Group #18018399 — subgroups

Subgroups — C₁₉×#12440 #18018399

Order 4864.

216 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 86 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
2	Q₈×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1670	C₂	1	2	yes	maximal
4	C₁₉×#1371	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1362	C₂	1	2	yes	maximal
6	C₂×C₁₉×Q₃₂	—	2	1	no	—
7	C₂×C₁₉×Q₃₂	—	2	1	no	—
8	C₁₉×#494	C₂²	1	2	yes	—
9	C₁₉×#475	—	2	2	no	—
10	C₈×Q₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×C₄⋊C₁₆	—	2	1	no	—
12	C₁₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₁₉	—	2	1	no	—
14	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#335	C₂²	1	2	yes	—
16	C₁₉×#332	—	2	2	no	—
17	C₂×Q₁₆×C₁₉	—	2	2	no	—
18	C₂×Q₁₆×C₁₉	—	2	2	no	—
19	C₁₉×C₄⋊Q₈	—	2	2	no	—
20	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
21	C₁₉×Q₃₂	—	4	2	no	—
22	C₁₉×Q₃₂	—	4	2	no	—
23	C₂×C₁₆×C₁₉	—	2	1	no	—
24	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
26	C₁₉×#106	—	4	2	no	—
27	C₁₉×C₄⋊C₈	—	2	1	no	—
28	C₁₉×C₄⋊C₈	—	2	2	no	—
29	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
30	C₁₉×#103	—	4	2	no	—
31	C₄×C₈×C₁₉	—	2	1	no	—
32	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₉	—	4	2	no	—
34	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
35	Q₁₆×C₁₉	—	4	2	no	—
36	Q₁₆×C₁₉	—	4	2	no	—
37	Q₁₆×C₁₉	—	8	2	no	—
38	C₂×C₈×C₁₉	—	2	1	no	—
39	C₂×C₈×C₁₉	—	4	2	no	—
40	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₁₉	—	2	1	no	—
43	C₄⋊C₄×C₁₉	—	4	2	no	—
44	C₄⋊C₄×C₁₉	—	4	2	no	—
45	C₄⋊C₄×C₁₉	D₈	1	1	yes	characteristic
46	C₄²×C₁₉	—	2	1	no	—
47	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₁₉	—	2	1	no	—
49	C₁₆×C₁₉	—	4	1	no	—
50	C₁₆×C₁₉	SD₁₆	1	2	yes	—
51	#12440	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	Q₈×C₁₉	—	2	1	no	—
53	Q₈×C₁₉	—	8	2	no	—
54	Q₈×C₁₉	—	8	2	no	—
55	Q₈×C₁₉	Q₃₂	1	2	yes	complemented
56	C₂×C₄×C₁₉	—	2	1	no	—
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	8	2	no	—
59	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
62	C₈×C₁₉	—	2	1	no	—
63	C₈×C₁₉	—	4	1	no	—
64	C₈×C₁₉	—	8	2	no	—
65	C₈×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₈×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
67	#2247	C₂×C₁₉	1	1	yes	characteristic
68	Q₈×C₁₆	C₂×C₁₉	1	1	yes	characteristic
69	#1670	C₂×C₁₉	1	2	yes	—
70	#1371	C₂×C₁₉	1	1	yes	characteristic
71	#1362	C₂×C₁₉	1	2	yes	—
72	C₂²×C₁₉	#466	1	1	yes	characteristic
73	C₄×C₁₉	—	2	2	no	—
74	C₄×C₁₉	—	2	1	no	—
75	C₄×C₁₉	—	4	1	no	—
76	C₄×C₁₉	—	16	2	no	—
77	C₄×C₁₉	#460	1	1	yes	characteristic
78	C₄×C₁₉	#472	1	1	yes	characteristic
79	C₄×C₁₉	C₂×Q₃₂	1	1	yes	characteristic
80	C₄×C₁₉	#508	1	1	yes	characteristic
81	C₂×Q₃₂	—	2	1	no	—
82	C₂×Q₃₂	—	2	1	no	—
83	#494	C₂²×C₁₉	1	2	yes	—
84	#475	—	2	2	no	—
85	C₈×Q₈	C₂²×C₁₉	1	1	yes	characteristic
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	C₂²×C₁₉	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
90	#335	C₂²×C₁₉	1	2	yes	—
91	#332	—	2	2	no	—
92	C₂×C₁₉	#1668	1	1	yes	characteristic
93	C₂×C₁₉	#2216	1	1	yes	characteristic
94	C₂×C₁₉	#2222	1	1	yes	characteristic
95	C₂×Q₁₆	—	2	2	no	—
96	C₂×Q₁₆	—	2	2	no	—
97	C₄⋊Q₈	—	2	2	no	—
98	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
99	Q₃₂	—	4	2	no	—
100	Q₃₂	—	4	2	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
103	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
104	#106	—	4	2	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
108	#103	—	4	2	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
111	C₁₉	#12440	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	D₈×C₁₉	1	1	yes	characteristic
114	Q₁₆	—	4	2	no	—
115	Q₁₆	—	4	2	no	—
116	Q₁₆	—	8	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
120	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	D₈×C₁₉	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	SD₁₆×C₁₉	1	2	yes	—
130	Q₈	—	2	1	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	8	2	no	—
133	Q₈	C₁₉×Q₃₂	1	2	yes	complemented
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
138	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
139	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	4	1	no	—
142	C₈	—	8	2	no	—
143	C₈	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
144	C₈	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂²	C₁₉×#466	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	16	2	no	—
150	C₄	C₁₉×#460	1	1	yes	characteristic
151	C₄	C₁₉×#472	1	1	yes	characteristic
152	C₄	C₂×C₁₉×Q₃₂	1	1	yes	characteristic
153	C₄	C₁₉×#508	1	1	yes	characteristic
154	C₂	C₁₉×#1668	1	1	yes	characteristic
155	C₂	C₁₉×#2216	1	1	yes	characteristic
156	C₂	C₁₉×#2222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2247

C₂

yes

maximal, characteristic

Q₈×C₁₆×C₁₉

C₂

yes

maximal, characteristic

C₁₉×#1670

C₂

yes

maximal

C₁₉×#1371

C₂

yes

maximal, characteristic

C₁₉×#1362

C₂

yes

maximal

C₂×C₁₉×Q₃₂

—

C₂×C₁₉×Q₃₂

—

C₁₉×#494

C₂²

yes

—

C₁₉×#475

—

C₈×Q₈×C₁₉

C₂²

yes

characteristic

C₁₉×C₄⋊C₁₆

—

C₁₉×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₆×C₁₉

—

C₄×C₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#335

C₂²

yes

—

C₁₉×#332

—

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₄×Q₈×C₁₉

D₄

yes

characteristic

C₁₉×Q₃₂

—

C₁₉×Q₃₂

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₁₉×#106

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

D₄

yes

characteristic

C₁₉×#103

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₂³

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

D₈

yes

characteristic

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₉

#41

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

D₈

yes

characteristic

C₄²×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

SD₁₆

yes

—

#12440

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

Q₃₂

yes

complemented

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₉

#135

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

C₂×SD₁₆

yes

C₈×C₁₉

yes

C₂×C₁₉

yes

characteristic

Q₈×C₁₆

C₂×C₁₉

yes

characteristic

#1670

C₂×C₁₉

yes

—

#1371

C₂×C₁₉

yes

characteristic

#1362

C₂×C₁₉

yes

—

C₂²×C₁₉

#466

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

C₂×Q₃₂

yes

characteristic

C₄×C₁₉

#508

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

C₂²×C₁₉

yes

—

#475

—

C₈×Q₈

C₂²×C₁₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²×C₁₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂²×C₁₉

yes

characteristic

#335

C₂²×C₁₉

yes

—

#332

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

D₄×C₁₉

yes

characteristic

Q₃₂

—

100

Q₃₂

—

101

C₂×C₁₆

—

102

C₂×C₁₆

D₄×C₁₉

yes

characteristic

103

C₂×C₁₆

D₄×C₁₉

yes

characteristic

104

#106

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

D₄×C₁₉

yes

characteristic

108

#103

—

109

C₄×C₈

—

110

C₄×C₈

C₂³×C₁₉

yes

characteristic

111

C₁₉

#12440

yes

characteristic, direct product factor, Sylow

112

C₂×Q₈

—

113

C₂×Q₈

D₈×C₁₉

yes

characteristic

114

Q₁₆

—

115

Q₁₆

—

116

Q₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

120

C₂×C₈

#41×C₁₉

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

D₈×C₁₉

yes

characteristic

125

C₄²

—

126

C₄²

C₂×D₄×C₁₉

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

SD₁₆×C₁₉

yes

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

C₁₉×Q₃₂

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₁₉×#121

yes

characteristic

138

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

139

C₂×C₄

C₁₉×#135

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

C₂×SD₁₆×C₁₉

yes

characteristic

144

C₈

C₁₉×AGL(1,ℤ/8)

yes

characteristic

145

C₂²

C₁₉×#466

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₁₉×#460

yes

characteristic

151

C₄

C₁₉×#472

yes

characteristic

152

C₄

C₂×C₁₉×Q₃₂

yes

characteristic

153

C₄

C₁₉×#508

yes

characteristic

154

C₂

C₁₉×#1668

yes

characteristic

155

C₂

C₁₉×#2216

yes

characteristic

156

C₂

C₁₉×#2222

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₉	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈×Q₈×C₁₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₉

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₉

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈×Q₈×C₁₉ ◁ G

N	H ≅ G/N	type
51. #12440	C₁₉	direct (N × H)
55. Q₈×C₁₉	Q₃₂	semidirect
111. C₁₉	#12440	direct (N × H)
133. Q₈	C₁₉×Q₃₂	semidirect

H ≅ G/N

type

51. #12440

C₁₉

direct (N × H)

55. Q₈×C₁₉

Q₃₂

semidirect

111. C₁₉

#12440

direct (N × H)

133. Q₈

C₁₉×Q₃₂

semidirect

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

subgroup	index
C₁₉×#2247	2
Q₈×C₁₆×C₁₉	2
C₁₉×#1670	2
C₁₉×#1371	2
C₁₉×#1362	2
#12440	19

subgroup

index

C₁₉×#2247

Q₈×C₁₆×C₁₉

C₁₉×#1670

C₁₉×#1371

C₁₉×#1362

#12440

Subgroups — C₁₉×#12440 #18018399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12440 #18018399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12440 #18018399