Group #14559607 — subgroups

Subgroups — C₁₉×#1750 #14559607

Order 2432.

336 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 168 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	C₂×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×C₂²⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₁₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#336	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
8	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
9	C₁₉×#117	—	2	2	no	—
10	C₁₉×#117	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
12	C₂×C₂²⋊C₄×C₁₉	C₄	1	2	yes	—
13	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×C₂²⋊C₈	C₄	1	8	yes	complemented
16	C₁₉×#97	—	2	2	no	—
17	C₄×C₈×C₁₉	—	2	2	no	—
18	C₂⁴×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₉	—	2	1	no	—
20	C₂²×C₄×C₁₉	—	2	1	no	—
21	C₂²×C₄×C₁₉	—	2	2	no	—
22	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₉	—	2	2	no	—
28	C₂×C₈×C₁₉	—	2	2	no	—
29	C₂×C₈×C₁₉	—	2	2	no	—
30	C₂×C₈×C₁₉	—	2	2	no	—
31	C₂×C₈×C₁₉	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₁₉	C₂×C₄	1	4	yes	—
33	C₄⋊C₄×C₁₉	—	2	4	no	—
34	C₂²⋊C₄×C₁₉	—	2	4	no	—
35	C₄²×C₁₉	—	2	1	no	—
36	C₄²×C₁₉	—	2	1	no	—
37	C₄²×C₁₉	D₄	1	2	yes	—
38	C₄²×C₁₉	D₄	1	2	yes	—
39	C₂³×C₁₉	—	2	1	no	—
40	C₂³×C₁₉	—	2	1	no	—
41	C₂³×C₁₉	—	2	4	no	—
42	C₂³×C₁₉	C₄²	1	2	yes	—
43	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉	—	2	1	no	—
45	C₂×C₄×C₁₉	—	2	1	no	—
46	C₂×C₄×C₁₉	—	2	4	no	—
47	C₂×C₄×C₁₉	—	2	4	no	—
48	C₂×C₄×C₁₉	—	2	4	no	—
49	C₂×C₄×C₁₉	—	2	2	no	—
50	C₂×C₄×C₁₉	—	2	4	no	—
51	C₂×C₄×C₁₉	C₄²	1	2	yes	—
52	C₂×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉	#41	1	2	yes	—
57	C₈×C₁₉	—	2	4	no	—
58	C₈×C₁₉	—	2	4	no	—
59	#1750	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×C₁₉	—	2	2	no	—
61	C₂²×C₁₉	—	2	4	no	—
62	C₂²×C₁₉	—	4	2	no	—
63	C₂²×C₁₉	—	4	2	no	—
64	C₂²×C₁₉	C₂×C₄²	1	1	yes	characteristic
65	C₂²×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₉	#117	1	1	yes	characteristic
67	C₂²×C₁₉	#131	1	4	yes	—
68	C₄×C₁₉	—	2	2	no	—
69	C₄×C₁₉	—	2	4	no	—
70	C₄×C₁₉	—	4	2	no	—
71	C₄×C₁₉	C₄×D₄	1	2	yes	—
72	C₄×C₁₉	C₄×D₄	1	2	yes	—
73	C₂×#117	C₂×C₁₉	1	1	yes	characteristic
74	C₂×C₂²⋊C₈	C₂×C₁₉	1	2	yes	—
75	C₂×#97	C₂×C₁₉	1	1	yes	characteristic, complemented
76	C₂×C₄×C₈	C₂×C₁₉	1	1	yes	characteristic, complemented
77	#336	C₂×C₁₉	1	2	yes	complemented
78	C₂×C₁₉	—	4	2	no	—
79	C₂×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₁₉	#405	1	2	yes	—
81	C₂×C₁₉	#408	1	2	yes	—
82	C₂×C₁₉	#433	1	2	yes	—
83	C₂³×C₄	C₂²×C₁₉	1	1	yes	characteristic
84	C₂²×C₈	C₂²×C₁₉	1	2	yes	—
85	C₂²×C₈	C₂²×C₁₉	1	2	yes	—
86	#117	—	2	2	no	—
87	#117	—	2	2	no	—
88	C₂×C₄⋊C₄	C₄×C₁₉	1	2	yes	—
89	C₂×C₂²⋊C₄	C₄×C₁₉	1	2	yes	—
90	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
91	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
92	C₂²⋊C₈	C₄×C₁₉	1	8	yes	complemented
93	#97	—	2	2	no	—
94	C₄×C₈	—	2	2	no	—
95	C₁₉	#1750	1	1	yes	characteristic, direct product factor, Sylow
96	C₂⁴	C₂×C₄×C₁₉	1	1	yes	characteristic
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	C₂×C₄×C₁₉	1	2	yes	—
101	C₂²×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
102	C₂²×C₄	C₂×C₄×C₁₉	1	4	yes	—
103	C₂²×C₄	C₂×C₄×C₁₉	1	2	yes	—
104	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	C₂×C₄×C₁₉	1	4	yes	—
110	C₂×C₈	C₂×C₄×C₁₉	1	4	yes	—
111	C₄⋊C₄	—	2	4	no	—
112	C₂²⋊C₄	—	2	4	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	D₄×C₁₉	1	2	yes	—
116	C₄²	D₄×C₁₉	1	2	yes	—
117	C₂³	—	2	1	no	—
118	C₂³	—	2	1	no	—
119	C₂³	—	2	4	no	—
120	C₂³	C₄²×C₁₉	1	2	yes	—
121	C₂³	C₂²×C₄×C₁₉	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	C₄²×C₁₉	1	2	yes	—
130	C₂×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
131	C₂×C₄	C₂²×C₄×C₁₉	1	2	yes	—
132	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
133	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
134	C₂×C₄	#41×C₁₉	1	2	yes	—
135	C₈	—	2	4	no	—
136	C₈	—	2	4	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	4	no	—
139	C₂²	—	4	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	C₂×C₄²×C₁₉	1	1	yes	characteristic
142	C₂²	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
143	C₂²	C₁₉×#117	1	1	yes	characteristic
144	C₂²	C₁₉×#131	1	4	yes	—
145	C₄	—	2	2	no	—
146	C₄	—	2	4	no	—
147	C₄	—	4	2	no	—
148	C₄	C₄×D₄×C₁₉	1	2	yes	—
149	C₄	C₄×D₄×C₁₉	1	2	yes	—
150	C₂	—	4	2	no	—
151	C₂	C₄×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
152	C₂	C₁₉×#405	1	2	yes	—
153	C₂	C₁₉×#408	1	2	yes	—
154	C₂	C₁₉×#433	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₉×#117

C₂

yes

maximal, characteristic

C₂×C₁₉×C₂²⋊C₈

C₂

yes

maximal

C₂×C₁₉×#97

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₈×C₁₉

C₂

yes

maximal, characteristic, complemented

C₁₉×#336

C₂

yes

maximal, complemented

C₂³×C₄×C₁₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉

C₂²

yes

—

C₂²×C₈×C₁₉

C₂²

yes

—

C₁₉×#117

—

C₁₉×#117

—

C₂×C₄⋊C₄×C₁₉

C₄

yes

—

C₂×C₂²⋊C₄×C₁₉

C₄

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₁₉×C₂²⋊C₈

C₄

yes

complemented

C₁₉×#97

—

C₄×C₈×C₁₉

—

C₂⁴×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×C₄

yes

—

C₂×C₈×C₁₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

D₄

yes

—

C₄²×C₁₉

D₄

yes

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₄²

yes

—

C₂³×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₄²

yes

—

C₂×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

#41

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

#1750

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×C₄²

yes

characteristic

C₂²×C₁₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₄×D₄

yes

—

C₄×C₁₉

C₄×D₄

yes

—

C₂×#117

C₂×C₁₉

yes

characteristic

C₂×C₂²⋊C₈

C₂×C₁₉

yes

—

C₂×#97

C₂×C₁₉

yes

characteristic, complemented

C₂×C₄×C₈

C₂×C₁₉

yes

characteristic, complemented

#336

C₂×C₁₉

yes

complemented

C₂×C₁₉

—

C₂×C₁₉

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₁₉

#405

yes

—

C₂×C₁₉

#408

yes

—

C₂×C₁₉

#433

yes

—

C₂³×C₄

C₂²×C₁₉

yes

characteristic

C₂²×C₈

C₂²×C₁₉

yes

—

C₂²×C₈

C₂²×C₁₉

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

C₄×C₁₉

yes

—

C₂×C₂²⋊C₄

C₄×C₁₉

yes

—

C₂×C₄²

C₂²×C₁₉

yes

characteristic

C₂×C₄²

C₂²×C₁₉

yes

characteristic

C₂²⋊C₈

C₄×C₁₉

yes

complemented

#97

—

C₄×C₈

—

C₁₉

#1750

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂×C₄×C₁₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂×C₄×C₁₉

yes

—

101

C₂²×C₄

C₂×C₄×C₁₉

yes

characteristic

102

C₂²×C₄

C₂×C₄×C₁₉

yes

—

103

C₂²×C₄

C₂×C₄×C₁₉

yes

—

104

C₂²×C₄

C₂³×C₁₉

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×C₄×C₁₉

yes

—

110

C₂×C₈

C₂×C₄×C₁₉

yes

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

D₄×C₁₉

yes

—

116

C₄²

D₄×C₁₉

yes

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₄²×C₁₉

yes

—

121

C₂³

C₂²×C₄×C₁₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₄²×C₁₉

yes

—

130

C₂×C₄

C₂²⋊C₄×C₁₉

yes

—

131

C₂×C₄

C₂²×C₄×C₁₉

yes

—

132

C₂×C₄

C₂×D₄×C₁₉

yes

characteristic

133

C₂×C₄

C₂×D₄×C₁₉

yes

characteristic

134

C₂×C₄

#41×C₁₉

yes

—

135

C₈

—

136

C₈

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₄²×C₁₉

yes

characteristic

142

C₂²

C₂×C₂²⋊C₄×C₁₉

yes

characteristic

143

C₂²

C₁₉×#117

yes

characteristic

144

C₂²

C₁₉×#131

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

C₄×D₄×C₁₉

yes

—

149

C₄

C₄×D₄×C₁₉

yes

—

150

C₂

—

151

C₂

C₄×C₂²⋊C₄×C₁₉

yes

characteristic

152

C₂

C₁₉×#405

yes

—

153

C₂

C₁₉×#408

yes

—

154

C₂

C₁₉×#433

yes

—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂³
commutator	C₂²	C₂×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₉
socle	C₂³×C₁₉	C₂²×C₄
prime radical	C₂⁴×C₁₉	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

subgroup

G/H

center

C₂²×C₄×C₁₉

C₂³

commutator

C₂²

C₂×C₄²×C₁₉

Frattini, cosocle

C₂²×C₄

C₂³×C₁₉

socle

C₂³×C₁₉

C₂²×C₄

prime radical

C₂⁴×C₁₉

C₂×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₁₉ ◁ G

N	H ≅ G/N	type
3. C₂×C₁₉×#97	C₂	semidirect
4. C₂×C₄×C₈×C₁₉	C₂	semidirect
5. C₁₉×#336	C₂	semidirect
15. C₁₉×C₂²⋊C₈	C₄	semidirect
59. #1750	C₁₉	direct (N × H)
75. C₂×#97	C₂×C₁₉	semidirect
76. C₂×C₄×C₈	C₂×C₁₉	semidirect
77. #336	C₂×C₁₉	semidirect
92. C₂²⋊C₈	C₄×C₁₉	semidirect
95. C₁₉	#1750	direct (N × H)

H ≅ G/N

type

3. C₂×C₁₉×#97

C₂

semidirect

4. C₂×C₄×C₈×C₁₉

C₂

semidirect

5. C₁₉×#336

C₂

semidirect

15. C₁₉×C₂²⋊C₈

C₄

semidirect

59. #1750

C₁₉

direct (N × H)

75. C₂×#97

C₂×C₁₉

semidirect

76. C₂×C₄×C₈

C₂×C₁₉

semidirect

77. #336

C₂×C₁₉

semidirect

92. C₂²⋊C₈

C₄×C₁₉

semidirect

95. C₁₉

#1750

direct (N × H)

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

subgroup	index
C₂×C₁₉×#117	2
C₂×C₁₉×C₂²⋊C₈	2
C₂×C₁₉×#97	2
C₂×C₄×C₈×C₁₉	2
C₁₉×#336	2
#1750	19

subgroup

index

C₂×C₁₉×#117

C₂×C₁₉×C₂²⋊C₈

C₂×C₁₉×#97

C₂×C₄×C₈×C₁₉

C₁₉×#336

#1750

Subgroups — C₁₉×#1750 #14559607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#1750 #14559607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#1750 #14559607