Group #18013257 — subgroups

Subgroups — C₁₉×#7298 #18013257

Order 4864.

216 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2162	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉×#108	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1567	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#108	—	2	1	no	—
5	C₁₉×#423	—	2	1	no	—
6	C₄×M₁₆×C₁₉	—	2	1	no	—
7	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
10	C₄×C₁₆×C₁₉	—	2	1	no	—
11	C₁₉×#341	C₄	1	2	yes	—
12	C₁₉×#335	—	2	1	no	—
13	C₁₉×#335	C₄	1	2	yes	—
14	C₂×M₁₆×C₁₉	—	2	2	no	—
15	C₂×M₁₆×C₁₉	—	4	1	no	—
16	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₁₉	—	2	2	no	—
20	C₁₉×#108	—	2	4	no	—
21	C₁₉×C₄⋊C₈	—	2	2	no	—
22	C₁₉×C₄⋊C₈	—	4	1	no	—
23	C₁₉×#97	—	4	1	no	—
24	C₄×C₈×C₁₉	—	4	1	no	—
25	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	Q₁₆	1	1	yes	characteristic
32	M₁₆×C₁₉	—	2	4	no	—
33	M₁₆×C₁₉	—	4	2	no	—
34	M₁₆×C₁₉	—	8	2	no	—
35	C₂×C₈×C₁₉	—	2	2	no	—
36	C₂×C₈×C₁₉	—	4	1	no	—
37	C₂×C₈×C₁₉	—	4	2	no	—
38	C₂×C₈×C₁₉	—	4	2	no	—
39	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₉	D₈	1	1	yes	characteristic
46	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
47	C₁₆×C₁₉	—	4	2	no	—
48	#7298	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₁₉	#103	1	1	yes	characteristic
50	C₂×C₄×C₁₉	—	2	1	no	—
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
59	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
60	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
61	C₈×C₁₉	—	2	1	no	—
62	C₈×C₁₉	—	2	1	no	—
63	C₈×C₁₉	—	2	1	no	—
64	C₈×C₁₉	—	4	4	no	—
65	C₈×C₁₉	—	8	2	no	—
66	C₈×C₁₉	#104	1	2	yes	—
67	#2162	C₂×C₁₉	1	1	yes	characteristic
68	C₄×#108	C₂×C₁₉	1	1	yes	characteristic
69	#1567	C₂×C₁₉	1	1	yes	characteristic
70	C₂²×C₁₉	—	2	1	no	—
71	C₂²×C₁₉	—	2	1	no	—
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	#340	1	1	yes	characteristic
74	C₄×C₁₉	—	2	1	no	—
75	C₄×C₁₉	—	2	1	no	—
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	#339	1	1	yes	characteristic
78	C₄×C₁₉	#344	1	1	yes	characteristic
79	C₄×C₁₉	#361	1	1	yes	characteristic
80	C₄×C₁₉	#362	1	1	yes	characteristic
81	C₄×C₁₉	#368	1	2	yes	—
82	C₂×#108	—	2	1	no	—
83	#423	—	2	1	no	—
84	C₄×M₁₆	—	2	1	no	—
85	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
86	C₄⋊C₁₆	C₄×C₁₉	1	2	yes	—
87	C₂²⋊C₁₆	—	2	1	no	—
88	C₄×C₁₆	—	2	1	no	—
89	#341	C₄×C₁₉	1	2	yes	—
90	#335	—	2	1	no	—
91	#335	C₄×C₁₉	1	2	yes	—
92	C₂×C₁₉	—	4	1	no	—
93	C₂×C₁₉	#1303	1	1	yes	characteristic
94	C₂×C₁₉	#1381	1	1	yes	characteristic
95	C₂×C₁₉	#1387	1	1	yes	characteristic
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	4	1	no	—
98	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
99	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
100	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
101	C₂×C₁₆	—	2	2	no	—
102	#108	—	2	4	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	4	1	no	—
105	#97	—	4	1	no	—
106	C₄×C₈	—	4	1	no	—
107	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
108	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
109	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
110	C₄×C₈	Q₈×C₁₉	1	1	yes	characteristic
111	C₁₉	#7298	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
113	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
114	C₂²×C₄	Q₁₆×C₁₉	1	1	yes	characteristic
115	M₁₆	—	2	4	no	—
116	M₁₆	—	4	2	no	—
117	M₁₆	—	8	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	C₄²×C₁₉	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
124	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
125	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
126	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
127	C₄²	C₄⋊C₄×C₁₉	1	1	yes	characteristic
128	C₄²	D₈×C₁₉	1	1	yes	characteristic
129	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
130	C₁₆	—	4	2	no	—
131	C₂³	C₁₉×#103	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	C₁₉×#95	1	1	yes	characteristic
138	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
139	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
140	C₂×C₄	C₁₉×#103	1	1	yes	characteristic
141	C₂×C₄	C₁₉×#106	1	1	yes	characteristic
142	C₂×C₄	C₁₉×#107	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	4	4	no	—
147	C₈	—	8	2	no	—
148	C₈	C₁₉×#104	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	C₁₉×#340	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	C₁₉×#339	1	1	yes	characteristic
157	C₄	C₁₉×#344	1	1	yes	characteristic
158	C₄	C₁₉×#361	1	1	yes	characteristic
159	C₄	C₁₉×#362	1	1	yes	characteristic
160	C₄	C₁₉×#368	1	2	yes	—
161	C₂	—	4	1	no	—
162	C₂	C₁₉×#1303	1	1	yes	characteristic
163	C₂	C₁₉×#1381	1	1	yes	characteristic
164	C₂	C₁₉×#1387	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2162

C₂

yes

maximal, characteristic

C₄×C₁₉×#108

C₂

yes

maximal, characteristic

C₁₉×#1567

C₂

yes

maximal, characteristic

C₂×C₁₉×#108

—

C₁₉×#423

—

C₄×M₁₆×C₁₉

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×C₄⋊C₁₆

C₄

yes

—

C₁₉×C₂²⋊C₁₆

—

C₄×C₁₆×C₁₉

—

C₁₉×#341

C₄

yes

—

C₁₉×#335

—

C₁₉×#335

C₄

yes

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

—

C₁₉×#108

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×#97

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

Q₈

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂²×C₄×C₁₉

Q₁₆

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄²

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

D₈

yes

characteristic

C₄²×C₁₉

SD₁₆

yes

characteristic

C₁₆×C₁₉

—

#7298

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

#103

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#104

yes

—

#2162

C₂×C₁₉

yes

characteristic

C₄×#108

C₂×C₁₉

yes

characteristic

#1567

C₂×C₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#340

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₂×#108

—

#423

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂²×C₁₉

yes

characteristic

C₄⋊C₁₆

C₄×C₁₉

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#341

C₄×C₁₉

yes

—

#335

—

#335

C₄×C₁₉

yes

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₁₉

yes

characteristic

C₂²×C₈

D₄×C₁₉

yes

characteristic

100

C₂×C₄²

D₄×C₁₉

yes

characteristic

101

C₂×C₁₆

—

102

#108

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

C₂×C₄×C₁₉

yes

characteristic

108

C₄×C₈

C₂×C₄×C₁₉

yes

characteristic

109

C₄×C₈

D₄×C₁₉

yes

characteristic

110

C₄×C₈

Q₈×C₁₉

yes

characteristic

111

C₁₉

#7298

yes

characteristic, direct product factor, Sylow

112

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

characteristic

113

C₂²×C₄

SD₁₆×C₁₉

yes

characteristic

114

C₂²×C₄

Q₁₆×C₁₉

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₄²×C₁₉

yes

characteristic

123

C₂×C₈

C₂²⋊C₄×C₁₉

yes

characteristic

124

C₂×C₈

C₄⋊C₄×C₁₉

yes

characteristic

125

C₂×C₈

C₄⋊C₄×C₁₉

yes

characteristic

126

C₄²

C₂²⋊C₄×C₁₉

yes

characteristic

127

C₄²

C₄⋊C₄×C₁₉

yes

characteristic

128

C₄²

D₈×C₁₉

yes

characteristic

129

C₄²

SD₁₆×C₁₉

yes

characteristic

130

C₁₆

—

131

C₂³

C₁₉×#103

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₁₉×#95

yes

characteristic

138

C₂×C₄

C₁₉×#102

yes

characteristic

139

C₂×C₄

C₁₉×#102

yes

characteristic

140

C₂×C₄

C₁₉×#103

yes

characteristic

141

C₂×C₄

C₁₉×#106

yes

characteristic

142

C₂×C₄

C₁₉×#107

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

C₁₉×#104

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₁₉×#340

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₁₉×#339

yes

characteristic

157

C₄

C₁₉×#344

yes

characteristic

158

C₄

C₁₉×#361

yes

characteristic

159

C₄

C₁₉×#362

yes

characteristic

160

C₄

C₁₉×#368

yes

—

161

C₂

—

162

C₂

C₁₉×#1303

yes

characteristic

163

C₂

C₁₉×#1381

yes

characteristic

164

C₂

C₁₉×#1387

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#102
commutator	C₂×C₈	C₄²×C₁₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₉
socle	C₂²×C₁₉	#340
prime radical	C₂³×C₁₉	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₉

#102

commutator

C₂×C₈

C₄²×C₁₉

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₁₉

C₂²×C₁₉

C₂³×C₁₉

G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

N	H ≅ G/N	type
48. #7298	C₁₉	direct (N × H)
111. C₁₉	#7298	direct (N × H)

H ≅ G/N

type

48. #7298

C₁₉

direct (N × H)

111. C₁₉

#7298

direct (N × H)

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

subgroup	index
C₁₉×#2162	2
C₄×C₁₉×#108	2
C₁₉×#1567	2
#7298	19

subgroup

index

C₁₉×#2162

C₄×C₁₉×#108

C₁₉×#1567

#7298

Subgroups — C₁₉×#7298 #18013257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#7298 #18013257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#7298 #18013257