Group #18025670 — subgroups

Subgroups — C₁₉×#19711 #18025670

Order 4864.

636 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 230 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×#3231	C₂	1	4	yes	maximal
2	C₁₉×#3170	C₂	1	2	yes	maximal
3	C₁₉×#1723	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1718	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1452	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#533	C₂²	1	4	yes	—
7	C₁₉×#500	C₂²	1	2	yes	—
8	C₁₉×#500	C₂²	1	4	yes	—
9	C₁₉×#498	C₂²	1	4	yes	—
10	C₁₉×#494	C₂²	1	2	yes	—
11	C₁₉×#487	C₂²	1	4	yes	—
12	C₁₉×#487	C₂²	1	4	yes	—
13	C₁₉×#487	C₂²	1	4	yes	—
14	C₁₉×#485	—	2	8	no	—
15	C₁₉×#479	—	2	8	no	—
16	C₁₉×#467	—	2	4	no	—
17	C₄×Q₁₆×C₁₉	—	2	4	no	—
18	C₁₉×#431	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#322	C₂²	1	4	yes	—
21	C₂×Q₁₆×C₁₉	—	2	4	no	—
22	C₂²×C₈×C₁₉	D₄	1	2	yes	—
23	C₁₉×C₄⋊Q₈	C₂³	1	4	yes	—
24	C₁₉×C₄⋊Q₈	C₂³	1	4	yes	—
25	C₁₉×#125	—	2	4	no	—
26	C₁₉×C₂²⋊Q₈	—	2	4	no	—
27	C₁₉×C₂²⋊Q₈	—	2	4	no	—
28	C₄×Q₈×C₁₉	—	2	4	no	—
29	C₄×Q₈×C₁₉	—	2	4	no	—
30	C₁₉×#117	—	2	4	no	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₁₉×#107	—	2	4	no	—
33	C₁₉×#107	—	2	8	no	—
34	C₁₉×#106	—	2	8	no	—
35	C₁₉×C₄⋊C₈	D₄	1	4	yes	—
36	C₁₉×#103	—	2	8	no	—
37	C₁₉×#103	—	2	8	no	—
38	C₁₉×#103	—	2	8	no	—
39	C₁₉×C₂²⋊C₈	D₄	1	4	yes	—
40	C₄×C₈×C₁₉	D₄	1	2	yes	—
41	C₄×C₈×C₁₉	C₂³	1	2	yes	—
42	C₄×C₈×C₁₉	C₂³	1	4	yes	—
43	C₂×Q₈×C₁₉	—	2	4	no	—
44	C₂×Q₈×C₁₉	—	2	4	no	—
45	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₁₉	—	4	8	no	—
48	C₂×C₈×C₁₉	—	4	2	no	—
49	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
50	C₂×C₈×C₁₉	C₂×D₄	1	8	yes	—
51	C₄⋊C₄×C₁₉	—	2	4	no	—
52	C₄⋊C₄×C₁₉	—	2	4	no	—
53	C₄⋊C₄×C₁₉	—	2	4	no	—
54	C₄⋊C₄×C₁₉	—	2	4	no	—
55	C₄⋊C₄×C₁₉	—	4	4	no	—
56	C₄⋊C₄×C₁₉	—	4	4	no	—
57	C₂²⋊C₄×C₁₉	—	4	4	no	—
58	C₄²×C₁₉	—	4	4	no	—
59	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₁₉	C₂×D₄	1	2	yes	—
61	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
62	#19711	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₁₉	#127	1	1	yes	characteristic
64	Q₈×C₁₉	—	4	8	no	—
65	Q₈×C₁₉	—	4	8	no	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	4	no	—
68	C₂×C₄×C₁₉	—	4	4	no	—
69	C₂×C₄×C₁₉	—	4	4	no	—
70	C₂×C₄×C₁₉	#127	1	2	yes	—
71	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
74	C₈×C₁₉	—	2	4	no	—
75	C₈×C₁₉	—	2	8	no	—
76	#3231	C₂×C₁₉	1	4	yes	—
77	#3170	C₂×C₁₉	1	2	yes	—
78	#1723	C₂×C₁₉	1	4	yes	complemented
79	#1718	C₂×C₁₉	1	4	yes	complemented
80	#1452	C₂×C₁₉	1	1	yes	characteristic
81	C₂²×C₁₉	—	2	1	no	—
82	C₂²×C₁₉	—	2	2	no	—
83	C₂²×C₁₉	C₂×#127	1	1	yes	characteristic
84	C₄×C₁₉	—	2	1	no	—
85	C₄×C₁₉	—	2	2	no	—
86	C₄×C₁₉	—	8	4	no	—
87	C₄×C₁₉	—	8	4	no	—
88	C₄×C₁₉	#575	1	2	yes	—
89	C₄×C₁₉	#578	1	4	yes	—
90	#533	C₂²×C₁₉	1	4	yes	—
91	#500	C₂²×C₁₉	1	2	yes	—
92	#500	C₂²×C₁₉	1	4	yes	—
93	#498	C₂²×C₁₉	1	4	yes	—
94	#494	C₂²×C₁₉	1	2	yes	—
95	#487	C₂²×C₁₉	1	4	yes	—
96	#487	C₂²×C₁₉	1	4	yes	—
97	#487	C₂²×C₁₉	1	4	yes	—
98	#485	—	2	8	no	—
99	#479	—	2	8	no	—
100	#467	—	2	4	no	—
101	C₄×Q₁₆	—	2	4	no	—
102	#431	C₂²×C₁₉	1	2	yes	—
103	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
104	#322	C₂²×C₁₉	1	4	yes	—
105	C₂×C₁₉	—	4	1	no	—
106	C₂×C₁₉	#3150	1	1	yes	characteristic
107	C₂×C₁₉	#3156	1	2	yes	—
108	C₂×Q₁₆	—	2	4	no	—
109	C₂²×C₈	D₄×C₁₉	1	2	yes	—
110	C₄⋊Q₈	C₂³×C₁₉	1	4	yes	—
111	C₄⋊Q₈	C₂³×C₁₉	1	4	yes	—
112	#125	—	2	4	no	—
113	C₂²⋊Q₈	—	2	4	no	—
114	C₂²⋊Q₈	—	2	4	no	—
115	C₄×Q₈	—	2	4	no	—
116	C₄×Q₈	—	2	4	no	—
117	#117	—	2	4	no	—
118	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
119	#107	—	2	4	no	—
120	#107	—	2	8	no	—
121	#106	—	2	8	no	—
122	C₄⋊C₈	D₄×C₁₉	1	4	yes	—
123	#103	—	2	8	no	—
124	#103	—	2	8	no	—
125	#103	—	2	8	no	—
126	C₂²⋊C₈	D₄×C₁₉	1	4	yes	—
127	C₄×C₈	D₄×C₁₉	1	2	yes	—
128	C₄×C₈	C₂³×C₁₉	1	2	yes	—
129	C₄×C₈	C₂³×C₁₉	1	4	yes	—
130	C₁₉	#19711	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	4	no	—
133	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
135	Q₁₆	—	4	8	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
138	C₂×C₈	C₂×D₄×C₁₉	1	8	yes	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₂²⋊C₄	—	4	4	no	—
146	C₄²	—	4	4	no	—
147	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
148	C₄²	C₂×D₄×C₁₉	1	2	yes	—
149	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
150	C₂³	C₁₉×#127	1	1	yes	characteristic
151	Q₈	—	4	8	no	—
152	Q₈	—	4	8	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	C₁₉×#127	1	2	yes	—
158	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
159	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
160	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
161	C₈	—	2	4	no	—
162	C₈	—	2	8	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	C₂×C₁₉×#127	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	8	4	no	—
169	C₄	—	8	4	no	—
170	C₄	C₁₉×#575	1	2	yes	—
171	C₄	C₁₉×#578	1	4	yes	—
172	C₂	—	4	1	no	—
173	C₂	C₁₉×#3150	1	1	yes	characteristic
174	C₂	C₁₉×#3156	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3231

C₂

yes

maximal

C₁₉×#3170

C₂

yes

maximal

C₁₉×#1723

C₂

yes

maximal, complemented

C₁₉×#1718

C₂

yes

maximal, complemented

C₁₉×#1452

C₂

yes

maximal, characteristic

C₁₉×#533

C₂²

yes

—

C₁₉×#500

C₂²

yes

—

C₁₉×#500

C₂²

yes

—

C₁₉×#498

C₂²

yes

—

C₁₉×#494

C₂²

yes

—

C₁₉×#487

C₂²

yes

—

C₁₉×#487

C₂²

yes

—

C₁₉×#487

C₂²

yes

—

C₁₉×#485

—

C₁₉×#479

—

C₁₉×#467

—

C₄×Q₁₆×C₁₉

—

C₁₉×#431

C₂²

yes

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#322

C₂²

yes

—

C₂×Q₁₆×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

—

C₁₉×C₄⋊Q₈

C₂³

yes

—

C₁₉×C₄⋊Q₈

C₂³

yes

—

C₁₉×#125

—

C₁₉×C₂²⋊Q₈

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₁₉×#117

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₁₉×#107

—

C₁₉×#107

—

C₁₉×#106

—

C₁₉×C₄⋊C₈

D₄

yes

—

C₁₉×#103

—

C₁₉×#103

—

C₁₉×#103

—

C₁₉×C₂²⋊C₈

D₄

yes

—

C₄×C₈×C₁₉

D₄

yes

—

C₄×C₈×C₁₉

C₂³

yes

—

C₄×C₈×C₁₉

C₂³

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

Q₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂⁴

yes

characteristic

#19711

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

#127

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#127

yes

—

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂²×D₄

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

#3231

C₂×C₁₉

yes

—

#3170

C₂×C₁₉

yes

—

#1723

C₂×C₁₉

yes

complemented

#1718

C₂×C₁₉

yes

complemented

#1452

C₂×C₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×#127

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#575

yes

—

C₄×C₁₉

#578

yes

—

#533

C₂²×C₁₉

yes

—

#500

C₂²×C₁₉

yes

—

#500

C₂²×C₁₉

yes

—

#498

C₂²×C₁₉

yes

—

#494

C₂²×C₁₉

yes

—

#487

C₂²×C₁₉

yes

—

#487

C₂²×C₁₉

yes

—

#487

C₂²×C₁₉

yes

—

#485

—

#479

—

100

#467

—

101

C₄×Q₁₆

—

102

#431

C₂²×C₁₉

yes

—

103

C₂×C₄×C₈

C₂²×C₁₉

yes

characteristic

104

#322

C₂²×C₁₉

yes

—

105

C₂×C₁₉

—

106

C₂×C₁₉

#3150

yes

characteristic

107

C₂×C₁₉

#3156

yes

—

108

C₂×Q₁₆

—

109

C₂²×C₈

D₄×C₁₉

yes

—

110

C₄⋊Q₈

C₂³×C₁₉

yes

—

111

C₄⋊Q₈

C₂³×C₁₉

yes

—

112

#125

—

113

C₂²⋊Q₈

—

114

C₂²⋊Q₈

—

115

C₄×Q₈

—

116

C₄×Q₈

—

117

#117

—

118

C₂×C₄²

C₂³×C₁₉

yes

characteristic

119

#107

—

120

#107

—

121

#106

—

122

C₄⋊C₈

D₄×C₁₉

yes

—

123

#103

—

124

#103

—

125

#103

—

126

C₂²⋊C₈

D₄×C₁₉

yes

—

127

C₄×C₈

D₄×C₁₉

yes

—

128

C₄×C₈

C₂³×C₁₉

yes

—

129

C₄×C₈

C₂³×C₁₉

yes

—

130

C₁₉

#19711

yes

characteristic, direct product factor, Sylow

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂²×C₄

C₂×D₄×C₁₉

yes

—

134

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

135

Q₁₆

—

136

C₂×C₈

—

137

C₂×C₈

C₂×D₄×C₁₉

yes

—

138

C₂×C₈

C₂×D₄×C₁₉

yes

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₂²⋊C₄

—

146

C₄²

—

147

C₄²

C₂×D₄×C₁₉

yes

characteristic

148

C₄²

C₂×D₄×C₁₉

yes

—

149

C₄²

C₂⁴×C₁₉

yes

characteristic

150

C₂³

C₁₉×#127

yes

characteristic

151

Q₈

—

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₁₉×#127

yes

—

158

C₂×C₄

C₁₉×#127

yes

characteristic

159

C₂×C₄

C₂²×D₄×C₁₉

yes

characteristic

160

C₂×C₄

C₂²×D₄×C₁₉

yes

—

161

C₈

—

162

C₈

—

163

C₂²

—

164

C₂²

—

165

C₂²

C₂×C₁₉×#127

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

C₁₉×#575

yes

—

171

C₄

C₁₉×#578

yes

—

172

C₂

—

173

C₂

C₁₉×#3150

yes

characteristic

174

C₂

C₁₉×#3156

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂³×C₁₉	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴×C₁₉

C₂³×C₁₉

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

N	H ≅ G/N	type
3. C₁₉×#1723	C₂	semidirect
4. C₁₉×#1718	C₂	semidirect
62. #19711	C₁₉	direct (N × H)
78. #1723	C₂×C₁₉	semidirect
79. #1718	C₂×C₁₉	semidirect
130. C₁₉	#19711	direct (N × H)

H ≅ G/N

type

3. C₁₉×#1723

C₂

semidirect

4. C₁₉×#1718

C₂

semidirect

62. #19711

C₁₉

direct (N × H)

78. #1723

C₂×C₁₉

semidirect

79. #1718

C₂×C₁₉

semidirect

130. C₁₉

#19711

direct (N × H)

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

subgroup	index
C₁₉×#3231	2
C₁₉×#3170	2
C₁₉×#1723	2
C₁₉×#1718	2
C₁₉×#1452	2
#19711	19

subgroup

index

C₁₉×#3231

C₁₉×#3170

C₁₉×#1723

C₁₉×#1718

C₁₉×#1452

#19711

Subgroups — C₁₉×#19711 #18025670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#19711 #18025670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#19711 #18025670