Group #18018272 — subgroups

Subgroups — C₁₉×#12313 #18018272

Order 4864.

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

normal subgroups only

This group has 356 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
2	C₁₉×#2214	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1713	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×#510	—	2	2	no	—
8	C₁₉×#500	—	2	2	no	—
9	C₁₉×#494	C₂²	1	1	yes	characteristic
10	C₁₉×#494	C₂²	1	1	yes	characteristic
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×#362	—	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	—	2	2	no	—
24	C₂×Q₁₆×C₁₉	—	4	2	no	—
25	C₁₉×#131	—	2	2	no	—
26	C₁₉×C₄⋊Q₈	—	2	1	no	—
27	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
28	C₁₉×Q₃₂	—	4	2	no	—
29	C₁₉×SD₃₂	—	4	2	no	—
30	C₁₉×M₃₂	—	2	2	no	—
31	C₁₉×#108	D₄	1	2	yes	—
32	C₁₉×#107	—	4	4	no	—
33	C₁₉×#104	—	2	2	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×Q₈×C₁₉	—	4	1	no	—
43	Q₁₆×C₁₉	—	4	4	no	—
44	Q₁₆×C₁₉	—	4	2	no	—
45	Q₁₆×C₁₉	—	4	2	no	—
46	Q₁₆×C₁₉	—	4	2	no	—
47	Q₁₆×C₁₉	—	4	2	no	—
48	Q₁₆×C₁₉	D₈	1	2	yes	—
49	SD₁₆×C₁₉	—	2	2	no	—
50	SD₁₆×C₁₉	—	4	4	no	—
51	D₈×C₁₉	D₈	1	2	yes	—
52	M₁₆×C₁₉	—	2	2	no	—
53	M₁₆×C₁₉	—	4	2	no	—
54	C₂×C₈×C₁₉	—	2	2	no	—
55	C₂×C₈×C₁₉	—	4	2	no	—
56	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
57	C₄⋊C₄×C₁₉	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₁₆×C₁₉	—	4	2	no	—
61	#12313	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	Q₈×C₁₉	—	2	2	no	—
63	Q₈×C₁₉	—	4	4	no	—
64	Q₈×C₁₉	—	8	2	no	—
65	Q₈×C₁₉	—	8	2	no	—
66	D₄×C₁₉	—	2	2	no	—
67	D₄×C₁₉	—	4	2	no	—
68	D₄×C₁₉	—	8	2	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	4	2	no	—
71	C₂×C₄×C₁₉	—	4	2	no	—
72	C₂×C₄×C₁₉	—	8	4	no	—
73	C₂×C₄×C₁₉	—	8	1	no	—
74	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
75	C₈×C₁₉	—	2	4	no	—
76	C₈×C₁₉	—	4	2	no	—
77	C₈×C₁₉	C₂×D₈	1	2	yes	—
78	C₈×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
79	#3293	C₂×C₁₉	1	1	yes	characteristic
80	#2214	C₂×C₁₉	1	2	yes	complemented
81	#1713	C₂×C₁₉	1	1	yes	characteristic, complemented
82	#1358	C₂×C₁₉	1	1	yes	characteristic, complemented
83	#1335	C₂×C₁₉	1	2	yes	complemented
84	C₂²×C₁₉	—	4	2	no	—
85	C₂²×C₁₉	#453	1	1	yes	characteristic
86	C₄×C₁₉	—	2	2	no	—
87	C₄×C₁₉	—	4	2	no	—
88	C₄×C₁₉	—	8	2	no	—
89	C₄×C₁₉	—	16	1	no	—
90	C₄×C₁₉	#447	1	2	yes	—
91	#578	—	2	2	no	—
92	#510	—	2	2	no	—
93	#500	—	2	2	no	—
94	#494	C₂²×C₁₉	1	1	yes	characteristic
95	#494	C₂²×C₁₉	1	1	yes	characteristic
96	#473	—	2	2	no	—
97	#456	—	2	2	no	—
98	#443	C₂²×C₁₉	1	2	yes	—
99	#364	C₂²×C₁₉	1	2	yes	—
100	#362	—	2	2	no	—
101	C₈²	C₂²×C₁₉	1	1	yes	characteristic
102	C₂×C₁₉	—	2	1	no	—
103	C₂×C₁₉	—	8	2	no	—
104	C₂×C₁₉	#1619	1	1	yes	characteristic
105	2₋¹⁺⁴	—	2	2	no	—
106	#137	—	2	4	no	—
107	#137	—	4	4	no	—
108	#135	—	4	2	no	—
109	#135	D₄×C₁₉	1	2	yes	—
110	C₂×Q₁₆	—	2	2	no	—
111	C₂×Q₁₆	—	2	2	no	—
112	C₂×Q₁₆	—	4	2	no	—
113	#131	—	2	2	no	—
114	C₄⋊Q₈	—	2	1	no	—
115	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
116	Q₃₂	—	4	2	no	—
117	SD₃₂	—	4	2	no	—
118	M₃₂	—	2	2	no	—
119	#108	D₄×C₁₉	1	2	yes	—
120	#107	—	4	4	no	—
121	#104	—	2	2	no	—
122	#101	—	2	2	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
125	C₁₉	#12313	1	1	yes	characteristic, direct product factor, Sylow
126	#41	—	2	2	no	—
127	#41	—	4	4	no	—
128	#41	—	8	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	4	4	no	—
131	C₂×Q₈	—	4	1	no	—
132	Q₁₆	—	4	4	no	—
133	Q₁₆	—	4	2	no	—
134	Q₁₆	—	4	2	no	—
135	Q₁₆	—	4	2	no	—
136	Q₁₆	—	4	2	no	—
137	Q₁₆	D₈×C₁₉	1	2	yes	—
138	SD₁₆	—	2	2	no	—
139	SD₁₆	—	4	4	no	—
140	D₈	D₈×C₁₉	1	2	yes	—
141	M₁₆	—	2	2	no	—
142	M₁₆	—	4	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
149	C₁₆	—	4	2	no	—
150	Q₈	—	2	2	no	—
151	Q₈	—	4	4	no	—
152	Q₈	—	8	2	no	—
153	Q₈	—	8	2	no	—
154	D₄	—	2	2	no	—
155	D₄	—	4	2	no	—
156	D₄	—	8	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₈	—	4	2	no	—
165	C₈	C₂×D₈×C₁₉	1	2	yes	—
166	C₈	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
167	C₂²	—	4	2	no	—
168	C₂²	C₁₉×#453	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	2	no	—
172	C₄	—	16	1	no	—
173	C₄	C₁₉×#447	1	2	yes	—
174	C₂	—	2	1	no	—
175	C₂	—	8	2	no	—
176	C₂	C₁₉×#1619	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3293

C₂

yes

maximal, characteristic

C₁₉×#2214

C₂

yes

maximal, complemented

C₁₉×#1713

C₂

yes

maximal, characteristic, complemented

C₁₉×#1358

C₂

yes

maximal, characteristic, complemented

C₁₉×#1335

C₂

yes

maximal, complemented

C₁₉×#578

—

C₁₉×#510

—

C₁₉×#500

—

C₁₉×#494

C₂²

yes

characteristic

C₁₉×#494

C₂²

yes

characteristic

C₁₉×#473

—

C₁₉×#456

—

C₁₉×#443

C₂²

yes

—

C₁₉×#364

C₂²

yes

—

C₁₉×#362

—

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×Q₁₆×C₁₉

—

C₁₉×#131

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×Q₃₂

—

C₁₉×SD₃₂

—

C₁₉×M₃₂

—

C₁₉×#108

D₄

yes

—

C₁₉×#107

—

C₁₉×#104

—

C₁₉×#101

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₂³

yes

characteristic

#41×C₁₉

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

D₈

yes

—

SD₁₆×C₁₉

—

SD₁₆×C₁₉

—

D₈×C₁₉

D₈

yes

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

#12313

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×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

#2214

C₂×C₁₉

yes

complemented

#1713

C₂×C₁₉

yes

characteristic, complemented

#1358

C₂×C₁₉

yes

characteristic, complemented

#1335

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

#453

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#447

yes

—

—

—

—

C₂²×C₁₉

yes

characteristic

#494

C₂²×C₁₉

yes

—

—

C₂²×C₁₉

yes

—

#364

C₂²×C₁₉

yes

—

100

#362

—

101

C₈²

C₂²×C₁₉

yes

characteristic

102

C₂×C₁₉

—

103

C₂×C₁₉

—

104

C₂×C₁₉

#1619

yes

characteristic

105

2₋¹⁺⁴

—

106

#137

—

107

#137

—

108

#135

—

109

#135

D₄×C₁₉

yes

—

110

C₂×Q₁₆

—

111

C₂×Q₁₆

—

112

C₂×Q₁₆

—

113

#131

—

114

C₄⋊Q₈

—

115

C₄⋊Q₈

D₄×C₁₉

yes

—

116

Q₃₂

—

117

SD₃₂

—

118

M₃₂

—

119

#108

D₄×C₁₉

yes

—

120

#107

—

121

#104

—

122

#101

—

123

C₄×C₈

—

124

C₄×C₈

C₂³×C₁₉

yes

characteristic

125

C₁₉

#12313

yes

characteristic, direct product factor, Sylow

126

#41

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

Q₁₆

—

133

Q₁₆

—

134

Q₁₆

—

135

Q₁₆

—

136

Q₁₆

—

137

Q₁₆

D₈×C₁₉

yes

—

138

SD₁₆

—

139

SD₁₆

—

140

D₈

D₈×C₁₉

yes

—

141

M₁₆

—

142

M₁₆

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

C₂×D₄×C₁₉

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄²

C₂×D₄×C₁₉

yes

characteristic

149

C₁₆

—

150

Q₈

—

151

Q₈

—

152

Q₈

—

153

Q₈

—

154

D₄

—

155

D₄

—

156

D₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₁₉×#120

yes

characteristic

163

C₈

—

164

C₈

—

165

C₈

C₂×D₈×C₁₉

yes

—

166

C₈

C₁₉×AGL(1,ℤ/8)

yes

—

167

C₂²

—

168

C₂²

C₁₉×#453

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

C₁₉×#447

yes

—

174

C₂

—

175

C₂

—

176

C₂

C₁₉×#1619

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₁₉×#3293	C₂
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

prime radical

C₁₉×#3293

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
2. C₁₉×#2214	C₂	semidirect
3. C₁₉×#1713	C₂	semidirect
4. C₁₉×#1358	C₂	semidirect
5. C₁₉×#1335	C₂	semidirect
61. #12313	C₁₉	direct (N × H)
80. #2214	C₂×C₁₉	semidirect
81. #1713	C₂×C₁₉	semidirect
82. #1358	C₂×C₁₉	semidirect
83. #1335	C₂×C₁₉	semidirect
125. C₁₉	#12313	direct (N × H)

H ≅ G/N

type

2. C₁₉×#2214

C₂

semidirect

3. C₁₉×#1713

C₂

semidirect

4. C₁₉×#1358

C₂

semidirect

5. C₁₉×#1335

C₂

semidirect

61. #12313

C₁₉

direct (N × H)

80. #2214

C₂×C₁₉

semidirect

81. #1713

C₂×C₁₉

semidirect

82. #1358

C₂×C₁₉

semidirect

83. #1335

C₂×C₁₉

semidirect

125. C₁₉

#12313

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×#2214	2
C₁₉×#1713	2
C₁₉×#1358	2
C₁₉×#1335	2
#12313	19

subgroup

index

C₁₉×#3293

C₁₉×#2214

C₁₉×#1713

C₁₉×#1358

C₁₉×#1335

#12313

Subgroups — C₁₉×#12313 #18018272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12313 #18018272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12313 #18018272