Group #18076032 — subgroups

Subgroups — #18076032

Order 4864.

384 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14561969	C₂	1	2	yes	maximal
2	#14561491	C₂	1	4	yes	maximal, complemented
3	C₁₉×#2298	C₂	1	1	yes	maximal, characteristic
4	#11926601	C₂²	1	4	yes	—
5	#11926270	—	2	2	no	—
6	#11926268	—	2	2	no	—
7	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
8	C₁₉×#388	—	2	2	no	—
9	C₁₉×#387	—	2	2	no	—
10	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
11	C₄×C₂²⋊C₄×C₁₉	C₄	1	2	yes	—
12	C₁₉×#376	C₄	1	1	yes	characteristic
13	C₄³×C₁₉	C₄	1	1	yes	characteristic
14	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₁₉×#117	—	2	4	no	—
16	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
17	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
19	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
20	C₂×C₄²×C₁₉	—	2	2	no	—
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
25	C₁₉×#95	—	2	2	no	—
26	C₁₉×#95	—	2	2	no	—
27	C₁₉×#95	—	2	2	no	—
28	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
29	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
30	#10597260	—	2	4	no	—
31	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	—	2	2	no	—
33	C₂²×C₄×C₁₉	—	2	2	no	—
34	C₂²×C₄×C₁₉	—	2	2	no	—
35	C₂²×C₄×C₁₉	—	4	2	no	—
36	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
40	C₄⋊C₄×C₁₉	—	2	4	no	—
41	C₄⋊C₄×C₁₉	—	4	4	no	—
42	C₂²⋊C₄×C₁₉	—	2	4	no	—
43	C₂²⋊C₄×C₁₉	—	4	4	no	—
44	C₄²×C₁₉	—	2	2	no	—
45	C₄²×C₁₉	—	4	2	no	—
46	C₄²×C₁₉	—	4	2	no	—
47	C₄²×C₁₉	—	4	4	no	—
48	C₄²×C₁₉	#41	1	4	yes	—
49	C₂×C₁₉⋊C₈	—	4	4	no	—
50	#8744	—	19	1	no	maximal, Sylow
51	C₂³×C₁₉	—	2	1	no	—
52	C₂³×C₁₉	—	2	2	no	—
53	C₂³×C₁₉	—	4	2	no	—
54	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉	—	2	2	no	—
56	C₂×C₄×C₁₉	—	2	4	no	—
57	C₂×C₄×C₁₉	—	2	2	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	4	2	no	—
61	C₂×C₄×C₁₉	—	4	4	no	—
62	C₂×C₄×C₁₉	—	4	2	no	—
63	C₂×C₄×C₁₉	—	4	2	no	—
64	C₂×C₄×C₁₉	—	4	2	no	—
65	C₂×C₄×C₁₉	—	4	2	no	—
66	C₂×C₄×C₁₉	—	4	2	no	—
67	C₂×C₄×C₁₉	—	4	2	no	—
68	C₂×C₄×C₁₉	#117	1	2	yes	—
69	C₂×C₄×C₁₉	#123	1	4	yes	—
70	C₁₉⋊C₈	—	8	4	no	—
71	#2298	D₁₉	1	1	yes	characteristic
72	#1464	—	19	2	no	—
73	#1285	—	19	4	no	—
74	C₂²×C₁₉	—	2	2	no	—
75	C₂²×C₁₉	—	4	1	no	—
76	C₂²×C₁₉	—	4	2	no	—
77	C₂²×C₁₉	—	8	2	no	—
78	C₂²×C₁₉	#381	1	1	yes	characteristic
79	C₂²×C₁₉	#413	1	2	yes	—
80	C₄×C₁₉	—	2	2	no	—
81	C₄×C₁₉	—	2	4	no	—
82	C₄×C₁₉	—	4	4	no	—
83	C₄×C₁₉	—	4	1	no	—
84	C₄×C₁₉	—	4	2	no	—
85	C₄×C₁₉	—	8	2	no	—
86	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
87	#432	—	19	4	no	—
88	#388	—	2	2	no	—
89	#387	—	2	2	no	—
90	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
91	C₄×C₂²⋊C₄	Dic₁₉	1	2	yes	—
92	#376	Dic₁₉	1	1	yes	characteristic
93	C₄³	Dic₁₉	1	1	yes	characteristic
94	#324	—	38	2	no	—
95	#323	—	38	2	no	—
96	C₂×C₁₉	—	2	2	no	—
97	C₂×C₁₉	—	8	1	no	—
98	C₂×C₁₉	#1822	1	1	yes	characteristic
99	C₂×C₁₉	#1827	1	2	yes	—
100	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
101	#117	—	2	4	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	2	yes	—
104	C₂×C₂²⋊C₄	—	2	2	no	—
105	C₂×C₂²⋊C₄	C₂²⋊D₁₉	1	2	yes	—
106	C₂×C₄²	—	2	2	no	—
107	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
109	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
110	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
111	C₄⋊C₈	—	38	4	no	—
112	C₂²⋊C₈	—	38	4	no	—
113	#97	—	38	4	no	—
114	#95	—	2	2	no	—
115	#95	—	2	2	no	—
116	#95	—	2	2	no	—
117	C₁₉	#8744	1	1	yes	characteristic, complemented, Sylow
118	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
124	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
125	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
127	C₂×C₈	—	76	4	no	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	4	no	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	4	4	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	4	2	no	—
134	C₄²	—	4	2	no	—
135	C₄²	—	4	4	no	—
136	C₄²	#64541	1	4	yes	—
137	C₂³	—	2	1	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	4	2	no	—
140	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	#10597372	1	2	yes	—
155	C₂×C₄	#10597375	1	4	yes	—
156	C₈	—	152	4	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	8	2	no	—
161	C₂²	#11926407	1	1	yes	characteristic
162	C₂²	#11926528	1	2	yes	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	4	4	no	—
166	C₄	—	4	1	no	—
167	C₄	—	4	2	no	—
168	C₄	—	8	2	no	—
169	C₂	—	2	2	no	—
170	C₂	—	8	1	no	—
171	C₂	#14563718	1	1	yes	characteristic
172	C₂	#14563739	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14561969

C₂

yes

maximal

#14561491

C₂

yes

maximal, complemented

C₁₉×#2298

C₂

yes

maximal, characteristic

#11926601

C₂²

yes

—

#11926270

—

#11926268

—

C₂×C₁₉×#117

C₂²

yes

characteristic

C₁₉×#388

—

C₁₉×#387

—

C₄×C₄⋊C₄×C₁₉

C₂²

yes

—

C₄×C₂²⋊C₄×C₁₉

C₄

yes

—

C₁₉×#376

C₄

yes

characteristic

C₄³×C₁₉

C₄

yes

characteristic

C₂³×C₄×C₁₉

C₂×C₄

yes

characteristic

C₁₉×#117

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₁₉×#95

—

C₁₉×#95

—

C₁₉×#95

—

C₂²⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

#10597260

—

C₂⁴×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

#41

yes

—

C₂×C₁₉⋊C₈

—

#8744

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂×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

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#117

yes

—

C₂×C₄×C₁₉

#123

yes

—

C₁₉⋊C₈

—

#2298

D₁₉

yes

characteristic

#1464

—

#1285

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#381

yes

characteristic

C₂²×C₁₉

#413

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₂×#117

C₂×D₁₉

yes

—

—

—

C₄×C₄⋊C₄

C₂×D₁₉

yes

—

C₄×C₂²⋊C₄

Dic₁₉

yes

—

#376

Dic₁₉

yes

characteristic

C₄³

Dic₁₉

yes

characteristic

#324

—

#323

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1822

yes

characteristic

C₂×C₁₉

#1827

yes

—

100

C₂³×C₄

C₂×Dic₁₉

yes

characteristic

101

#117

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

—

104

C₂×C₂²⋊C₄

—

105

C₂×C₂²⋊C₄

C₂²⋊D₁₉

yes

—

106

C₂×C₄²

—

107

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

108

C₂×C₄²

C₂×Dic₁₉

yes

—

109

C₂×C₄²

C₂×Dic₁₉

yes

—

110

C₂×C₄²

C₂²×D₁₉

yes

characteristic

111

C₄⋊C₈

—

112

C₂²⋊C₈

—

113

#97

—

114

#95

—

115

#95

—

116

#95

—

117

C₁₉

#8744

yes

characteristic, complemented, Sylow

118

C₂⁴

C₂²⋊Dic₁₉

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

124

C₂²×C₄

C₂²⋊Dic₁₉

yes

—

125

C₂²×C₄

C₂²×Dic₁₉

yes

characteristic

126

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

—

127

C₂×C₈

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

#64541

yes

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

C₂×C₂²⋊Dic₁₉

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#10597372

yes

—

155

C₂×C₄

#10597375

yes

—

156

C₈

—

152

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

#11926407

yes

characteristic

162

C₂²

#11926528

yes

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₂

—

170

C₂

—

171

C₂

#14563718

yes

characteristic

172

C₂

#14563739

yes

—

	subgroup	G/H
center	C₂²	#11926407
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2298	C₂
socle	C₂²×C₁₉	#381
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2298	D₁₉
hypocenter	C₁₉	#8744
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2298

subgroup

G/H

center

C₂²

#11926407

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#2298

C₂

socle

C₂²×C₁₉

#381

prime radical

C₂⁴×C₁₉

C₂²⋊C₄

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2298

N	H ≅ G/N	type
2. #14561491	C₂	semidirect
117. C₁₉	#8744	semidirect

H ≅ G/N

type

2. #14561491

C₂

semidirect

117. C₁₉

#8744

semidirect

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

subgroup	index
#14561969	2
#14561491	2
C₁₉×#2298	2
#8744	19

subgroup

index

#14561969

#14561491

C₁₉×#2298

#8744

Subgroups — #18076032

Conjugacy classes of subgroups

Complements

Maximal subgroups