Group #18018270 — subgroups

Subgroups — C₁₉×#12311 #18018270

Order 4864.

366 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3293	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#2213	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1709	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1358	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1335	C₂	1	2	yes	maximal, complemented
6	C₁₉×#578	—	2	2	no	—
7	C₁₉×#509	—	2	2	no	—
8	C₁₉×#494	C₂²	1	1	yes	characteristic
9	C₁₉×#493	C₂²	1	1	yes	characteristic
10	C₁₉×#486	—	2	2	no	—
11	C₁₉×#473	—	2	2	no	—
12	C₁₉×#456	—	2	2	no	—
13	C₁₉×#443	C₂²	1	2	yes	—
14	C₁₉×#364	C₂²	1	2	yes	—
15	C₁₉×#361	—	2	2	no	—
16	C₈²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×2₋¹⁺⁴	—	2	2	no	—
18	C₁₉×#137	—	2	4	no	—
19	C₁₉×#137	—	4	4	no	—
20	C₁₉×#135	—	4	2	no	—
21	C₁₉×#135	D₄	1	2	yes	—
22	C₂×Q₁₆×C₁₉	—	2	2	no	—
23	C₂×Q₁₆×C₁₉	—	4	2	no	—
24	C₂×D₈×C₁₉	—	2	2	no	—
25	C₁₉×#131	—	2	2	no	—
26	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
27	C₁₉×#127	—	2	1	no	—
28	C₁₉×SD₃₂	—	4	2	no	—
29	C₁₉×D₁₆	—	4	2	no	—
30	C₁₉×M₃₂	—	2	2	no	—
31	C₁₉×#108	D₄	1	2	yes	—
32	C₁₉×#104	—	2	2	no	—
33	C₁₉×#102	—	4	4	no	—
34	C₁₉×#101	—	2	2	no	—
35	C₄×C₈×C₁₉	—	2	1	no	—
36	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
37	#41×C₁₉	—	2	2	no	—
38	#41×C₁₉	—	4	4	no	—
39	#41×C₁₉	—	8	2	no	—
40	C₂×Q₈×C₁₉	—	2	2	no	—
41	C₂×Q₈×C₁₉	—	4	4	no	—
42	C₂×D₄×C₁₉	—	4	2	no	—
43	C₂×D₄×C₁₉	—	4	1	no	—
44	Q₁₆×C₁₉	—	4	4	no	—
45	Q₁₆×C₁₉	—	4	2	no	—
46	Q₁₆×C₁₉	—	4	2	no	—
47	Q₁₆×C₁₉	D₈	1	2	yes	—
48	SD₁₆×C₁₉	—	2	2	no	—
49	SD₁₆×C₁₉	—	4	4	no	—
50	D₈×C₁₉	—	4	2	no	—
51	D₈×C₁₉	—	4	2	no	—
52	D₈×C₁₉	D₈	1	2	yes	—
53	M₁₆×C₁₉	—	2	2	no	—
54	M₁₆×C₁₉	—	4	2	no	—
55	C₂×C₈×C₁₉	—	2	2	no	—
56	C₂×C₈×C₁₉	—	4	2	no	—
57	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₁₆×C₁₉	—	4	2	no	—
61	#12311	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₁₉	—	8	1	no	—
63	Q₈×C₁₉	—	2	2	no	—
64	Q₈×C₁₉	—	4	4	no	—
65	Q₈×C₁₉	—	8	2	no	—
66	D₄×C₁₉	—	2	2	no	—
67	D₄×C₁₉	—	4	2	no	—
68	D₄×C₁₉	—	8	4	no	—
69	D₄×C₁₉	—	8	2	no	—
70	D₄×C₁₉	—	8	2	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	4	2	no	—
73	C₂×C₄×C₁₉	—	4	2	no	—
74	C₂×C₄×C₁₉	—	8	4	no	—
75	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
76	C₈×C₁₉	—	4	2	no	—
77	C₈×C₁₉	—	4	2	no	—
78	C₈×C₁₉	C₂×D₈	1	2	yes	—
79	C₈×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
80	#3293	C₂×C₁₉	1	1	yes	characteristic, complemented
81	#2213	C₂×C₁₉	1	2	yes	complemented
82	#1709	C₂×C₁₉	1	1	yes	characteristic, complemented
83	#1358	C₂×C₁₉	1	1	yes	characteristic, complemented
84	#1335	C₂×C₁₉	1	2	yes	complemented
85	C₂²×C₁₉	—	4	2	no	—
86	C₂²×C₁₉	—	16	2	no	—
87	C₂²×C₁₉	—	16	1	no	—
88	C₂²×C₁₉	#453	1	1	yes	characteristic
89	C₄×C₁₉	—	2	2	no	—
90	C₄×C₁₉	—	4	2	no	—
91	C₄×C₁₉	—	8	2	no	—
92	C₄×C₁₉	#447	1	2	yes	—
93	#578	—	2	2	no	—
94	#509	—	2	2	no	—
95	#494	C₂²×C₁₉	1	1	yes	characteristic
96	#493	C₂²×C₁₉	1	1	yes	characteristic
97	#486	—	2	2	no	—
98	#473	—	2	2	no	—
99	#456	—	2	2	no	—
100	#443	C₂²×C₁₉	1	2	yes	—
101	#364	C₂²×C₁₉	1	2	yes	—
102	#361	—	2	2	no	—
103	C₈²	C₂²×C₁₉	1	1	yes	characteristic
104	C₂×C₁₉	—	2	1	no	—
105	C₂×C₁₉	—	8	2	no	—
106	C₂×C₁₉	—	32	1	no	—
107	C₂×C₁₉	#1619	1	1	yes	characteristic
108	2₋¹⁺⁴	—	2	2	no	—
109	#137	—	2	4	no	—
110	#137	—	4	4	no	—
111	#135	—	4	2	no	—
112	#135	D₄×C₁₉	1	2	yes	—
113	C₂×Q₁₆	—	2	2	no	—
114	C₂×Q₁₆	—	4	2	no	—
115	C₂×D₈	—	2	2	no	—
116	#131	—	2	2	no	—
117	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
118	#127	—	2	1	no	—
119	SD₃₂	—	4	2	no	—
120	D₁₆	—	4	2	no	—
121	M₃₂	—	2	2	no	—
122	#108	D₄×C₁₉	1	2	yes	—
123	#104	—	2	2	no	—
124	#102	—	4	4	no	—
125	#101	—	2	2	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
128	C₁₉	#12311	1	1	yes	characteristic, direct product factor, Sylow
129	#41	—	2	2	no	—
130	#41	—	4	4	no	—
131	#41	—	8	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×D₄	—	4	2	no	—
135	C₂×D₄	—	4	1	no	—
136	Q₁₆	—	4	4	no	—
137	Q₁₆	—	4	2	no	—
138	Q₁₆	—	4	2	no	—
139	Q₁₆	D₈×C₁₉	1	2	yes	—
140	SD₁₆	—	2	2	no	—
141	SD₁₆	—	4	4	no	—
142	D₈	—	4	2	no	—
143	D₈	—	4	2	no	—
144	D₈	D₈×C₁₉	1	2	yes	—
145	M₁₆	—	2	2	no	—
146	M₁₆	—	4	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
152	C₁₆	—	4	2	no	—
153	C₂³	—	8	1	no	—
154	Q₈	—	2	2	no	—
155	Q₈	—	4	4	no	—
156	Q₈	—	8	2	no	—
157	D₄	—	2	2	no	—
158	D₄	—	4	2	no	—
159	D₄	—	8	4	no	—
160	D₄	—	8	2	no	—
161	D₄	—	8	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	8	4	no	—
166	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
167	C₈	—	4	2	no	—
168	C₈	—	4	2	no	—
169	C₈	C₂×D₈×C₁₉	1	2	yes	—
170	C₈	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
171	C₂²	—	4	2	no	—
172	C₂²	—	16	2	no	—
173	C₂²	—	16	1	no	—
174	C₂²	C₁₉×#453	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	8	2	no	—
178	C₄	C₁₉×#447	1	2	yes	—
179	C₂	—	2	1	no	—
180	C₂	—	8	2	no	—
181	C₂	—	32	1	no	—
182	C₂	C₁₉×#1619	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3293

C₂

yes

maximal, characteristic, complemented

C₁₉×#2213

C₂

yes

maximal, complemented

C₁₉×#1709

C₂

yes

maximal, characteristic, complemented

C₁₉×#1358

C₂

yes

maximal, characteristic, complemented

C₁₉×#1335

C₂

yes

maximal, complemented

C₁₉×#578

—

C₁₉×#509

—

C₁₉×#494

C₂²

yes

characteristic

C₁₉×#493

C₂²

yes

characteristic

C₁₉×#486

—

C₁₉×#473

—

C₁₉×#456

—

C₁₉×#443

C₂²

yes

—

C₁₉×#364

C₂²

yes

—

C₁₉×#361

—

C₈²×C₁₉

C₂²

yes

characteristic

C₁₉×2₋¹⁺⁴

—

C₁₉×#137

—

C₁₉×#137

—

C₁₉×#135

—

C₁₉×#135

D₄

yes

—

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂×D₈×C₁₉

—

C₁₉×#131

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#127

—

C₁₉×SD₃₂

—

C₁₉×D₁₆

—

C₁₉×M₃₂

—

C₁₉×#108

D₄

yes

—

C₁₉×#104

—

C₁₉×#102

—

C₁₉×#101

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₂³

yes

characteristic

#41×C₁₉

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

D₈

yes

—

SD₁₆×C₁₉

—

SD₁₆×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

D₈

yes

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

#12311

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#120

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

C₂×D₈

yes

—

C₈×C₁₉

AGL(1,ℤ/8)

yes

—

#3293

C₂×C₁₉

yes

characteristic, complemented

#2213

C₂×C₁₉

yes

complemented

#1709

C₂×C₁₉

yes

characteristic, complemented

#1358

C₂×C₁₉

yes

characteristic, complemented

#1335

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#453

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#447

yes

—

#578

—

#509

—

#494

C₂²×C₁₉

yes

characteristic

#493

C₂²×C₁₉

yes

—

—

—

100

#443

C₂²×C₁₉

yes

—

101

#364

C₂²×C₁₉

yes

—

102

#361

—

103

C₈²

C₂²×C₁₉

yes

characteristic

104

C₂×C₁₉

—

105

C₂×C₁₉

—

106

C₂×C₁₉

—

107

C₂×C₁₉

#1619

yes

characteristic

108

2₋¹⁺⁴

—

109

#137

—

110

#137

—

111

#135

—

112

#135

D₄×C₁₉

yes

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×D₈

—

116

#131

—

117

C₄⋊Q₈

D₄×C₁₉

yes

—

118

#127

—

119

SD₃₂

—

120

D₁₆

—

121

M₃₂

—

122

#108

D₄×C₁₉

yes

—

123

#104

—

124

#102

—

125

#101

—

126

C₄×C₈

—

127

C₄×C₈

C₂³×C₁₉

yes

characteristic

128

C₁₉

#12311

yes

characteristic, direct product factor, Sylow

129

#41

—

130

#41

—

131

#41

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×D₄

—

135

C₂×D₄

—

136

Q₁₆

—

137

Q₁₆

—

138

Q₁₆

—

139

Q₁₆

D₈×C₁₉

yes

—

140

SD₁₆

—

141

SD₁₆

—

142

D₈

—

143

D₈

—

144

D₈

D₈×C₁₉

yes

—

145

M₁₆

—

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

C₂×D₄×C₁₉

yes

—

150

C₄⋊C₄

—

151

C₄²

C₂×D₄×C₁₉

yes

characteristic

152

C₁₆

—

153

C₂³

—

154

Q₈

—

155

Q₈

—

156

Q₈

—

157

D₄

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

C₁₉×#120

yes

characteristic

167

C₈

—

168

C₈

—

169

C₈

C₂×D₈×C₁₉

yes

—

170

C₈

C₁₉×AGL(1,ℤ/8)

yes

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₁₉×#453

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

C₁₉×#447

yes

—

179

C₂

—

180

C₂

—

181

C₂

—

182

C₂

C₁₉×#1619

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
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

#1619

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₉

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×#3293	C₂	semidirect
2. C₁₉×#2213	C₂	semidirect
3. C₁₉×#1709	C₂	semidirect
4. C₁₉×#1358	C₂	semidirect
5. C₁₉×#1335	C₂	semidirect
61. #12311	C₁₉	direct (N × H)
80. #3293	C₂×C₁₉	semidirect
81. #2213	C₂×C₁₉	semidirect
82. #1709	C₂×C₁₉	semidirect
83. #1358	C₂×C₁₉	semidirect
84. #1335	C₂×C₁₉	semidirect
128. C₁₉	#12311	direct (N × H)

H ≅ G/N

type

1. C₁₉×#3293

C₂

semidirect

2. C₁₉×#2213

C₂

semidirect

3. C₁₉×#1709

C₂

semidirect

4. C₁₉×#1358

C₂

semidirect

5. C₁₉×#1335

C₂

semidirect

61. #12311

C₁₉

direct (N × H)

80. #3293

C₂×C₁₉

semidirect

81. #2213

C₂×C₁₉

semidirect

82. #1709

C₂×C₁₉

semidirect

83. #1358

C₂×C₁₉

semidirect

84. #1335

C₂×C₁₉

semidirect

128. C₁₉

#12311

direct (N × H)

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

subgroup	index
C₁₉×#3293	2
C₁₉×#2213	2
C₁₉×#1709	2
C₁₉×#1358	2
C₁₉×#1335	2
#12311	19

subgroup

index

C₁₉×#3293

C₁₉×#2213

C₁₉×#1709

C₁₉×#1358

C₁₉×#1335

#12311

Subgroups — C₁₉×#12311 #18018270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12311 #18018270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12311 #18018270