Group #18069171 — subgroups

Subgroups — #18069171

Order 4864.

188 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 47 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	#14564574	C₂	1	1	yes	maximal, characteristic
2	#14561575	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₃₈	C₄	1	1	yes	characteristic
5	C₂×#10597260	C₄	1	1	yes	characteristic
6	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
7	#11926298	—	2	1	no	—
8	#11926298	—	2	1	no	—
9	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
11	C₂²×Dic₃₈	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₉	C₈	1	1	yes	characteristic
16	C₂×Dic₁₉⋊C₄	—	2	1	no	—
17	C₂×C₄×Dic₁₉	—	2	1	no	—
18	C₂²×C₁₉⋊C₈	—	2	1	no	—
19	C₄×Dic₃₈	—	4	1	no	—
20	C₄×Dic₃₈	—	4	1	no	—
21	C₂×C₁₆×C₁₉	—	4	1	no	—
22	C₁₉×C₄⋊C₈	—	4	1	no	—
23	C₂×C₁₉⋊C₁₆	—	4	1	no	—
24	#10597260	—	4	1	no	—
25	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
28	C₂²×Dic₁₉	—	2	1	no	—
29	C₂²×Dic₁₉	—	2	1	no	—
30	C₂×Dic₃₈	—	4	1	no	—
31	C₂×Dic₃₈	—	4	2	no	—
32	C₂×C₈×C₁₉	—	2	1	no	—
33	C₂×C₈×C₁₉	—	2	1	no	—
34	C₂×C₈×C₁₉	—	4	1	no	—
35	C₂×C₈×C₁₉	—	4	1	no	—
36	C₄²×C₁₉	—	2	1	no	—
37	C₄²×C₁₉	—	2	1	no	—
38	C₄⋊Dic₁₉	—	4	1	no	—
39	Dic₁₉⋊C₄	—	4	2	no	—
40	C₄×Dic₁₉	—	4	1	no	—
41	C₄×Dic₁₉	—	4	1	no	—
42	C₂×C₁₉⋊C₈	—	2	2	no	—
43	C₂×C₁₉⋊C₈	—	4	1	no	—
44	C₂×C₁₉⋊C₈	—	4	1	no	—
45	C₁₆×C₁₉	—	8	1	no	—
46	C₁₉⋊C₁₆	—	8	1	no	—
47	#7061	—	19	1	no	maximal, Sylow
48	C₂³×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₉	—	2	1	no	—
50	C₂×C₄×C₁₉	—	2	1	no	—
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	1	no	—
55	C₂×C₄×C₁₉	—	4	1	no	—
56	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
58	C₂×Dic₁₉	—	2	2	no	—
59	C₂×Dic₁₉	—	4	1	no	—
60	C₂×Dic₁₉	—	4	1	no	—
61	C₂×Dic₁₉	—	4	1	no	—
62	C₂×Dic₁₉	—	4	1	no	—
63	C₂×Dic₁₉	—	4	1	no	—
64	Dic₃₈	—	4	2	no	—
65	Dic₃₈	—	4	2	no	—
66	C₈×C₁₉	—	4	1	no	—
67	C₈×C₁₉	—	4	1	no	—
68	C₁₉⋊C₈	—	4	2	no	—
69	#1985	—	19	1	no	—
70	#1322	—	19	1	no	—
71	#1322	D₁₉	1	1	yes	characteristic
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	—	2	1	no	—
74	C₂²×C₁₉	#323	1	1	yes	characteristic
75	C₂²×C₁₉	#350	1	1	yes	characteristic
76	C₂²×C₁₉	#350	1	1	yes	characteristic
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	—	4	1	no	—
79	C₄×C₁₉	—	4	1	no	—
80	C₄×C₁₉	#354	1	1	yes	characteristic
81	C₄×C₁₉	#356	1	1	yes	characteristic
82	Dic₁₉	—	4	2	no	—
83	Dic₁₉	—	8	1	no	—
84	C₂×C₄×Q₈	—	19	1	no	—
85	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
86	C₂×#97	—	19	1	no	—
87	C₂²⋊C₁₆	—	2	1	no	—
88	C₂²⋊C₁₆	—	38	1	no	—
89	#336	—	38	1	no	—
90	#336	—	38	1	no	—
91	C₂×C₁₉	—	2	1	no	—
92	C₂×C₁₉	—	2	1	no	—
93	C₂×C₁₉	#1315	1	1	yes	characteristic
94	C₂×C₁₉	#1319	1	1	yes	characteristic
95	C₂×C₁₉	#1320	1	1	yes	characteristic
96	C₂²×Q₈	—	19	1	no	—
97	C₂²×C₈	—	38	1	no	—
98	C₂²×C₈	D₇₆	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
100	C₄×Q₈	—	76	1	no	—
101	C₄×Q₈	—	76	1	no	—
102	C₂×C₄⋊C₄	—	19	1	no	—
103	C₂×C₄⋊C₄	—	38	1	no	—
104	C₂×C₄²	—	38	1	no	—
105	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
106	C₂×C₁₆	—	4	1	no	—
107	C₂×C₁₆	—	76	1	no	—
108	C₄⋊C₈	—	4	1	no	—
109	#97	—	76	1	no	—
110	C₁₉	#7061	1	1	yes	characteristic, complemented, Sylow
111	C₂×Q₈	—	76	1	no	—
112	C₂×Q₈	—	76	2	no	—
113	C₂²×C₄	—	38	1	no	—
114	C₂²×C₄	—	38	1	no	—
115	C₂²×C₄	C₈×D₁₉	1	1	yes	characteristic
116	C₂²×C₄	#64515	1	1	yes	characteristic
117	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	38	2	no	—
123	C₂×C₈	—	76	1	no	—
124	C₂×C₈	—	76	1	no	—
125	C₄⋊C₄	—	76	1	no	—
126	C₄⋊C₄	—	76	2	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	76	1	no	—
130	C₄²	—	76	1	no	—
131	C₁₆	—	8	1	no	—
132	C₁₆	—	152	1	no	—
133	C₂³	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
134	Q₈	—	76	2	no	—
135	Q₈	—	76	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	38	2	no	—
144	C₂×C₄	—	76	1	no	—
145	C₂×C₄	—	76	1	no	—
146	C₂×C₄	—	76	1	no	—
147	C₂×C₄	—	76	1	no	—
148	C₂×C₄	—	76	1	no	—
149	C₂×C₄	#10597263	1	1	yes	characteristic
150	C₂×C₄	#10597280	1	1	yes	characteristic
151	C₈	—	4	1	no	—
152	C₈	—	4	1	no	—
153	C₈	—	76	2	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#11926267	1	1	yes	characteristic
157	C₂²	#11926327	1	1	yes	characteristic
158	C₂²	#11926329	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	76	2	no	—
163	C₄	—	152	1	no	—
164	C₄	#11926338	1	1	yes	characteristic
165	C₄	#11926342	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	#14561560	1	1	yes	characteristic
169	C₂	#14561568	1	1	yes	characteristic
170	C₂	#14561570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564574

C₂

yes

maximal, characteristic

#14561575

C₂

yes

maximal, characteristic

C₁₉×#1322

C₂

yes

maximal, characteristic

C₂×C₄×Dic₃₈

C₄

yes

characteristic

C₂×#10597260

C₄

yes

characteristic

C₂²⋊(C₁₉⋊C₁₆)

—

#11926298

—

#11926298

—

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₁₉×C₂²⋊C₁₆

—

C₂²×Dic₃₈

C₈

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₉

C₈

yes

characteristic

C₂×Dic₁₉⋊C₄

—

C₂×C₄×Dic₁₉

—

C₂²×C₁₉⋊C₈

—

C₄×Dic₃₈

—

C₄×Dic₃₈

—

C₂×C₁₆×C₁₉

—

C₁₉×C₄⋊C₈

—

C₂×C₁₉⋊C₁₆

—

#10597260

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

C₄×Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₁₆×C₁₉

—

C₁₉⋊C₁₆

—

#7061

—

maximal, Sylow

C₂³×C₁₉

C₂²⋊C₈

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

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#1985

—

#1322

—

#1322

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

Dic₁₉

—

Dic₁₉

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

C₂×#97

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#336

—

#336

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

D₇₆

yes

characteristic

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

100

C₄×Q₈

—

101

C₄×Q₈

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄²

—

105

C₂×C₄²

C₄×D₁₉

yes

characteristic

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₄⋊C₈

—

109

#97

—

110

C₁₉

#7061

yes

characteristic, complemented, Sylow

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₈×D₁₉

yes

characteristic

116

C₂²×C₄

#64515

yes

characteristic

117

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₁₆

—

132

C₁₆

—

152

—

133

C₂³

(C₂×D₁₉)⋊C₈

yes

characteristic

134

Q₈

—

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

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

#10597263

yes

characteristic

150

C₂×C₄

#10597280

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₂²

—

155

C₂²

—

156

C₂²

#11926267

yes

characteristic

157

C₂²

#11926327

yes

characteristic

158

C₂²

#11926329

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

152

—

164

C₄

#11926338

yes

characteristic

165

C₄

#11926342

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

#14561560

yes

characteristic

169

C₂

#14561568

yes

characteristic

170

C₂

#14561570

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597263
commutator	C₂²×C₄×C₁₉	C₂×C₈
Frattini	C₂×C₄⋊C₈	C₂×D₁₉
Fitting	C₁₉×#1322	C₂
socle	C₂²×C₁₉	#323
prime radical	C₂³×C₁₉	C₂²⋊C₈
cosocle	C₂×C₁₉×C₄⋊C₈	C₂²
hypercenter	#1322	D₁₉
hypocenter	C₁₉	#7061
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

subgroup

G/H

center

C₂×C₄

#10597263

commutator

C₂²×C₄×C₁₉

C₂×C₈

Frattini

C₂×C₄⋊C₈

C₂×D₁₉

Fitting

C₁₉×#1322

C₂

socle

C₂²×C₁₉

#323

prime radical

C₂³×C₁₉

C₂²⋊C₈

cosocle

C₂×C₁₉×C₄⋊C₈

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

N	H ≅ G/N	type
110. C₁₉	#7061	semidirect

H ≅ G/N

type

110. C₁₉

#7061

semidirect

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

subgroup	index
#14564574	2
#14561575	2
C₁₉×#1322	2
#7061	19

subgroup

index

#14564574

#14561575

C₁₉×#1322

#7061

Subgroups — #18069171

Conjugacy classes of subgroups

Complements

Maximal subgroups