Group #18018410 — subgroups

Subgroups — C₁₉×#12451 #18018410

Order 4864.

224 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2252	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1673	C₂	1	2	yes	maximal
4	C₁₉×#1371	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1364	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×C₁₉	—	2	1	no	—
7	C₁₉×#500	C₂²	1	2	yes	—
8	C₁₉×#461	—	2	2	no	—
9	C₈×D₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#366	—	2	1	no	—
11	C₁₉×#366	—	2	1	no	—
12	C₁₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
14	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#335	C₂²	1	2	yes	—
16	C₁₉×#332	—	2	2	no	—
17	C₂×SD₁₆×C₁₉	—	4	2	no	—
18	C₂²×C₈×C₁₉	—	2	1	no	—
19	C₁₉×C₄⋊Q₈	—	2	2	no	—
20	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₉	—	2	1	no	—
22	C₂×C₁₆×C₁₉	—	4	2	no	—
23	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
25	C₁₉×#107	—	2	2	no	—
26	C₁₉×#107	—	2	2	no	—
27	C₁₉×C₄⋊C₈	—	2	2	no	—
28	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
29	C₁₉×#103	—	4	2	no	—
30	C₁₉×C₂²⋊C₈	—	2	1	no	—
31	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₁₉	—	4	2	no	—
33	C₂×D₄×C₁₉	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	—	2	1	no	—
35	SD₁₆×C₁₉	—	8	4	no	—
36	C₂×C₈×C₁₉	—	2	1	no	—
37	C₂×C₈×C₁₉	—	4	1	no	—
38	C₂×C₈×C₁₉	—	4	1	no	—
39	C₂×C₈×C₁₉	—	4	2	no	—
40	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₁₉	—	4	2	no	—
43	C₄⋊C₄×C₁₉	—	4	2	no	—
44	C₄⋊C₄×C₁₉	D₈	1	1	yes	characteristic
45	C₂²⋊C₄×C₁₉	—	2	1	no	—
46	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
47	C₁₆×C₁₉	—	2	1	no	—
48	C₁₆×C₁₉	—	4	1	no	—
49	C₁₆×C₁₉	SD₁₆	1	2	yes	complemented
50	#12451	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₁₉	—	2	1	no	—
52	Q₈×C₁₉	—	8	2	no	—
53	Q₈×C₁₉	—	8	2	no	—
54	D₄×C₁₉	—	2	1	no	—
55	D₄×C₁₉	D₁₆	1	2	yes	—
56	C₂×C₄×C₁₉	—	2	1	no	—
57	C₂×C₄×C₁₉	—	4	1	no	—
58	C₂×C₄×C₁₉	—	4	1	no	—
59	C₂×C₄×C₁₉	—	8	2	no	—
60	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
63	C₈×C₁₉	—	2	1	no	—
64	C₈×C₁₉	—	4	1	no	—
65	C₈×C₁₉	—	8	2	no	—
66	C₈×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
67	C₈×C₁₉	#137	1	1	yes	characteristic
68	#2252	C₂×C₁₉	1	1	yes	characteristic, complemented
69	D₄×C₁₆	C₂×C₁₉	1	1	yes	characteristic
70	#1673	C₂×C₁₉	1	2	yes	—
71	#1371	C₂×C₁₉	1	1	yes	characteristic, complemented
72	#1364	C₂×C₁₉	1	2	yes	complemented
73	C₂²×C₁₉	—	2	2	no	—
74	C₂²×C₁₉	—	4	1	no	—
75	C₂²×C₁₉	—	4	1	no	—
76	C₂²×C₁₉	#466	1	1	yes	characteristic
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	—	4	1	no	—
79	C₄×C₁₉	—	16	2	no	—
80	C₄×C₁₉	#461	1	1	yes	characteristic
81	C₄×C₁₉	#473	1	1	yes	characteristic
82	C₄×C₁₉	C₂×D₁₆	1	1	yes	characteristic
83	C₄×C₁₉	#508	1	1	yes	characteristic
84	C₂²×C₁₆	—	2	1	no	—
85	#500	C₂²×C₁₉	1	2	yes	—
86	#461	—	2	2	no	—
87	C₈×D₄	C₂²×C₁₉	1	1	yes	characteristic
88	#366	—	2	1	no	—
89	#366	—	2	1	no	—
90	C₄⋊C₁₆	C₂²×C₁₉	1	1	yes	characteristic
91	C₂²⋊C₁₆	—	2	1	no	—
92	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
93	#335	C₂²×C₁₉	1	2	yes	—
94	#332	—	2	2	no	—
95	C₂×C₁₉	—	4	2	no	—
96	C₂×C₁₉	#1673	1	1	yes	characteristic
97	C₂×C₁₉	#2215	1	1	yes	characteristic
98	C₂×C₁₉	#2222	1	1	yes	characteristic
99	C₂×SD₁₆	—	4	2	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₄⋊Q₈	—	2	2	no	—
102	C₄×D₄	D₄×C₁₉	1	1	yes	characteristic
103	C₂×C₁₆	—	2	1	no	—
104	C₂×C₁₆	—	4	2	no	—
105	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
106	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
107	#107	—	2	2	no	—
108	#107	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
111	#103	—	4	2	no	—
112	C₂²⋊C₈	—	2	1	no	—
113	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
114	C₁₉	#12451	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×Q₈	—	4	2	no	—
116	C₂×D₄	D₈×C₁₉	1	1	yes	characteristic
117	C₂²×C₄	—	2	1	no	—
118	SD₁₆	—	8	4	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
124	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	D₈×C₁₉	1	1	yes	characteristic
128	C₂²⋊C₄	—	2	1	no	—
129	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
130	C₁₆	—	2	1	no	—
131	C₁₆	—	4	1	no	—
132	C₁₆	SD₁₆×C₁₉	1	2	yes	complemented
133	C₂³	—	2	1	no	—
134	Q₈	—	8	2	no	—
135	Q₈	—	8	2	no	—
136	D₄	—	2	1	no	—
137	D₄	C₁₉×D₁₆	1	2	yes	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
143	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
144	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	4	1	no	—
147	C₈	—	8	2	no	—
148	C₈	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
149	C₈	C₁₉×#137	1	1	yes	characteristic
150	C₂²	—	2	2	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	C₁₉×#466	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	16	2	no	—
157	C₄	C₁₉×#461	1	1	yes	characteristic
158	C₄	C₁₉×#473	1	1	yes	characteristic
159	C₄	C₂×C₁₉×D₁₆	1	1	yes	characteristic
160	C₄	C₁₉×#508	1	1	yes	characteristic
161	C₂	—	4	2	no	—
162	C₂	C₁₉×#1673	1	1	yes	characteristic
163	C₂	C₁₉×#2215	1	1	yes	characteristic
164	C₂	C₁₉×#2222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2252

C₂

yes

maximal, characteristic, complemented

D₄×C₁₆×C₁₉

C₂

yes

maximal, characteristic

C₁₉×#1673

C₂

yes

maximal

C₁₉×#1371

C₂

yes

maximal, characteristic, complemented

C₁₉×#1364

C₂

yes

maximal, complemented

C₂²×C₁₆×C₁₉

—

C₁₉×#500

C₂²

yes

—

C₁₉×#461

—

C₈×D₄×C₁₉

C₂²

yes

characteristic

C₁₉×#366

—

C₁₉×#366

—

C₁₉×C₄⋊C₁₆

C₂²

yes

characteristic

C₁₉×C₂²⋊C₁₆

—

C₄×C₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#335

C₂²

yes

—

C₁₉×#332

—

C₂×SD₁₆×C₁₉

—

C₂²×C₈×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₄×D₄×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₁₉×#107

—

C₁₉×#107

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

D₄

yes

characteristic

C₁₉×#103

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

C₂³

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

—

SD₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₉

#41

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

D₈

yes

characteristic

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

SD₁₆

yes

complemented

#12451

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

D₁₆

yes

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₉

#135

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

C₂×SD₁₆

yes

C₈×C₁₉

yes

C₂×C₁₉

yes

characteristic, complemented

D₄×C₁₆

C₂×C₁₉

yes

characteristic

#1673

C₂×C₁₉

yes

—

#1371

C₂×C₁₉

yes

characteristic, complemented

#1364

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#466

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

C₂×D₁₆

yes

characteristic

C₄×C₁₉

#508

yes

characteristic

C₂²×C₁₆

—

#500

C₂²×C₁₉

yes

—

#461

—

C₈×D₄

C₂²×C₁₉

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

C₂²×C₁₉

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂²×C₁₉

yes

characteristic

#335

C₂²×C₁₉

yes

—

#332

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×SD₁₆

—

100

C₂²×C₈

—

101

C₄⋊Q₈

—

102

C₄×D₄

D₄×C₁₉

yes

characteristic

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

D₄×C₁₉

yes

characteristic

106

C₂×C₁₆

D₄×C₁₉

yes

characteristic

107

#107

—

108

#107

—

109

C₄⋊C₈

—

110

C₄⋊C₈

D₄×C₁₉

yes

characteristic

111

#103

—

112

C₂²⋊C₈

—

113

C₄×C₈

C₂³×C₁₉

yes

characteristic

114

C₁₉

#12451

yes

characteristic, direct product factor, Sylow

115

C₂×Q₈

—

116

C₂×D₄

D₈×C₁₉

yes

characteristic

117

C₂²×C₄

—

118

SD₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

124

C₂×C₈

#41×C₁₉

yes

characteristic

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

D₈×C₁₉

yes

characteristic

128

C₂²⋊C₄

—

129

C₄²

C₂×D₄×C₁₉

yes

characteristic

130

C₁₆

—

131

C₁₆

—

132

C₁₆

SD₁₆×C₁₉

yes

complemented

133

C₂³

—

134

Q₈

—

135

Q₈

—

136

D₄

—

137

D₄

C₁₉×D₁₆

yes

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₁₉×#121

yes

characteristic

143

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

144

C₂×C₄

C₁₉×#135

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

C₂×SD₁₆×C₁₉

yes

characteristic

149

C₈

C₁₉×#137

yes

characteristic

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₁₉×#466

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

C₁₉×#461

yes

characteristic

158

C₄

C₁₉×#473

yes

characteristic

159

C₄

C₂×C₁₉×D₁₆

yes

characteristic

160

C₄

C₁₉×#508

yes

characteristic

161

C₂

—

162

C₂

C₁₉×#1673

yes

characteristic

163

C₂

C₁₉×#2215

yes

characteristic

164

C₂

C₁₉×#2222

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₂×D₄×C₁₉	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈×D₄×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₉

prime radical

C₂×D₄×C₁₉

D₈

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈×D₄×C₁₉ ◁ G

N	H ≅ G/N	type
1. C₁₉×#2252	C₂	semidirect
4. C₁₉×#1371	C₂	semidirect
5. C₁₉×#1364	C₂	semidirect
49. C₁₆×C₁₉	SD₁₆	semidirect
50. #12451	C₁₉	direct (N × H)
68. #2252	C₂×C₁₉	semidirect
71. #1371	C₂×C₁₉	semidirect
72. #1364	C₂×C₁₉	semidirect
114. C₁₉	#12451	direct (N × H)
132. C₁₆	SD₁₆×C₁₉	semidirect

H ≅ G/N

type

1. C₁₉×#2252

C₂

semidirect

4. C₁₉×#1371

C₂

semidirect

5. C₁₉×#1364

C₂

semidirect

49. C₁₆×C₁₉

SD₁₆

semidirect

50. #12451

C₁₉

direct (N × H)

68. #2252

C₂×C₁₉

semidirect

71. #1371

C₂×C₁₉

semidirect

72. #1364

C₂×C₁₉

semidirect

114. C₁₉

#12451

direct (N × H)

132. C₁₆

SD₁₆×C₁₉

semidirect

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

subgroup	index
C₁₉×#2252	2
D₄×C₁₆×C₁₉	2
C₁₉×#1673	2
C₁₉×#1371	2
C₁₉×#1364	2
#12451	19

subgroup

index

C₁₉×#2252

D₄×C₁₆×C₁₉

C₁₉×#1673

C₁₉×#1371

C₁₉×#1364

#12451

Subgroups — C₁₉×#12451 #18018410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12451 #18018410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12451 #18018410