Group #18018396 — subgroups

Subgroups — C₁₉×#12437 #18018396

Order 4864.

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

normal subgroups only

This group has 214 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×#2247	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#2214	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×#1358	C₂	1	2	yes	maximal, complemented
6	C₁₉×#510	—	2	2	no	—
7	C₂×C₁₉×Q₃₂	—	2	1	no	—
8	C₂×C₁₉×Q₃₂	—	2	1	no	—
9	C₁₉×#504	—	2	1	no	—
10	C₁₉×#494	C₂²	1	2	yes	—
11	C₁₉×#443	C₂²	1	1	yes	characteristic
12	C₁₉×#364	C₂²	1	1	yes	characteristic
13	C₁₉×#364	C₂²	1	2	yes	—
14	C₁₉×#362	—	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	C₂×Q₁₆×C₁₉	—	2	2	no	—
19	C₂×Q₁₆×C₁₉	—	2	2	no	—
20	C₁₉×#131	—	2	1	no	—
21	C₁₉×C₄⋊Q₈	—	2	2	no	—
22	C₁₉×Q₃₂	—	4	2	no	—
23	C₁₉×Q₃₂	—	4	2	no	—
24	C₁₉×Q₃₂	—	4	1	no	—
25	C₁₉×Q₃₂	—	4	1	no	—
26	C₁₉×SD₃₂	—	4	2	no	—
27	C₁₉×M₃₂	—	2	1	no	—
28	C₁₉×M₃₂	—	2	2	no	—
29	C₁₉×M₃₂	—	4	1	no	—
30	C₂×C₁₆×C₁₉	—	4	1	no	—
31	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
33	C₁₉×#108	D₄	1	1	yes	characteristic
34	C₁₉×#104	—	2	1	no	—
35	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
36	#41×C₁₉	—	2	1	no	—
37	C₂×Q₈×C₁₉	—	4	2	no	—
38	Q₁₆×C₁₉	—	4	2	no	—
39	Q₁₆×C₁₉	—	4	2	no	—
40	Q₁₆×C₁₉	—	4	2	no	—
41	Q₁₆×C₁₉	—	4	2	no	—
42	Q₁₆×C₁₉	D₈	1	1	yes	characteristic
43	SD₁₆×C₁₉	—	2	1	no	—
44	D₈×C₁₉	D₈	1	1	yes	characteristic
45	M₁₆×C₁₉	—	2	1	no	—
46	M₁₆×C₁₉	—	4	1	no	—
47	C₂×C₈×C₁₉	—	4	1	no	—
48	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₁₉	—	8	2	no	—
51	C₄²×C₁₉	#41	1	1	yes	characteristic
52	C₁₆×C₁₉	—	2	1	no	—
53	C₁₆×C₁₉	—	4	1	no	—
54	C₁₆×C₁₉	—	4	2	no	—
55	C₁₆×C₁₉	D₈	1	1	yes	characteristic
56	C₁₆×C₁₉	D₈	1	1	yes	characteristic
57	#12437	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	Q₈×C₁₉	—	2	1	no	—
59	Q₈×C₁₉	—	8	2	no	—
60	Q₈×C₁₉	—	8	2	no	—
61	D₄×C₁₉	—	2	1	no	—
62	D₄×C₁₉	—	4	1	no	—
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	4	1	no	—
65	C₂×C₄×C₁₉	—	8	2	no	—
66	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
67	C₈×C₁₉	—	4	1	no	—
68	C₈×C₁₉	C₂×D₈	1	1	yes	characteristic
69	C₈×C₁₉	C₂×D₈	1	1	yes	characteristic
70	C₈×C₁₉	#135	1	1	yes	characteristic
71	C₈×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
72	#2247	C₂×C₁₉	1	1	yes	characteristic, complemented
73	#2214	C₂×C₁₉	1	2	yes	—
74	#2170	C₂×C₁₉	1	1	yes	characteristic
75	#1373	C₂×C₁₉	1	1	yes	characteristic, complemented
76	#1358	C₂×C₁₉	1	2	yes	complemented
77	C₂²×C₁₉	—	4	1	no	—
78	C₂²×C₁₉	#472	1	1	yes	characteristic
79	C₄×C₁₉	—	2	2	no	—
80	C₄×C₁₉	—	4	1	no	—
81	C₄×C₁₉	—	16	2	no	—
82	C₄×C₁₉	#459	1	1	yes	characteristic
83	C₄×C₁₉	#466	1	1	yes	characteristic
84	#510	—	2	2	no	—
85	C₂×Q₃₂	—	2	1	no	—
86	C₂×Q₃₂	—	2	1	no	—
87	#504	—	2	1	no	—
88	#494	C₂²×C₁₉	1	2	yes	—
89	#443	C₂²×C₁₉	1	1	yes	characteristic
90	#364	C₂²×C₁₉	1	1	yes	characteristic
91	#364	C₂²×C₁₉	1	2	yes	—
92	#362	—	2	2	no	—
93	#350	—	2	1	no	—
94	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
95	C₂×C₁₉	—	2	1	no	—
96	C₂×C₁₉	—	8	1	no	—
97	C₂×C₁₉	#1667	1	1	yes	characteristic
98	#135	D₄×C₁₉	1	1	yes	characteristic
99	C₂×Q₁₆	—	2	2	no	—
100	C₂×Q₁₆	—	2	2	no	—
101	#131	—	2	1	no	—
102	C₄⋊Q₈	—	2	2	no	—
103	Q₃₂	—	4	2	no	—
104	Q₃₂	—	4	2	no	—
105	Q₃₂	—	4	1	no	—
106	Q₃₂	—	4	1	no	—
107	SD₃₂	—	4	2	no	—
108	M₃₂	—	2	1	no	—
109	M₃₂	—	2	2	no	—
110	M₃₂	—	4	1	no	—
111	C₂×C₁₆	—	4	1	no	—
112	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
113	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
114	#108	D₄×C₁₉	1	1	yes	characteristic
115	#104	—	2	1	no	—
116	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
117	C₁₉	#12437	1	1	yes	characteristic, direct product factor, Sylow
118	#41	—	2	1	no	—
119	C₂×Q₈	—	4	2	no	—
120	Q₁₆	—	4	2	no	—
121	Q₁₆	—	4	2	no	—
122	Q₁₆	—	4	2	no	—
123	Q₁₆	—	4	2	no	—
124	Q₁₆	D₈×C₁₉	1	1	yes	characteristic
125	SD₁₆	—	2	1	no	—
126	D₈	D₈×C₁₉	1	1	yes	characteristic
127	M₁₆	—	2	1	no	—
128	M₁₆	—	4	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
131	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
132	C₄⋊C₄	—	8	2	no	—
133	C₄²	#41×C₁₉	1	1	yes	characteristic
134	C₁₆	—	2	1	no	—
135	C₁₆	—	4	1	no	—
136	C₁₆	—	4	2	no	—
137	C₁₆	D₈×C₁₉	1	1	yes	characteristic
138	C₁₆	D₈×C₁₉	1	1	yes	characteristic
139	Q₈	—	2	1	no	—
140	Q₈	—	8	2	no	—
141	Q₈	—	8	2	no	—
142	D₄	—	2	1	no	—
143	D₄	—	4	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	C₂×D₈×C₁₉	1	1	yes	characteristic
150	C₈	C₂×D₈×C₁₉	1	1	yes	characteristic
151	C₈	C₁₉×#135	1	1	yes	characteristic
152	C₈	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂²	—	4	1	no	—
154	C₂²	C₁₉×#472	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	4	1	no	—
157	C₄	—	16	2	no	—
158	C₄	C₁₉×#459	1	1	yes	characteristic
159	C₄	C₁₉×#466	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	8	1	no	—
162	C₂	C₁₉×#1667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2247

C₂

yes

maximal, characteristic, complemented

C₁₉×#2214

C₂

yes

maximal

C₁₉×#2170

C₂

yes

maximal, characteristic

C₁₉×#1373

C₂

yes

maximal, characteristic, complemented

C₁₉×#1358

C₂

yes

maximal, complemented

C₁₉×#510

—

C₂×C₁₉×Q₃₂

—

C₂×C₁₉×Q₃₂

—

C₁₉×#504

—

C₁₉×#494

C₂²

yes

—

C₁₉×#443

C₂²

yes

characteristic

C₁₉×#364

C₂²

yes

characteristic

C₁₉×#364

C₂²

yes

—

C₁₉×#362

—

C₁₉×#350

—

C₄×C₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#135

D₄

yes

characteristic

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₁₉×#131

—

C₁₉×C₄⋊Q₈

—

C₁₉×Q₃₂

—

C₁₉×Q₃₂

—

C₁₉×Q₃₂

—

C₁₉×Q₃₂

—

C₁₉×SD₃₂

—

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×#104

—

C₄×C₈×C₁₉

C₂³

yes

characteristic

#41×C₁₉

—

C₂×Q₈×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

D₈

yes

characteristic

SD₁₆×C₁₉

—

D₈×C₁₉

D₈

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

—

C₄²×C₁₉

#41

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

D₈

yes

characteristic

C₁₆×C₁₉

D₈

yes

characteristic

#12437

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₁₉

—

Q₈×C₁₉

—

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×D₈

yes

characteristic

C₈×C₁₉

C₂×D₈

yes

C₈×C₁₉

yes

C₈×C₁₉

yes

C₂×C₁₉

yes

characteristic, complemented

#2214

C₂×C₁₉

yes

—

#2170

C₂×C₁₉

yes

characteristic

#1373

C₂×C₁₉

yes

characteristic, complemented

#1358

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

#472

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#504

—

#494

C₂²×C₁₉

yes

—

#443

C₂²×C₁₉

yes

characteristic

#364

C₂²×C₁₉

yes

characteristic

#364

C₂²×C₁₉

yes

—

#362

—

#350

—

C₄×C₁₆

C₂²×C₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1667

yes

characteristic

#135

D₄×C₁₉

yes

characteristic

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

#131

—

102

C₄⋊Q₈

—

103

Q₃₂

—

104

Q₃₂

—

105

Q₃₂

—

106

Q₃₂

—

107

SD₃₂

—

108

M₃₂

—

109

M₃₂

—

110

M₃₂

—

111

C₂×C₁₆

—

112

C₂×C₁₆

D₄×C₁₉

yes

characteristic

113

C₂×C₁₆

D₄×C₁₉

yes

characteristic

114

#108

D₄×C₁₉

yes

characteristic

115

#104

—

116

C₄×C₈

C₂³×C₁₉

yes

characteristic

117

C₁₉

#12437

yes

characteristic, direct product factor, Sylow

118

#41

—

119

C₂×Q₈

—

120

Q₁₆

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

124

Q₁₆

D₈×C₁₉

yes

characteristic

125

SD₁₆

—

126

D₈

D₈×C₁₉

yes

characteristic

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

131

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

132

C₄⋊C₄

—

133

C₄²

#41×C₁₉

yes

characteristic

134

C₁₆

—

135

C₁₆

—

136

C₁₆

—

137

C₁₆

D₈×C₁₉

yes

characteristic

138

C₁₆

D₈×C₁₉

yes

characteristic

139

Q₈

—

140

Q₈

—

141

Q₈

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₁₉×#121

yes

characteristic

148

C₈

—

149

C₈

C₂×D₈×C₁₉

yes

characteristic

150

C₈

C₂×D₈×C₁₉

yes

characteristic

151

C₈

C₁₉×#135

yes

characteristic

152

C₈

C₁₉×AGL(1,ℤ/8)

yes

characteristic

153

C₂²

—

154

C₂²

C₁₉×#472

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

C₁₉×#459

yes

characteristic

159

C₄

C₁₉×#466

yes

characteristic

160

C₂

—

161

C₂

—

162

C₂

C₁₉×#1667

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1667
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

#1667

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×#2247	C₂	semidirect
4. C₁₉×#1373	C₂	semidirect
5. C₁₉×#1358	C₂	semidirect
57. #12437	C₁₉	direct (N × H)
72. #2247	C₂×C₁₉	semidirect
75. #1373	C₂×C₁₉	semidirect
76. #1358	C₂×C₁₉	semidirect
117. C₁₉	#12437	direct (N × H)

H ≅ G/N

type

1. C₁₉×#2247

C₂

semidirect

4. C₁₉×#1373

C₂

semidirect

5. C₁₉×#1358

C₂

semidirect

57. #12437

C₁₉

direct (N × H)

72. #2247

C₂×C₁₉

semidirect

75. #1373

C₂×C₁₉

semidirect

76. #1358

C₂×C₁₉

semidirect

117. C₁₉

#12437

direct (N × H)

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

subgroup	index
C₁₉×#2247	2
C₁₉×#2214	2
C₁₉×#2170	2
C₁₉×#1373	2
C₁₉×#1358	2
#12437	19

subgroup

index

C₁₉×#2247

C₁₉×#2214

C₁₉×#2170

C₁₉×#1373

C₁₉×#1358

#12437

Subgroups — C₁₉×#12437 #18018396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12437 #18018396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12437 #18018396