Group #18018417 — subgroups

Subgroups — C₁₉×#12458 #18018417

Order 4864.

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

normal subgroups only

This group has 192 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×#1676	C₂	1	2	yes	maximal
4	C₁₉×#1371	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1365	C₂	1	2	yes	maximal
6	C₁₉×#500	C₂²	1	2	yes	—
7	C₁₉×#477	—	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×#333	—	2	2	no	—
17	C₁₉×C₄⋊Q₈	—	2	2	no	—
18	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₁₉	—	2	1	no	—
20	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
22	C₁₉×#107	—	2	2	no	—
23	C₁₉×#107	—	2	2	no	—
24	C₁₉×#107	—	4	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	C₂×C₈×C₁₉	—	2	1	no	—
34	C₂×C₈×C₁₉	—	4	2	no	—
35	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₄⋊C₄×C₁₉	—	4	2	no	—
39	C₄⋊C₄×C₁₉	—	4	2	no	—
40	C₄⋊C₄×C₁₉	D₈	1	1	yes	characteristic
41	C₄²×C₁₉	—	2	1	no	—
42	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₁₉	—	2	1	no	—
44	C₁₆×C₁₉	—	4	1	no	—
45	C₁₆×C₁₉	Q₁₆	1	2	yes	—
46	#12458	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	Q₈×C₁₉	—	2	1	no	—
48	Q₈×C₁₉	—	16	2	no	—
49	Q₈×C₁₉	Q₃₂	1	2	yes	—
50	C₂×C₄×C₁₉	—	2	1	no	—
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	—	8	2	no	—
53	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
54	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
56	C₈×C₁₉	—	2	1	no	—
57	C₈×C₁₉	—	4	1	no	—
58	C₈×C₁₉	—	8	2	no	—
59	C₈×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₁₉	#137	1	1	yes	characteristic
61	#2252	C₂×C₁₉	1	1	yes	characteristic
62	Q₈×C₁₆	C₂×C₁₉	1	1	yes	characteristic
63	#1676	C₂×C₁₉	1	2	yes	—
64	#1371	C₂×C₁₉	1	1	yes	characteristic
65	#1365	C₂×C₁₉	1	2	yes	—
66	C₂²×C₁₉	#466	1	1	yes	characteristic
67	C₄×C₁₉	—	2	2	no	—
68	C₄×C₁₉	—	2	1	no	—
69	C₄×C₁₉	—	4	1	no	—
70	C₄×C₁₉	—	16	2	no	—
71	C₄×C₁₉	#462	1	1	yes	characteristic
72	C₄×C₁₉	#473	1	1	yes	characteristic
73	C₄×C₁₉	C₂×Q₃₂	1	1	yes	characteristic
74	C₄×C₁₉	#508	1	1	yes	characteristic
75	#500	C₂²×C₁₉	1	2	yes	—
76	#477	—	2	2	no	—
77	C₈×Q₈	C₂²×C₁₉	1	1	yes	characteristic
78	#366	—	2	1	no	—
79	#366	—	2	1	no	—
80	C₄⋊C₁₆	—	2	1	no	—
81	C₄⋊C₁₆	C₂²×C₁₉	1	1	yes	characteristic
82	C₄×C₁₆	—	2	1	no	—
83	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
84	#335	C₂²×C₁₉	1	2	yes	—
85	#333	—	2	2	no	—
86	C₂×C₁₉	#1674	1	1	yes	characteristic
87	C₂×C₁₉	#2216	1	1	yes	characteristic
88	C₂×C₁₉	#2221	1	1	yes	characteristic
89	C₄⋊Q₈	—	2	2	no	—
90	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
93	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
94	#107	—	2	2	no	—
95	#107	—	2	2	no	—
96	#107	—	4	2	no	—
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
100	#103	—	4	2	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
103	C₁₉	#12458	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	D₈×C₁₉	1	1	yes	characteristic
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
109	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	D₈×C₁₉	1	1	yes	characteristic
114	C₄²	—	2	1	no	—
115	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	4	1	no	—
118	C₁₆	Q₁₆×C₁₉	1	2	yes	—
119	Q₈	—	2	1	no	—
120	Q₈	—	16	2	no	—
121	Q₈	C₁₉×Q₃₂	1	2	yes	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
126	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
127	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	4	1	no	—
130	C₈	—	8	2	no	—
131	C₈	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
132	C₈	C₁₉×#137	1	1	yes	characteristic
133	C₂²	C₁₉×#466	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	1	no	—
137	C₄	—	16	2	no	—
138	C₄	C₁₉×#462	1	1	yes	characteristic
139	C₄	C₁₉×#473	1	1	yes	characteristic
140	C₄	C₂×C₁₉×Q₃₂	1	1	yes	characteristic
141	C₄	C₁₉×#508	1	1	yes	characteristic
142	C₂	C₁₉×#1674	1	1	yes	characteristic
143	C₂	C₁₉×#2216	1	1	yes	characteristic
144	C₂	C₁₉×#2221	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×#1676

C₂

yes

maximal

C₁₉×#1371

C₂

yes

maximal, characteristic

C₁₉×#1365

C₂

yes

maximal

C₁₉×#500

C₂²

yes

—

C₁₉×#477

—

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×#333

—

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×#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

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

—

#12458

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

Q₃₂

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

#1676

C₂×C₁₉

yes

—

#1371

C₂×C₁₉

yes

characteristic

#1365

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

C₄×C₁₉

yes

C₂²×C₁₉

yes

—

#477

—

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

—

#333

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₄⋊Q₈

—

C₄×Q₈

D₄×C₁₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₉

yes

characteristic

C₂×C₁₆

D₄×C₁₉

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄×C₁₉

yes

characteristic

100

#103

—

101

C₄×C₈

—

102

C₄×C₈

C₂³×C₁₉

yes

characteristic

103

C₁₉

#12458

yes

characteristic, direct product factor, Sylow

104

C₂×Q₈

—

105

C₂×Q₈

D₈×C₁₉

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

109

C₂×C₈

#41×C₁₉

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

D₈×C₁₉

yes

characteristic

114

C₄²

—

115

C₄²

C₂×D₄×C₁₉

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₁₆

Q₁₆×C₁₉

yes

—

119

Q₈

—

120

Q₈

—

121

Q₈

C₁₉×Q₃₂

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₁₉×#121

yes

characteristic

126

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

127

C₂×C₄

C₁₉×#135

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

C₂×Q₁₆×C₁₉

yes

characteristic

132

C₈

C₁₉×#137

yes

characteristic

133

C₂²

C₁₉×#466

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

C₁₉×#462

yes

characteristic

139

C₄

C₁₉×#473

yes

characteristic

140

C₄

C₂×C₁₉×Q₃₂

yes

characteristic

141

C₄

C₁₉×#508

yes

characteristic

142

C₂

C₁₉×#1674

yes

characteristic

143

C₂

C₁₉×#2216

yes

characteristic

144

C₂

C₁₉×#2221

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. #12458	C₁₉	direct (N × H)
103. C₁₉	#12458	direct (N × H)

H ≅ G/N

type

46. #12458

C₁₉

direct (N × H)

103. C₁₉

#12458

direct (N × H)

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×#1676	2
C₁₉×#1371	2
C₁₉×#1365	2
#12458	19

subgroup

index

C₁₉×#2252

Q₈×C₁₆×C₁₉

C₁₉×#1676

C₁₉×#1371

C₁₉×#1365

#12458

Subgroups — C₁₉×#12458 #18018417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12458 #18018417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12458 #18018417