Group #18018471 — subgroups

Subgroups — C₁₉×#12512 #18018471

Order 4864.

268 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2241	C₂	1	2	yes	maximal, complemented
2	C₁₉×#2240	C₂	1	2	yes	maximal
3	C₁₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1709	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#500	—	2	1	no	—
7	C₁₉×#500	C₂²	1	2	yes	—
8	C₁₉×#494	—	2	1	no	—
9	C₁₉×#494	C₂²	1	1	yes	characteristic
10	C₁₉×#493	C₂²	1	1	yes	characteristic
11	C₁₉×#486	—	2	1	no	—
12	C₁₉×#486	—	2	1	no	—
13	C₁₉×#358	—	2	4	no	—
14	C₁₉×#357	—	2	4	no	—
15	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
16	C₈²×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₉	—	2	2	no	—
18	C₂×Q₁₆×C₁₉	—	4	2	no	—
19	C₂×D₈×C₁₉	—	2	2	no	—
20	C₁₉×C₄⋊Q₈	—	2	1	no	—
21	C₁₉×C₄⋊Q₈	—	2	2	no	—
22	C₁₉×#127	—	2	1	no	—
23	C₂×C₁₆×C₁₉	D₄	1	4	yes	—
24	C₁₉×#107	—	2	2	no	—
25	C₁₉×#107	—	2	2	no	—
26	C₁₉×#107	—	4	2	no	—
27	C₁₉×#102	—	4	2	no	—
28	C₁₉×#102	—	4	2	no	—
29	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₁₉	—	4	1	no	—
33	C₂×Q₈×C₁₉	—	4	2	no	—
34	C₂×D₄×C₁₉	—	4	1	no	—
35	C₂×D₄×C₁₉	—	4	1	no	—
36	C₂×D₄×C₁₉	—	4	1	no	—
37	Q₁₆×C₁₉	—	4	4	no	—
38	Q₁₆×C₁₉	—	8	4	no	—
39	D₈×C₁₉	—	4	4	no	—
40	C₂×C₈×C₁₉	D₈	1	2	yes	—
41	C₂×C₈×C₁₉	D₈	1	2	yes	—
42	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
43	C₄⋊C₄×C₁₉	—	4	2	no	—
44	C₄⋊C₄×C₁₉	—	4	1	no	—
45	C₄⋊C₄×C₁₉	—	4	1	no	—
46	C₄⋊C₄×C₁₉	—	4	2	no	—
47	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₁₉	—	2	4	no	—
49	#12512	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₁₉	—	8	1	no	—
51	Q₈×C₁₉	—	8	2	no	—
52	Q₈×C₁₉	—	8	4	no	—
53	D₄×C₁₉	—	8	2	no	—
54	D₄×C₁₉	—	8	2	no	—
55	D₄×C₁₉	—	8	2	no	—
56	C₂×C₄×C₁₉	—	8	1	no	—
57	C₂×C₄×C₁₉	—	8	2	no	—
58	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
61	C₈×C₁₉	—	2	2	no	—
62	C₈×C₁₉	—	2	2	no	—
63	C₈×C₁₉	#135	1	4	yes	—
64	#2241	C₂×C₁₉	1	2	yes	complemented
65	#2240	C₂×C₁₉	1	2	yes	—
66	#1713	C₂×C₁₉	1	1	yes	characteristic, complemented
67	#1709	C₂×C₁₉	1	1	yes	characteristic
68	#1311	C₂×C₁₉	1	1	yes	characteristic, complemented
69	C₂²×C₁₉	—	16	1	no	—
70	C₂²×C₁₉	—	16	1	no	—
71	C₂²×C₁₉	—	16	1	no	—
72	C₂²×C₁₉	#493	1	1	yes	characteristic
73	C₄×C₁₉	—	16	1	no	—
74	C₄×C₁₉	—	16	2	no	—
75	C₄×C₁₉	#495	1	2	yes	—
76	C₄×C₁₉	#509	1	2	yes	—
77	C₄×C₁₉	#510	1	2	yes	—
78	#500	—	2	1	no	—
79	#500	C₂²×C₁₉	1	2	yes	—
80	#494	—	2	1	no	—
81	#494	C₂²×C₁₉	1	1	yes	characteristic
82	#493	C₂²×C₁₉	1	1	yes	characteristic
83	#486	—	2	1	no	—
84	#486	—	2	1	no	—
85	#358	—	2	4	no	—
86	#357	—	2	4	no	—
87	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
88	C₈²	C₂²×C₁₉	1	1	yes	characteristic
89	C₂×C₁₉	—	32	1	no	—
90	C₂×C₁₉	#1709	1	1	yes	characteristic
91	C₂×C₁₉	#2250	1	1	yes	characteristic
92	C₂×C₁₉	#2251	1	1	yes	characteristic
93	C₂×Q₁₆	—	2	2	no	—
94	C₂×Q₁₆	—	4	2	no	—
95	C₂×D₈	—	2	2	no	—
96	C₄⋊Q₈	—	2	1	no	—
97	C₄⋊Q₈	—	2	2	no	—
98	#127	—	2	1	no	—
99	C₂×C₁₆	D₄×C₁₉	1	4	yes	—
100	#107	—	2	2	no	—
101	#107	—	2	2	no	—
102	#107	—	4	2	no	—
103	#102	—	4	2	no	—
104	#102	—	4	2	no	—
105	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
106	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
107	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
108	C₁₉	#12512	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×Q₈	—	4	1	no	—
110	C₂×Q₈	—	4	2	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	4	1	no	—
113	C₂×D₄	—	4	1	no	—
114	Q₁₆	—	4	4	no	—
115	Q₁₆	—	8	4	no	—
116	D₈	—	4	4	no	—
117	C₂×C₈	D₈×C₁₉	1	2	yes	—
118	C₂×C₈	D₈×C₁₉	1	2	yes	—
119	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
125	C₁₆	—	2	4	no	—
126	C₂³	—	8	1	no	—
127	Q₈	—	8	2	no	—
128	Q₈	—	8	4	no	—
129	D₄	—	8	2	no	—
130	D₄	—	8	2	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
135	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
136	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	C₁₉×#135	1	4	yes	—
140	C₂²	—	16	1	no	—
141	C₂²	—	16	1	no	—
142	C₂²	—	16	1	no	—
143	C₂²	C₁₉×#493	1	1	yes	characteristic
144	C₄	—	16	1	no	—
145	C₄	—	16	2	no	—
146	C₄	C₁₉×#495	1	2	yes	—
147	C₄	C₁₉×#509	1	2	yes	—
148	C₄	C₁₉×#510	1	2	yes	—
149	C₂	—	32	1	no	—
150	C₂	C₁₉×#1709	1	1	yes	characteristic
151	C₂	C₁₉×#2250	1	1	yes	characteristic
152	C₂	C₁₉×#2251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2241

C₂

yes

maximal, complemented

C₁₉×#2240

C₂

yes

maximal

C₁₉×#1713

C₂

yes

maximal, characteristic, complemented

C₁₉×#1709

C₂

yes

maximal, characteristic

C₁₉×#1311

C₂

yes

maximal, characteristic, complemented

C₁₉×#500

—

C₁₉×#500

C₂²

yes

—

C₁₉×#494

—

C₁₉×#494

C₂²

yes

characteristic

C₁₉×#493

C₂²

yes

characteristic

C₁₉×#486

—

C₁₉×#486

—

C₁₉×#358

—

C₁₉×#357

—

C₄×C₁₆×C₁₉

C₂²

yes

—

C₈²×C₁₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂×D₈×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×#127

—

C₂×C₁₆×C₁₉

D₄

yes

—

C₁₉×#107

—

C₁₉×#107

—

C₁₉×#107

—

C₁₉×#102

—

C₁₉×#102

—

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

C₂³

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

D₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

#12512

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#135

yes

—

#2241

C₂×C₁₉

yes

complemented

#2240

C₂×C₁₉

yes

—

#1713

C₂×C₁₉

yes

characteristic, complemented

#1709

C₂×C₁₉

yes

characteristic

#1311

C₂×C₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#493

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#495

yes

—

C₄×C₁₉

#509

yes

—

C₄×C₁₉

#510

yes

—

#500

—

#500

C₂²×C₁₉

yes

—

#494

—

#494

C₂²×C₁₉

yes

characteristic

#493

C₂²×C₁₉

yes

—

—

—

—

C₄×C₁₆

C₂²×C₁₉

yes

—

C₈²

C₂²×C₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₄×C₁₉

yes

—

100

#107

—

101

#107

—

102

#107

—

103

#102

—

104

#102

—

105

C₄×C₈

D₄×C₁₉

yes

characteristic

106

C₄×C₈

D₄×C₁₉

yes

characteristic

107

C₄×C₈

C₂³×C₁₉

yes

characteristic

108

C₁₉

#12512

yes

characteristic, direct product factor, Sylow

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

Q₁₆

—

115

Q₁₆

—

116

D₈

—

117

C₂×C₈

D₈×C₁₉

yes

—

118

C₂×C₈

D₈×C₁₉

yes

—

119

C₂×C₈

C₂×D₄×C₁₉

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

C₂×D₄×C₁₉

yes

characteristic

125

C₁₆

—

126

C₂³

—

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₁₉×#127

yes

characteristic

135

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

136

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

C₁₉×#135

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₁₉×#493

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

C₁₉×#495

yes

—

147

C₄

C₁₉×#509

yes

—

148

C₄

C₁₉×#510

yes

—

149

C₂

—

150

C₂

C₁₉×#1709

yes

characteristic

151

C₂

C₁₉×#2250

yes

characteristic

152

C₂

C₁₉×#2251

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₁₉×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈²×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₉

prime radical

C₁₉×#493

C₂²

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈²×C₁₉ ◁ G

N	H ≅ G/N	type
1. C₁₉×#2241	C₂	semidirect
3. C₁₉×#1713	C₂	semidirect
5. C₁₉×#1311	C₂	semidirect
49. #12512	C₁₉	direct (N × H)
64. #2241	C₂×C₁₉	semidirect
66. #1713	C₂×C₁₉	semidirect
68. #1311	C₂×C₁₉	semidirect
108. C₁₉	#12512	direct (N × H)

H ≅ G/N

type

1. C₁₉×#2241

C₂

semidirect

3. C₁₉×#1713

C₂

semidirect

5. C₁₉×#1311

C₂

semidirect

49. #12512

C₁₉

direct (N × H)

64. #2241

C₂×C₁₉

semidirect

66. #1713

C₂×C₁₉

semidirect

68. #1311

C₂×C₁₉

semidirect

108. C₁₉

#12512

direct (N × H)

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

subgroup	index
C₁₉×#2241	2
C₁₉×#2240	2
C₁₉×#1713	2
C₁₉×#1709	2
C₁₉×#1311	2
#12512	19

subgroup

index

C₁₉×#2241

C₁₉×#2240

C₁₉×#1713

C₁₉×#1709

C₁₉×#1311

#12512

Subgroups — C₁₉×#12512 #18018471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12512 #18018471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12512 #18018471