Group #18017993 — subgroups

Subgroups — C₁₉×#12034 #18017993

Order 4864.

204 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2162	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1568	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1564	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1372	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1371	C₂	1	2	yes	maximal, complemented
6	C₁₉×#432	—	2	1	no	—
7	C₁₉×#431	—	2	1	no	—
8	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₁₉×C₄⋊C₁₆	C₄	1	2	yes	—
10	C₁₉×C₂²⋊C₁₆	C₄	1	2	yes	—
11	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#335	C₂²	1	2	yes	—
14	C₁₉×#335	C₂²	1	2	yes	—
15	C₁₉×#334	—	2	1	no	—
16	C₁₉×#334	—	2	1	no	—
17	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
22	C₁₉×C₄⋊C₈	—	2	2	no	—
23	C₁₉×C₄⋊C₈	—	2	2	no	—
24	C₁₉×C₂²⋊C₈	—	4	1	no	—
25	C₁₉×C₂²⋊C₈	—	4	1	no	—
26	C₁₉×#97	—	4	1	no	—
27	C₄×C₈×C₁₉	—	4	1	no	—
28	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	Q₁₆	1	1	yes	characteristic
35	C₂×C₈×C₁₉	—	2	2	no	—
36	C₂×C₈×C₁₉	—	4	1	no	—
37	C₂×C₈×C₁₉	—	4	2	no	—
38	C₂×C₈×C₁₉	—	4	2	no	—
39	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₉	D₈	1	1	yes	characteristic
45	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
46	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
47	C₁₆×C₁₉	—	2	2	no	—
48	C₁₆×C₁₉	—	2	2	no	—
49	#12034	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₁₉	#107	1	1	yes	characteristic
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	2	no	—
54	C₂×C₄×C₁₉	—	4	1	no	—
55	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
56	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
61	C₈×C₁₉	—	2	1	no	—
62	C₈×C₁₉	—	2	1	no	—
63	C₈×C₁₉	—	2	1	no	—
64	C₈×C₁₉	—	8	2	no	—
65	C₈×C₁₉	—	8	2	no	—
66	C₈×C₁₉	#131	1	2	yes	—
67	#2162	C₂×C₁₉	1	1	yes	characteristic
68	#1568	C₂×C₁₉	1	1	yes	characteristic
69	#1564	C₂×C₁₉	1	1	yes	characteristic
70	#1372	C₂×C₁₉	1	2	yes	complemented
71	#1371	C₂×C₁₉	1	2	yes	complemented
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	—	2	1	no	—
74	C₂²×C₁₉	—	2	1	no	—
75	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	—	2	1	no	—
79	C₄×C₁₉	—	2	1	no	—
80	C₄×C₁₉	#424	1	1	yes	characteristic
81	C₄×C₁₉	#430	1	1	yes	characteristic
82	C₄×C₁₉	#508	1	2	yes	—
83	#432	—	2	1	no	—
84	#431	—	2	1	no	—
85	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
86	C₄⋊C₁₆	C₄×C₁₉	1	2	yes	—
87	C₂²⋊C₁₆	C₄×C₁₉	1	2	yes	—
88	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
89	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
90	#335	C₂²×C₁₉	1	2	yes	—
91	#335	C₂²×C₁₉	1	2	yes	—
92	#334	—	2	1	no	—
93	#334	—	2	1	no	—
94	C₂×C₁₉	—	4	1	no	—
95	C₂×C₁₉	#1571	1	1	yes	characteristic
96	C₂×C₁₉	#2158	1	1	yes	characteristic
97	C₂×C₁₉	#2161	1	1	yes	characteristic
98	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
99	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
100	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
101	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
102	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₂²⋊C₈	—	4	1	no	—
106	C₂²⋊C₈	—	4	1	no	—
107	#97	—	4	1	no	—
108	C₄×C₈	—	4	1	no	—
109	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
110	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
111	C₄×C₈	Q₈×C₁₉	1	1	yes	characteristic
112	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
113	C₁₉	#12034	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
115	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
116	C₂²×C₄	Q₁₆×C₁₉	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
122	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
123	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
124	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
125	C₄²	C₄⋊C₄×C₁₉	1	1	yes	characteristic
126	C₄²	D₈×C₁₉	1	1	yes	characteristic
127	C₄²	Q₁₆×C₁₉	1	1	yes	characteristic
128	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
129	C₁₆	—	2	2	no	—
130	C₁₆	—	2	2	no	—
131	C₂³	C₁₉×#107	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₁₉×#107	1	1	yes	characteristic
137	C₂×C₄	C₁₉×#107	1	1	yes	characteristic
138	C₂×C₄	C₁₉×#107	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
140	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
141	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	8	2	no	—
146	C₈	—	8	2	no	—
147	C₈	C₁₉×#131	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	C₂×C₁₉×#107	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	C₁₉×#424	1	1	yes	characteristic
157	C₄	C₁₉×#430	1	1	yes	characteristic
158	C₄	C₁₉×#508	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	C₁₉×#1571	1	1	yes	characteristic
161	C₂	C₁₉×#2158	1	1	yes	characteristic
162	C₂	C₁₉×#2161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2162

C₂

yes

maximal, characteristic

C₁₉×#1568

C₂

yes

maximal, characteristic

C₁₉×#1564

C₂

yes

maximal, characteristic

C₁₉×#1372

C₂

yes

maximal, complemented

C₁₉×#1371

C₂

yes

maximal, complemented

C₁₉×#432

—

C₁₉×#431

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×C₄⋊C₁₆

C₄

yes

—

C₁₉×C₂²⋊C₁₆

C₄

yes

—

C₄×C₁₆×C₁₉

C₂²

yes

characteristic

C₄×C₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#335

C₂²

yes

—

C₁₉×#335

C₂²

yes

—

C₁₉×#334

—

C₁₉×#334

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

Q₈

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

C₂×C₄

yes

—

C₂×C₁₆×C₁₉

C₂×C₄

yes

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

Q₈

yes

characteristic

C₄×C₈×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

Q₁₆

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×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×Q₈

yes

characteristic

C₄²×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

D₈

yes

characteristic

C₄²×C₁₉

Q₁₆

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

—

#12034

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

#107

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₁₆

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#131

yes

—

#2162

C₂×C₁₉

yes

characteristic

#1568

C₂×C₁₉

yes

characteristic

#1564

C₂×C₁₉

yes

characteristic

#1372

C₂×C₁₉

yes

complemented

#1371

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×#107

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

#432

—

#431

—

C₂×C₄×C₈

C₂²×C₁₉

yes

characteristic

C₄⋊C₁₆

C₄×C₁₉

yes

—

C₂²⋊C₁₆

C₄×C₁₉

yes

—

C₄×C₁₆

C₂²×C₁₉

yes

characteristic

C₄×C₁₆

C₂²×C₁₉

yes

characteristic

#335

C₂²×C₁₉

yes

—

#335

C₂²×C₁₉

yes

—

#334

—

#334

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×C₈

C₂×C₄×C₁₉

yes

characteristic

C₂²×C₈

Q₈×C₁₉

yes

characteristic

100

C₂×C₄²

D₄×C₁₉

yes

characteristic

101

C₂×C₁₆

C₂×C₄×C₁₉

yes

—

102

C₂×C₁₆

C₂×C₄×C₁₉

yes

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

—

106

C₂²⋊C₈

—

107

#97

—

108

C₄×C₈

—

109

C₄×C₈

C₂×C₄×C₁₉

yes

characteristic

110

C₄×C₈

D₄×C₁₉

yes

characteristic

111

C₄×C₈

Q₈×C₁₉

yes

characteristic

112

C₄×C₈

C₂³×C₁₉

yes

characteristic

113

C₁₉

#12034

yes

characteristic, direct product factor, Sylow

114

C₂²×C₄

C₄⋊C₄×C₁₉

yes

characteristic

115

C₂²×C₄

D₈×C₁₉

yes

characteristic

116

C₂²×C₄

Q₁₆×C₁₉

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₄⋊C₄×C₁₉

yes

characteristic

122

C₂×C₈

C₄⋊C₄×C₁₉

yes

characteristic

123

C₂×C₈

C₂²×C₄×C₁₉

yes

characteristic

124

C₂×C₈

C₂×Q₈×C₁₉

yes

characteristic

125

C₄²

C₄⋊C₄×C₁₉

yes

characteristic

126

C₄²

D₈×C₁₉

yes

characteristic

127

C₄²

Q₁₆×C₁₉

yes

characteristic

128

C₄²

C₂×D₄×C₁₉

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₂³

C₁₉×#107

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₁₉×#107

yes

characteristic

137

C₂×C₄

C₁₉×#107

yes

characteristic

138

C₂×C₄

C₁₉×#107

yes

characteristic

139

C₂×C₄

C₂×C₄⋊C₄×C₁₉

yes

characteristic

140

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

141

C₂×C₄

C₂×Q₁₆×C₁₉

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

C₁₉×#131

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₂×C₁₉×#107

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₁₉×#424

yes

characteristic

157

C₄

C₁₉×#430

yes

characteristic

158

C₄

C₁₉×#508

yes

—

159

C₂

—

160

C₂

C₁₉×#1571

yes

characteristic

161

C₂

C₁₉×#2158

yes

characteristic

162

C₂

C₁₉×#2161

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×#107
prime radical	C₂³×C₁₉	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₉

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₁₉

Frattini, cosocle

C₄×C₈

C₂³×C₁₉

C₂²×C₁₉

C₂³×C₁₉

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

N	H ≅ G/N	type
4. C₁₉×#1372	C₂	semidirect
5. C₁₉×#1371	C₂	semidirect
49. #12034	C₁₉	direct (N × H)
70. #1372	C₂×C₁₉	semidirect
71. #1371	C₂×C₁₉	semidirect
113. C₁₉	#12034	direct (N × H)

H ≅ G/N

type

4. C₁₉×#1372

C₂

semidirect

5. C₁₉×#1371

C₂

semidirect

49. #12034

C₁₉

direct (N × H)

70. #1372

C₂×C₁₉

semidirect

71. #1371

C₂×C₁₉

semidirect

113. C₁₉

#12034

direct (N × H)

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

subgroup	index
C₁₉×#2162	2
C₁₉×#1568	2
C₁₉×#1564	2
C₁₉×#1372	2
C₁₉×#1371	2
#12034	19

subgroup

index

C₁₉×#2162

C₁₉×#1568

C₁₉×#1564

C₁₉×#1372

C₁₉×#1371

#12034

Subgroups — C₁₉×#12034 #18017993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12034 #18017993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12034 #18017993