Group #18018416 — subgroups

Subgroups — C₁₉×#12457 #18018416

Order 4864.

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

normal subgroups only

This group has 204 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×#2252	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1674	C₂	1	2	yes	maximal
4	C₁₉×#1371	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1364	C₂	1	2	yes	maximal
6	C₁₉×#500	C₂²	1	2	yes	—
7	C₁₉×#462	—	2	2	no	—
8	C₈×Q₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₁₉×#366	—	2	1	no	—
10	C₁₉×#366	—	2	1	no	—
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	—	4	2	no	—
18	C₁₉×C₄⋊Q₈	—	2	2	no	—
19	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₉	—	2	1	no	—
21	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
23	C₁₉×#107	—	2	2	no	—
24	C₁₉×#107	—	2	2	no	—
25	C₁₉×C₄⋊C₈	—	2	1	no	—
26	C₁₉×C₄⋊C₈	—	2	2	no	—
27	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
28	C₁₉×#103	—	4	2	no	—
29	C₄×C₈×C₁₉	—	2	1	no	—
30	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₁₉	—	4	2	no	—
32	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
33	Q₁₆×C₁₉	—	8	4	no	—
34	C₂×C₈×C₁₉	—	2	1	no	—
35	C₂×C₈×C₁₉	—	4	2	no	—
36	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₄⋊C₄×C₁₉	—	4	2	no	—
40	C₄⋊C₄×C₁₉	—	4	2	no	—
41	C₄⋊C₄×C₁₉	D₈	1	1	yes	characteristic
42	C₄²×C₁₉	—	2	1	no	—
43	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₁₉	—	2	1	no	—
45	C₁₆×C₁₉	—	4	1	no	—
46	C₁₆×C₁₉	Q₁₆	1	2	yes	complemented
47	#12457	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	Q₈×C₁₉	—	2	1	no	—
49	Q₈×C₁₉	—	8	2	no	—
50	Q₈×C₁₉	—	8	2	no	—
51	Q₈×C₁₉	D₁₆	1	2	yes	—
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	8	2	no	—
55	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
56	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
58	C₈×C₁₉	—	2	1	no	—
59	C₈×C₁₉	—	4	1	no	—
60	C₈×C₁₉	—	8	2	no	—
61	C₈×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₈×C₁₉	#137	1	1	yes	characteristic
63	#2252	C₂×C₁₉	1	1	yes	characteristic
64	Q₈×C₁₆	C₂×C₁₉	1	1	yes	characteristic
65	#1674	C₂×C₁₉	1	2	yes	—
66	#1371	C₂×C₁₉	1	1	yes	characteristic
67	#1364	C₂×C₁₉	1	2	yes	—
68	C₂²×C₁₉	#466	1	1	yes	characteristic
69	C₄×C₁₉	—	2	2	no	—
70	C₄×C₁₉	—	2	1	no	—
71	C₄×C₁₉	—	4	1	no	—
72	C₄×C₁₉	—	16	2	no	—
73	C₄×C₁₉	#462	1	1	yes	characteristic
74	C₄×C₁₉	#473	1	1	yes	characteristic
75	C₄×C₁₉	C₂×D₁₆	1	1	yes	characteristic
76	C₄×C₁₉	#508	1	1	yes	characteristic
77	#500	C₂²×C₁₉	1	2	yes	—
78	#462	—	2	2	no	—
79	C₈×Q₈	C₂²×C₁₉	1	1	yes	characteristic
80	#366	—	2	1	no	—
81	#366	—	2	1	no	—
82	C₄⋊C₁₆	—	2	1	no	—
83	C₄⋊C₁₆	C₂²×C₁₉	1	1	yes	characteristic
84	C₄×C₁₆	—	2	1	no	—
85	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
86	#335	C₂²×C₁₉	1	2	yes	—
87	#332	—	2	2	no	—
88	C₂×C₁₉	#1674	1	1	yes	characteristic
89	C₂×C₁₉	#2215	1	1	yes	characteristic
90	C₂×C₁₉	#2222	1	1	yes	characteristic
91	C₂×Q₁₆	—	4	2	no	—
92	C₄⋊Q₈	—	2	2	no	—
93	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
96	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
97	#107	—	2	2	no	—
98	#107	—	2	2	no	—
99	C₄⋊C₈	—	2	1	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
102	#103	—	4	2	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
105	C₁₉	#12457	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	D₈×C₁₉	1	1	yes	characteristic
108	Q₁₆	—	8	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
112	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	D₈×C₁₉	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	Q₁₆×C₁₉	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	8	2	no	—
124	Q₈	—	8	2	no	—
125	Q₈	C₁₉×D₁₆	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	8	2	no	—
129	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
130	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
131	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	8	2	no	—
135	C₈	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
136	C₈	C₁₉×#137	1	1	yes	characteristic
137	C₂²	C₁₉×#466	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	16	2	no	—
142	C₄	C₁₉×#462	1	1	yes	characteristic
143	C₄	C₁₉×#473	1	1	yes	characteristic
144	C₄	C₂×C₁₉×D₁₆	1	1	yes	characteristic
145	C₄	C₁₉×#508	1	1	yes	characteristic
146	C₂	C₁₉×#1674	1	1	yes	characteristic
147	C₂	C₁₉×#2215	1	1	yes	characteristic
148	C₂	C₁₉×#2222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2252

C₂

yes

maximal, characteristic

Q₈×C₁₆×C₁₉

C₂

yes

maximal, characteristic

C₁₉×#1674

C₂

yes

maximal

C₁₉×#1371

C₂

yes

maximal, characteristic

C₁₉×#1364

C₂

yes

maximal

C₁₉×#500

C₂²

yes

—

C₁₉×#462

—

C₈×Q₈×C₁₉

C₂²

yes

characteristic

C₁₉×#366

—

C₁₉×#366

—

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

—

C₁₉×C₄⋊Q₈

—

C₄×Q₈×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₁₉×#107

—

C₁₉×#107

—

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

—

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

Q₁₆

yes

complemented

#12457

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

D₁₆

yes

—

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×Q₁₆

yes

C₈×C₁₉

yes

C₂×C₁₉

yes

characteristic

Q₈×C₁₆

C₂×C₁₉

yes

characteristic

#1674

C₂×C₁₉

yes

—

#1371

C₂×C₁₉

yes

characteristic

#1364

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×D₁₆

yes

C₄×C₁₉

yes

C₂²×C₁₉

yes

—

#462

—

C₈×Q₈

C₂²×C₁₉

yes

characteristic

#366

—

#366

—

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

—

C₄⋊Q₈

—

C₄×Q₈

D₄×C₁₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₉

yes

characteristic

C₂×C₁₆

D₄×C₁₉

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

D₄×C₁₉

yes

characteristic

102

#103

—

103

C₄×C₈

—

104

C₄×C₈

C₂³×C₁₉

yes

characteristic

105

C₁₉

#12457

yes

characteristic, direct product factor, Sylow

106

C₂×Q₈

—

107

C₂×Q₈

D₈×C₁₉

yes

characteristic

108

Q₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

112

C₂×C₈

#41×C₁₉

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

D₈×C₁₉

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₄×C₁₉

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

Q₁₆×C₁₉

yes

complemented

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

C₁₉×D₁₆

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₁₉×#121

yes

characteristic

130

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

131

C₂×C₄

C₁₉×#135

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

C₂×Q₁₆×C₁₉

yes

characteristic

136

C₈

C₁₉×#137

yes

characteristic

137

C₂²

C₁₉×#466

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

C₁₉×#462

yes

characteristic

143

C₄

C₁₉×#473

yes

characteristic

144

C₄

C₂×C₁₉×D₁₆

yes

characteristic

145

C₄

C₁₉×#508

yes

characteristic

146

C₂

C₁₉×#1674

yes

characteristic

147

C₂

C₁₉×#2215

yes

characteristic

148

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
46. C₁₆×C₁₉	Q₁₆	semidirect
47. #12457	C₁₉	direct (N × H)
105. C₁₉	#12457	direct (N × H)
121. C₁₆	Q₁₆×C₁₉	semidirect

H ≅ G/N

type

46. C₁₆×C₁₉

Q₁₆

semidirect

47. #12457

C₁₉

direct (N × H)

105. C₁₉

#12457

direct (N × H)

121. C₁₆

Q₁₆×C₁₉

semidirect

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

subgroup	index
C₁₉×#2252	2
Q₈×C₁₆×C₁₉	2
C₁₉×#1674	2
C₁₉×#1371	2
C₁₉×#1364	2
#12457	19

subgroup

index

C₁₉×#2252

Q₈×C₁₆×C₁₉

C₁₉×#1674

C₁₉×#1371

C₁₉×#1364

#12457

Subgroups — C₁₉×#12457 #18018416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12457 #18018416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12457 #18018416