Group #18069458 — subgroups

Subgroups — #18069458

Order 4864.

246 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 127 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×C₄×C₁₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#14562383	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₉×#108	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₉⋊C₁₆	—	2	1	no	—
5	#11926597	—	2	1	no	—
6	C₄×C₁₉⋊C₁₆	—	2	1	no	—
7	C₄×C₁₉⋊C₁₆	C₄	1	2	yes	—
8	#11926297	C₄	1	2	yes	—
9	#11926295	—	2	1	no	—
10	C₂×C₁₉×#108	C₄	1	2	yes	—
11	C₄×M₁₆×C₁₉	—	2	1	no	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#341	—	2	1	no	—
14	C₂×M₁₆×C₁₉	—	2	2	no	—
15	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
18	C₁₉×#108	—	2	2	no	—
19	C₁₉×#108	C₈	1	4	yes	complemented
20	C₁₉×#97	—	4	1	no	—
21	C₄×C₈×C₁₉	—	4	1	no	—
22	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
26	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
27	C₂×C₁₉⋊C₁₆	—	2	4	no	—
28	C₂×C₁₉⋊C₁₆	—	2	2	no	—
29	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
30	C₄×C₁₉⋊C₈	—	4	1	no	—
31	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	Q₁₆	1	1	yes	characteristic
34	M₁₆×C₁₉	—	2	4	no	—
35	M₁₆×C₁₉	—	4	2	no	—
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	—	4	2	no	—
38	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
44	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₉	D₈	1	1	yes	characteristic
47	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
48	C₂×C₁₉⋊C₈	—	4	2	no	—
49	C₁₉⋊C₁₆	—	2	4	no	—
50	#7172	—	19	1	no	maximal, Sylow
51	C₂³×C₁₉	#103	1	1	yes	characteristic
52	C₂×C₄×C₁₉	—	2	2	no	—
53	C₂×C₄×C₁₉	—	2	2	no	—
54	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
55	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
59	C₂×C₄×C₁₉	#104	1	2	yes	—
60	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
62	C₈×C₁₉	—	2	1	no	—
63	C₈×C₁₉	—	2	1	no	—
64	C₈×C₁₉	—	4	4	no	—
65	C₈×C₁₉	C₄×C₈	1	1	yes	characteristic
66	C₈×C₁₉	#97	1	1	yes	characteristic
67	C₈×C₁₉	C₄⋊C₈	1	2	yes	—
68	C₁₉⋊C₈	—	8	2	no	—
69	C₂×C₄×C₁₆	—	19	1	no	—
70	C₄×#108	D₁₉	1	1	yes	characteristic
71	#1564	—	19	1	no	—
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	—	2	1	no	—
74	C₂²×C₁₉	Q₈⋊C₈	1	2	yes	—
75	C₂²×C₁₉	#340	1	1	yes	characteristic
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	D₄⋊C₈	1	2	yes	—
79	C₄×C₁₉	#336	1	1	yes	characteristic
80	C₄×C₁₉	#339	1	1	yes	characteristic
81	C₄×C₁₉	#359	1	2	yes	—
82	C₄×C₁₉	#368	1	2	yes	—
83	C₂²×C₁₆	—	38	1	no	—
84	#431	—	38	1	no	—
85	C₂×#108	Dic₁₉	1	2	yes	—
86	C₄×M₁₆	—	2	1	no	—
87	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
88	C₄×C₁₆	—	19	2	no	—
89	C₄×C₁₆	—	38	1	no	—
90	#341	—	2	1	no	—
91	#335	—	19	2	no	—
92	#334	—	38	1	no	—
93	C₂×C₁₉	—	2	2	no	—
94	C₂×C₁₉	#1279	1	1	yes	characteristic
95	C₂×C₁₉	#1381	1	1	yes	characteristic
96	C₂×C₁₉	#1382	1	1	yes	characteristic
97	C₂×M₁₆	—	2	2	no	—
98	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
100	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
101	C₂×C₁₆	—	38	4	no	—
102	C₂×C₁₆	—	38	2	no	—
103	#108	—	2	2	no	—
104	#108	C₁₉⋊C₈	1	4	yes	complemented
105	C₄⋊C₈	—	38	2	no	—
106	C₂²⋊C₈	—	76	1	no	—
107	#97	—	4	1	no	—
108	C₄×C₈	—	4	1	no	—
109	C₄×C₈	—	76	1	no	—
110	C₄×C₈	Dic₃₈	1	1	yes	characteristic
111	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
112	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
113	C₄×C₈	D₇₆	1	1	yes	characteristic
114	C₁₉	#7172	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₄	#64527	1	1	yes	characteristic
116	C₂²×C₄	#64528	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	M₁₆	—	4	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	76	2	no	—
123	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
124	C₂×C₈	#64520	1	2	yes	—
125	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
126	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
128	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
129	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
130	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
131	C₄²	#64525	1	1	yes	characteristic
132	C₄²	#64526	1	1	yes	characteristic
133	C₁₆	—	38	4	no	—
134	C₂³	#10597292	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#10597264	1	1	yes	characteristic
138	C₂×C₄	#10597265	1	1	yes	characteristic
139	C₂×C₄	#10597266	1	1	yes	characteristic
140	C₂×C₄	#10597267	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
142	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
143	C₂×C₄	#10597289	1	1	yes	characteristic
144	C₂×C₄	#10597294	1	2	yes	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	4	4	no	—
148	C₈	—	152	2	no	—
149	C₈	C₄×C₁₉⋊C₈	1	1	yes	characteristic
150	C₈	#10597260	1	1	yes	characteristic
151	C₈	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	Q₈⋊(C₁₉⋊C₈)	1	2	yes	—
155	C₂²	#11926306	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	D₄⋊(C₁₉⋊C₈)	1	2	yes	—
159	C₄	#11926299	1	1	yes	characteristic
160	C₄	#11926303	1	1	yes	characteristic
161	C₄	#11926350	1	2	yes	—
162	C₄	#11926373	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#14561476	1	1	yes	characteristic
165	C₂	#14561728	1	1	yes	characteristic
166	C₂	#14561732	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₉⋊C₁₆

C₂

yes

maximal, characteristic

#14562383

C₂

yes

maximal, characteristic

C₄×C₁₉×#108

C₂

yes

maximal, characteristic

C₂²×C₁₉⋊C₁₆

—

#11926597

—

C₄×C₁₉⋊C₁₆

—

C₄×C₁₉⋊C₁₆

C₄

yes

—

#11926297

C₄

yes

—

#11926295

—

C₂×C₁₉×#108

C₄

yes

—

C₄×M₁₆×C₁₉

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#341

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₁₉×#108

—

C₁₉×#108

C₈

yes

complemented

C₁₉×#97

—

C₄×C₈×C₁₉

—

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

C₂²⋊(C₁₉⋊C₈)

—

C₂×C₁₉⋊C₁₆

—

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

M₁₆×C₁₉

—

M₁₆×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

C₂×C₈×C₁₉

C₂×C₈

yes

—

C₂×C₈×C₁₉

M₁₆

yes

—

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

D₈

yes

characteristic

C₄²×C₁₉

SD₁₆

yes

characteristic

C₂×C₁₉⋊C₈

—

C₁₉⋊C₁₆

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#95

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

—

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

C₄×C₈

yes

characteristic

C₈×C₁₉

#97

yes

characteristic

C₈×C₁₉

C₄⋊C₈

yes

—

C₁₉⋊C₈

—

C₂×C₄×C₁₆

—

C₄×#108

D₁₉

yes

characteristic

#1564

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

Q₈⋊C₈

yes

—

C₂²×C₁₉

#340

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

D₄⋊C₈

yes

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₄×C₁₉

#368

yes

—

C₂²×C₁₆

—

#431

—

C₂×#108

Dic₁₉

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

#341

—

#335

—

#334

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×M₁₆

—

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

100

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

#108

—

104

#108

C₁₉⋊C₈

yes

complemented

105

C₄⋊C₈

—

106

C₂²⋊C₈

—

107

#97

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

Dic₃₈

yes

characteristic

111

C₄×C₈

C₄×D₁₉

yes

characteristic

112

C₄×C₈

C₄×D₁₉

yes

characteristic

113

C₄×C₈

D₇₆

yes

characteristic

114

C₁₉

#7172

yes

characteristic, complemented, Sylow

115

C₂²×C₄

#64527

yes

characteristic

116

C₂²×C₄

#64528

yes

characteristic

117

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂×C₁₉⋊C₈

yes

—

124

C₂×C₈

#64520

yes

—

125

C₂×C₈

C₄×Dic₁₉

yes

characteristic

126

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

127

C₂×C₈

C₄⋊Dic₁₉

yes

characteristic

128

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

129

C₄²

Dic₁₉⋊C₄

yes

characteristic

130

C₄²

(C₂×D₁₉)⋊C₄

yes

characteristic

131

C₄²

#64525

yes

characteristic

132

C₄²

#64526

yes

characteristic

133

C₁₆

—

134

C₂³

#10597292

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#10597264

yes

characteristic

138

C₂×C₄

#10597265

yes

characteristic

139

C₂×C₄

#10597266

yes

characteristic

140

C₂×C₄

#10597267

yes

characteristic

141

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

—

142

C₂×C₄

C₁₉⋊#95

yes

characteristic

143

C₂×C₄

#10597289

yes

characteristic

144

C₂×C₄

#10597294

yes

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

152

—

149

C₈

C₄×C₁₉⋊C₈

yes

characteristic

150

C₈

#10597260

yes

characteristic

151

C₈

C₄⋊(C₁₉⋊C₈)

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

Q₈⋊(C₁₉⋊C₈)

yes

—

155

C₂²

#11926306

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

D₄⋊(C₁₉⋊C₈)

yes

—

159

C₄

#11926299

yes

characteristic

160

C₄

#11926303

yes

characteristic

161

C₄

#11926350

yes

—

162

C₄

#11926373

yes

—

163

C₂

—

164

C₂

#14561476

yes

characteristic

165

C₂

#14561728

yes

characteristic

166

C₂

#14561732

yes

characteristic

	subgroup	G/H
center	C₄²	#64525
commutator	C₈×C₁₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₄×C₁₉×#108	C₂
socle	C₂²×C₁₉	#340
prime radical	C₂³×C₁₉	#103
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	C₄×#108	D₁₉
hypocenter	C₁₉	#7172
LCS	G ▷ C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₄²

#64525

commutator

C₈×C₁₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₄×C₁₉×#108

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₄×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
19. C₁₉×#108	C₈	semidirect
104. #108	C₁₉⋊C₈	semidirect
114. C₁₉	#7172	semidirect

H ≅ G/N

type

19. C₁₉×#108

C₈

semidirect

104. #108

C₁₉⋊C₈

semidirect

114. C₁₉

#7172

semidirect

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

subgroup	index
C₂×C₄×C₁₉⋊C₁₆	2
#14562383	2
C₄×C₁₉×#108	2
#7172	19

subgroup

index

C₂×C₄×C₁₉⋊C₁₆

#14562383

C₄×C₁₉×#108

#7172

Subgroups — #18069458

Conjugacy classes of subgroups

Complements

Maximal subgroups