Group #18069586 — subgroups

Subgroups — #18069586

Order 4864.

188 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 83 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	#14565434	C₂	1	1	yes	maximal, characteristic
2	#14563773	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1568	C₂	1	1	yes	maximal, characteristic
4	#11926568	—	2	1	no	—
5	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
6	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
7	C₄×C₁₉⋊C₁₆	C₄	1	2	yes	—
8	#11926309	—	2	1	no	—
9	#11926309	C₄	1	2	yes	—
10	C₁₉×#432	—	2	1	no	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#335	C₄	1	2	yes	—
13	C₁₉×#334	—	2	1	no	—
14	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
17	C₂×#64520	—	2	2	no	—
18	C₁₉×C₄⋊C₈	—	2	2	no	—
19	C₁₉×C₂²⋊C₈	—	4	1	no	—
20	C₁₉×#97	—	4	1	no	—
21	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
25	C₂×C₁₉⋊C₁₆	—	2	2	no	—
26	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
27	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
28	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	Q₁₆	1	1	yes	characteristic
31	C₂×C₈×C₁₉	—	4	1	no	—
32	C₂×C₈×C₁₉	—	4	1	no	—
33	C₂×C₈×C₁₉	—	4	2	no	—
34	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₁₉	D₈	1	1	yes	characteristic
41	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
42	#64520	—	4	2	no	—
43	#64520	—	4	2	no	—
44	C₂×C₁₉⋊C₈	—	4	2	no	—
45	C₁₉⋊C₁₆	—	4	2	no	—
46	#7220	—	19	1	no	maximal, Sylow
47	C₂³×C₁₉	#103	1	1	yes	characteristic
48	C₂×C₄×C₁₉	—	2	1	no	—
49	C₂×C₄×C₁₉	—	2	1	no	—
50	C₂×C₄×C₁₉	—	2	2	no	—
51	C₂×C₄×C₁₉	—	4	1	no	—
52	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
53	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
54	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
55	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
56	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
58	C₈×C₁₉	—	2	1	no	—
59	C₈×C₁₉	—	2	1	no	—
60	C₈×C₁₉	—	2	1	no	—
61	C₈×C₁₉	—	2	1	no	—
62	C₈×C₁₉	—	8	2	no	—
63	C₁₉⋊C₈	—	8	2	no	—
64	#2162	—	19	1	no	—
65	#1837	—	19	1	no	—
66	#1568	D₁₉	1	1	yes	characteristic
67	C₂²×C₁₉	—	2	1	no	—
68	C₂²×C₁₉	—	2	1	no	—
69	C₂²×C₁₉	—	2	1	no	—
70	C₂²×C₁₉	#340	1	1	yes	characteristic
71	C₄×C₁₉	—	2	1	no	—
72	C₄×C₁₉	—	2	1	no	—
73	C₄×C₁₉	—	2	1	no	—
74	C₄×C₁₉	—	2	1	no	—
75	C₄×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
76	C₄×C₁₉	#344	1	1	yes	characteristic
77	C₄×C₁₉	#359	1	2	yes	—
78	#432	—	2	1	no	—
79	#423	—	38	1	no	—
80	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
81	C₄⋊C₁₆	—	38	1	no	—
82	C₂²⋊C₁₆	—	38	1	no	—
83	C₄×C₁₆	—	19	2	no	—
84	#341	—	19	2	no	—
85	#341	—	38	1	no	—
86	#335	Dic₁₉	1	2	yes	—
87	#334	—	2	1	no	—
88	C₂×C₁₉	—	4	1	no	—
89	C₂×C₁₉	#1286	1	1	yes	characteristic
90	C₂×C₁₉	#1382	1	1	yes	characteristic
91	C₂×C₁₉	#1385	1	1	yes	characteristic
92	C₂×M₁₆	—	38	2	no	—
93	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
94	C₂²×C₈	D₇₆	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
96	C₂×C₁₆	—	38	2	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	76	1	no	—
99	C₄⋊C₈	—	76	1	no	—
100	C₂²⋊C₈	—	4	1	no	—
101	#97	—	4	1	no	—
102	C₄×C₈	Dic₃₈	1	1	yes	characteristic
103	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
106	C₁₉	#7220	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
108	C₂²×C₄	#64526	1	1	yes	characteristic
109	C₂²×C₄	#64528	1	1	yes	characteristic
110	M₁₆	—	76	2	no	—
111	M₁₆	—	76	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	76	2	no	—
116	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
117	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
119	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
120	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
121	C₄²	#64525	1	1	yes	characteristic
122	C₄²	#64527	1	1	yes	characteristic
123	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
124	C₁₆	—	76	2	no	—
125	C₂³	#10597267	1	1	yes	characteristic
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#10597264	1	1	yes	characteristic
131	C₂×C₄	#10597265	1	1	yes	characteristic
132	C₂×C₄	#10597266	1	1	yes	characteristic
133	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
134	C₂×C₄	#10597289	1	1	yes	characteristic
135	C₂×C₄	#10597292	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	8	2	no	—
141	C₈	—	152	2	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#11926306	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	#11926301	1	1	yes	characteristic
151	C₄	#11926316	1	1	yes	characteristic
152	C₄	#11926351	1	2	yes	—
153	C₂	—	4	1	no	—
154	C₂	#14561495	1	1	yes	characteristic
155	C₂	#14561730	1	1	yes	characteristic
156	C₂	#14561738	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565434

C₂

yes

maximal, characteristic

#14563773

C₂

yes

maximal, characteristic

C₁₉×#1568

C₂

yes

maximal, characteristic

#11926568

—

C₄⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

—

C₄×C₁₉⋊C₁₆

C₄

yes

—

#11926309

—

#11926309

C₄

yes

—

C₁₉×#432

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#335

C₄

yes

—

C₁₉×#334

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×#64520

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

Q₈

yes

characteristic

C₂×C₁₉⋊C₁₆

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂²×C₄×C₁₉

Q₁₆

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄²

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

D₈

yes

characteristic

C₄²×C₁₉

SD₁₆

yes

characteristic

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₁₉⋊C₁₆

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#2162

—

#1837

—

#1568

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#340

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

#432

—

#423

—

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#341

—

#341

—

#335

Dic₁₉

yes

—

#334

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×M₁₆

—

C₂²×C₈

C₄×D₁₉

yes

characteristic

C₂²×C₈

D₇₆

yes

characteristic

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

#97

—

102

C₄×C₈

Dic₃₈

yes

characteristic

103

C₄×C₈

C₄×D₁₉

yes

characteristic

104

C₄×C₈

C₂×Dic₁₉

yes

characteristic

105

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

106

C₁₉

#7220

yes

characteristic, complemented, Sylow

107

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

108

C₂²×C₄

#64526

yes

characteristic

109

C₂²×C₄

#64528

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄×Dic₁₉

yes

characteristic

117

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

118

C₂×C₈

C₄⋊Dic₁₉

yes

characteristic

119

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

120

C₄²

Dic₁₉⋊C₄

yes

characteristic

121

C₄²

#64525

yes

characteristic

122

C₄²

#64527

yes

characteristic

123

C₄²

C₂²⋊Dic₁₉

yes

characteristic

124

C₁₆

—

125

C₂³

#10597267

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#10597264

yes

characteristic

131

C₂×C₄

#10597265

yes

characteristic

132

C₂×C₄

#10597266

yes

characteristic

133

C₂×C₄

C₁₉⋊#95

yes

characteristic

134

C₂×C₄

#10597289

yes

characteristic

135

C₂×C₄

#10597292

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

152

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#11926306

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#11926301

yes

characteristic

151

C₄

#11926316

yes

characteristic

152

C₄

#11926351

yes

—

153

C₂

—

154

C₂

#14561495

yes

characteristic

155

C₂

#14561730

yes

characteristic

156

C₂

#14561738

yes

characteristic

	subgroup	G/H
center	C₂²	#11926306
commutator	C₂×C₈×C₁₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#1568	C₂
socle	C₂²×C₁₉	#340
prime radical	C₂³×C₁₉	#103
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#1568	D₁₉
hypocenter	C₁₉	#7220
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

subgroup

G/H

center

C₂²

#11926306

commutator

C₂×C₈×C₁₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₁₉×#1568

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₄×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

N	H ≅ G/N	type
106. C₁₉	#7220	semidirect

H ≅ G/N

type

106. C₁₉

#7220

semidirect

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

subgroup	index
#14565434	2
#14563773	2
C₁₉×#1568	2
#7220	19

subgroup

index

#14565434

#14563773

C₁₉×#1568

#7220

Subgroups — #18069586

Conjugacy classes of subgroups

Complements

Maximal subgroups