁₉×#367

C₂²

yes

characteristic

C₁₉×#357

—

C₂×D₈×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₁₉×#121

—

C₁₉×D₁₆

—

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

C₂³

yes

characteristic

C₁₉×#108

—

C₁₉×#108

—

C₁₉×#107

—

C₁₉×#102

—

C₁₉×C₃₂

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

D₈

yes

characteristic

D₈×C₁₉

—

D₈×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

characteristic

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

#41

yes

—

#13717

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

SD₃₂

yes

characteristic

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

SD₃₂

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

C₂×C₁₉

yes

characteristic

#2161

C₂×C₁₉

yes

characteristic, complemented

#1426

C₂×C₁₉

yes

complemented

#1419

C₂×C₁₉

yes

complemented

#1400

C₂×C₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×SD₃₂

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×D₁₆

C₂²×C₁₉

yes

characteristic

C₂×M₃₂

C₂²×C₁₉

yes

characteristic

#466

—

C₂×#108

—

M₆₄

C₂²×C₁₉

yes

—

#368

—

#368

C₂²×C₁₉

yes

—

#367

C₂²×C₁₉

yes

characteristic

#357

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#2233

yes

characteristic

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₁₉

yes

characteristic

#121

—

D₁₆

—

M₃₂

—

C₂×C₁₆

D₄×C₁₉

yes

characteristic

C₂×C₁₆

C₂³×C₁₉

yes

—

—

—

—

C₃₂

—

100

C₁₉

#13717

yes

characteristic, direct product factor, Sylow

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂²×C₄

D₈×C₁₉

yes

characteristic

104

D₈

—

105

D₈

—

106

M₁₆

—

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

D₈×C₁₉

yes

characteristic

112

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₁₆

—

116

C₁₆

#41×C₁₉

yes

—

117

C₂³

—

118

C₂³

C₁₉×SD₃₂

yes

characteristic

119

D₄

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₁₉×SD₃₂

yes

characteristic

126

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

C₁₉×#123

yes

characteristic

130

C₈

C₁₉×#137

yes

characteristic

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂×C₁₉×SD₃₂

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

C₁₉×#480

yes

characteristic

140

C₄

C₁₉×#509

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

C₁₉×#2233

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₉
prime radical	C₁₉×#2218	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₈×C₁₉ ◁ C₂×C₁₉×M₃₂ ◁ G

subgroup

G/H

center, socle

C₂×C₁₉

#2233

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₁₉

prime radical

C₁₉×#2218

C₂

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₈×C₁₉ ◁ C₂×C₁₉×M₃₂ ◁ G

N	H ≅ G/N	type
2. C₁₉×#2161	C₂	semidirect
3. C₁₉×#1426	C₂	semidirect
4. C₁₉×#1419	C₂	semidirect
5. C₁₉×#1400	C₂	semidirect
44. #13717	C₁₉	direct (N × H)
60. #2161	C₂×C₁₉	semidirect
61. #1426	C₂×C₁₉	semidirect
62. #1419	C₂×C₁₉	semidirect
63. #1400	C₂×C₁₉	semidirect
100. C₁₉	#13717	direct (N × H)

H ≅ G/N

type

2. C₁₉×#2161

C₂

semidirect

3. C₁₉×#1426

C₂

semidirect

4. C₁₉×#1419

C₂

semidirect

5. C₁₉×#1400

C₂

semidirect

44. #13717

C₁₉

direct (N × H)

60. #2161

C₂×C₁₉

semidirect

61. #1426

C₂×C₁₉

semidirect

62. #1419

C₂×C₁₉

semidirect

63. #1400

C₂×C₁₉

semidirect

100. C₁₉

#13717

direct (N × H)

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

subgroup	index
C₁₉×#2218	2
C₁₉×#2161	2
C₁₉×#1426	2
C₁₉×#1419	2
C₁₉×#1400	2
#13717	19

subgroup

index

C₁₉×#2218

C₁₉×#2161

C₁₉×#1426

C₁₉×#1419

C₁₉×#1400

#13717

Subgroups — C₁₉×#13717 #18019676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#13717 #18019676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13717 #18019676