Group #18069513 — subgroups

Subgroups — #18069513

Order 4864.

222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂×#11926266	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2162	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
6	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
7	C₂²⋊(C₁₉⋊C₁₆)	C₄	1	2	yes	—
8	C₄×C₁₉⋊C₁₆	—	2	1	no	—
9	#11926266	—	2	1	no	—
10	#11926266	—	2	1	no	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×C₄⋊C₁₆	—	2	1	no	—
13	C₁₉×C₂²⋊C₁₆	C₄	1	2	yes	—
14	C₄×C₁₆×C₁₉	—	2	1	no	—
15	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₁₉⋊C₈	C₈	1	2	yes	—
19	C₂²×C₁₉⋊C₈	C₈	1	2	yes	—
20	C₂×C₁₆×C₁₉	—	2	2	no	—
21	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₉	D₄	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	—	2	1	no	—
27	C₄×C₁₉⋊C₈	—	2	1	no	—
28	C₄×C₁₉⋊C₈	—	2	1	no	—
29	C₄×C₁₉⋊C₈	—	2	1	no	—
30	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₉	—	4	1	no	—
34	C₂×C₈×C₁₉	—	4	1	no	—
35	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
42	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
43	C₂×C₁₉⋊C₈	—	2	2	no	—
44	C₂×C₁₉⋊C₈	—	2	2	no	—
45	C₂×C₁₉⋊C₈	—	2	2	no	—
46	C₂×C₁₉⋊C₈	—	4	2	no	—
47	C₂×C₁₉⋊C₈	C₁₆	1	4	yes	complemented
48	C₁₆×C₁₉	—	4	2	no	—
49	C₁₉⋊C₁₆	—	4	2	no	—
50	#7195	—	19	1	no	maximal, Sylow
51	C₂³×C₁₉	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₂×C₁₆	1	2	yes	—
62	C₂×C₄×C₁₉	M₃₂	1	2	yes	—
63	C₈×C₁₉	—	2	1	no	—
64	C₈×C₁₉	—	2	1	no	—
65	C₈×C₁₉	—	2	1	no	—
66	C₈×C₁₉	—	2	1	no	—
67	C₁₉⋊C₈	—	2	4	no	—
68	C₁₉⋊C₈	—	4	2	no	—
69	#2162	—	19	1	no	—
70	#2162	D₁₉	1	1	yes	characteristic
71	C₂×#322	—	19	1	no	—
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	—	2	1	no	—
74	C₂²×C₁₉	#336	1	1	yes	characteristic
75	C₂²×C₁₉	C₄×C₁₆	1	1	yes	characteristic
76	C₂²×C₁₉	#346	1	1	yes	characteristic
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	—	2	1	no	—
79	C₄×C₁₉	#344	1	1	yes	characteristic
80	C₄×C₁₉	#344	1	1	yes	characteristic
81	C₄×C₁₉	C₂²⋊C₁₆	1	2	yes	—
82	C₄×C₁₉	#349	1	2	yes	—
83	C₄×C₁₉	C₄⋊C₁₆	1	2	yes	—
84	C₂×C₄×C₈	—	19	1	no	—
85	C₂×C₄×C₈	—	19	1	no	—
86	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
87	C₄⋊C₁₆	—	2	1	no	—
88	C₄⋊C₁₆	—	38	1	no	—
89	C₂²⋊C₁₆	—	19	2	no	—
90	C₂²⋊C₁₆	Dic₁₉	1	2	yes	—
91	C₄×C₁₆	—	2	1	no	—
92	C₄×C₁₆	—	38	1	no	—
93	#322	—	38	1	no	—
94	#322	—	38	1	no	—
95	C₂×C₁₉	—	2	1	no	—
96	C₂×C₁₉	—	2	1	no	—
97	C₂×C₁₉	#1281	1	1	yes	characteristic
98	C₂×C₁₉	#1374	1	1	yes	characteristic
99	C₂×C₁₉	#1377	1	1	yes	characteristic
100	C₂²×C₈	—	19	2	no	—
101	C₂²×C₈	—	19	2	no	—
102	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
103	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
104	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	38	2	no	—
107	C₄×C₈	—	38	1	no	—
108	C₄×C₈	—	38	1	no	—
109	C₄×C₈	—	38	1	no	—
110	C₄×C₈	—	38	1	no	—
111	C₄×C₈	Dic₃₈	1	1	yes	characteristic
112	C₄×C₈	D₇₆	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
115	C₁₉	#7195	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₄	C₈×D₁₉	1	1	yes	characteristic
117	C₂²×C₄	C₈×D₁₉	1	1	yes	characteristic
118	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	19	4	no	—
122	C₂×C₈	—	38	2	no	—
123	C₂×C₈	—	38	2	no	—
124	C₂×C₈	—	38	2	no	—
125	C₂×C₈	—	76	2	no	—
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	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
130	C₄²	#64515	1	1	yes	characteristic
131	C₄²	#64515	1	1	yes	characteristic
132	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
133	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
134	C₁₆	—	4	2	no	—
135	C₁₆	—	76	2	no	—
136	C₂³	C₈×Dic₁₉	1	1	yes	characteristic
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	C₁₆×D₁₉	1	2	yes	—
141	C₂×C₄	#10597255	1	2	yes	—
142	C₂×C₄	Dic₁₉⋊C₈	1	1	yes	characteristic
143	C₂×C₄	Dic₁₉⋊C₈	1	1	yes	characteristic
144	C₂×C₄	#10597272	1	1	yes	characteristic
145	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
146	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
147	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	38	4	no	—
153	C₈	—	76	2	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#11926298	1	1	yes	characteristic
157	C₂²	C₁₆×Dic₁₉	1	1	yes	characteristic
158	C₂²	#11926319	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#11926314	1	1	yes	characteristic
162	C₄	#11926316	1	1	yes	characteristic
163	C₄	(C₂×D₁₉)⋊C₁₆	1	2	yes	—
164	C₄	#11926325	1	2	yes	—
165	C₄	Dic₁₉⋊C₁₆	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	#14561479	1	1	yes	characteristic
169	C₂	#14561712	1	1	yes	characteristic
170	C₂	#14561718	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565434

C₂

yes

maximal, characteristic

C₂×#11926266

C₂

yes

maximal, characteristic

C₁₉×#2162

C₂

yes

maximal, characteristic

C₂×C₄×C₁₉⋊C₈

C₄

yes

characteristic

C₂×C₄×C₁₉⋊C₈

C₄

yes

characteristic

C₄⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

C₄

yes

—

C₄×C₁₉⋊C₁₆

—

#11926266

—

#11926266

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×C₄⋊C₁₆

—

C₁₉×C₂²⋊C₁₆

C₄

yes

—

C₄×C₁₆×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₁₉⋊C₈

C₈

yes

—

C₂²×C₁₉⋊C₈

C₈

yes

—

C₂×C₁₆×C₁₉

—

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

Q₈

yes

characteristic

C₂×C₁₉⋊C₁₆

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

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

M₁₆

yes

characteristic

C₄²×C₁₉

M₁₆

yes

characteristic

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

C₁₆

yes

complemented

C₁₆×C₁₉

—

C₁₉⋊C₁₆

—

#7195

—

maximal, Sylow

C₂³×C₁₉

C₄×C₈

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₂×C₁₆

yes

—

C₂×C₄×C₁₉

M₃₂

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2162

—

#2162

D₁₉

yes

characteristic

C₂×#322

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#336

yes

characteristic

C₂²×C₁₉

C₄×C₁₆

yes

characteristic

C₂²×C₁₉

#346

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

C₂²⋊C₁₆

yes

—

C₄×C₁₉

#349

yes

—

C₄×C₁₉

C₄⋊C₁₆

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

Dic₁₉

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

—

#322

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

C₄×D₁₉

yes

characteristic

103

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

104

C₂×C₄²

C₄×D₁₉

yes

characteristic

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

Dic₃₈

yes

characteristic

112

C₄×C₈

D₇₆

yes

characteristic

113

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

114

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

115

C₁₉

#7195

yes

characteristic, complemented, Sylow

116

C₂²×C₄

C₈×D₁₉

yes

characteristic

117

C₂²×C₄

C₈×D₁₉

yes

characteristic

118

C₂²×C₄

C₄×Dic₁₉

yes

characteristic

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

127

C₂×C₈

C₄⋊Dic₁₉

yes

characteristic

128

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

129

C₂×C₈

C₂²⋊Dic₁₉

yes

characteristic

130

C₄²

#64515

yes

characteristic

131

C₄²

#64515

yes

characteristic

132

C₄²

Dic₁₉⋊C₄

yes

characteristic

133

C₄²

(C₂×D₁₉)⋊C₄

yes

characteristic

134

C₁₆

—

135

C₁₆

—

136

C₂³

C₈×Dic₁₉

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₁₆×D₁₉

yes

—

141

C₂×C₄

#10597255

yes

—

142

C₂×C₄

Dic₁₉⋊C₈

yes

characteristic

143

C₂×C₄

Dic₁₉⋊C₈

yes

characteristic

144

C₂×C₄

#10597272

yes

characteristic

145

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

146

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

147

C₂×C₄

C₁₉⋊#95

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₂²

—

155

C₂²

—

156

C₂²

#11926298

yes

characteristic

157

C₂²

C₁₆×Dic₁₉

yes

characteristic

158

C₂²

#11926319

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

#11926314

yes

characteristic

162

C₄

#11926316

yes

characteristic

163

C₄

(C₂×D₁₉)⋊C₁₆

yes

—

164

C₄

#11926325

yes

—

165

C₄

Dic₁₉⋊C₁₆

yes

—

166

C₂

—

167

C₂

—

168

C₂

#14561479

yes

characteristic

169

C₂

#14561712

yes

characteristic

170

C₂

#14561718

yes

characteristic

	subgroup	G/H
center	C₄²	(C₂×D₁₉)⋊C₄
commutator	C₂²×C₁₉	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#2162	C₂
socle	C₂²×C₁₉	#336
prime radical	C₂³×C₁₉	C₄×C₈
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#2162	D₁₉
hypocenter	C₁₉	#7195
LCS	G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #2162

subgroup

G/H

center

C₄²

(C₂×D₁₉)⋊C₄

commutator

C₂²×C₁₉

C₄×C₁₆

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₁₉×#2162

C₂

socle

C₂²×C₁₉

#336

prime radical

C₂³×C₁₉

C₄×C₈

cosocle

C₂×C₄×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #2162

N	H ≅ G/N	type
47. C₂×C₁₉⋊C₈	C₁₆	semidirect
115. C₁₉	#7195	semidirect

H ≅ G/N

type

47. C₂×C₁₉⋊C₈

C₁₆

semidirect

115. C₁₉

#7195

semidirect

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

subgroup	index
#14565434	2
C₂×#11926266	2
C₁₉×#2162	2
#7195	19

subgroup

index

#14565434

C₂×#11926266

C₁₉×#2162

#7195

Subgroups — #18069513

Conjugacy classes of subgroups

Complements

Maximal subgroups