Group #18070128 — subgroups

Subgroups — #18070128

Order 4864.

196 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 41 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	#14564786	C₂	1	1	yes	maximal, characteristic
2	#14561762	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1319	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597316	C₄	1	1	yes	characteristic
5	C₂×#10597292	—	2	1	no	—
6	C₂×#10597293	—	2	1	no	—
7	#11926306	—	2	1	no	—
8	#11926299	C₄	1	1	yes	characteristic
9	C₁₉×#391	C₂²	1	1	yes	characteristic
10	C₁₉×#324	—	2	1	no	—
11	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₉	—	2	1	no	—
15	C₂×Dic₁₉⋊C₄	—	4	1	no	—
16	C₂×#64520	—	4	1	no	—
17	C₂²×C₁₉⋊C₈	—	2	1	no	—
18	#10597316	—	4	2	no	—
19	C₁₉×C₂²⋊C₈	—	4	1	no	—
20	C₁₉×#95	—	4	1	no	—
21	#10597293	—	2	2	no	—
22	#10597293	—	4	1	no	—
23	#10597292	—	4	2	no	—
24	C₂×Q₈×C₁₉	—	2	1	no	—
25	C₂×Q₈×C₁₉	—	4	1	no	—
26	C₂×Q₈×C₁₉	—	4	1	no	—
27	C₂×Q₈×C₁₉	SD₁₆	1	1	yes	characteristic
28	C₂×Q₈×C₁₉	Q₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	—	2	1	no	—
30	C₂²×C₄×C₁₉	—	2	1	no	—
31	C₂²×C₄×C₁₉	—	2	1	no	—
32	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×Dic₁₉	—	4	1	no	—
34	C₂×C₈×C₁₉	—	8	1	no	—
35	C₄⋊C₄×C₁₉	—	4	1	no	—
36	C₄²×C₁₉	—	4	1	no	—
37	C₄⋊Dic₁₉	—	2	2	no	—
38	C₄⋊Dic₁₉	—	4	1	no	—
39	Dic₁₉⋊C₄	—	4	2	no	—
40	Dic₁₉⋊C₄	—	4	2	no	—
41	#64520	—	4	2	no	—
42	#64520	—	8	1	no	—
43	C₂×C₁₉⋊C₈	—	4	1	no	—
44	C₂×C₁₉⋊C₈	—	4	1	no	—
45	C₂×C₁₉⋊C₈	—	4	1	no	—
46	C₂×C₁₉⋊C₈	—	8	1	no	—
47	#7429	—	19	1	no	maximal, Sylow
48	C₂³×C₁₉	#99	1	1	yes	characteristic
49	Q₈×C₁₉	—	4	1	no	—
50	Q₈×C₁₉	—	4	1	no	—
51	Q₈×C₁₉	—	8	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	—	2	1	no	—
56	C₂×C₄×C₁₉	—	4	1	no	—
57	C₂×C₄×C₁₉	—	4	1	no	—
58	C₂×C₄×C₁₉	—	4	1	no	—
59	C₂×C₄×C₁₉	—	4	1	no	—
60	C₂×C₄×C₁₉	—	4	1	no	—
61	C₂×C₄×C₁₉	—	4	1	no	—
62	C₂×C₄×C₁₉	—	8	1	no	—
63	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
64	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
65	C₂×Dic₁₉	—	4	2	no	—
66	C₂×Dic₁₉	—	8	1	no	—
67	C₂×Dic₁₉	—	8	1	no	—
68	C₈×C₁₉	—	16	1	no	—
69	C₁₉⋊C₈	—	8	1	no	—
70	C₁₉⋊C₈	—	8	2	no	—
71	#2026	—	19	1	no	—
72	#1394	—	19	1	no	—
73	#1319	D₁₉	1	1	yes	characteristic
74	C₂²×C₁₉	—	2	1	no	—
75	C₂²×C₁₉	—	2	1	no	—
76	C₂²×C₁₉	#328	1	1	yes	characteristic
77	C₂²×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
78	C₂²×C₁₉	#354	1	1	yes	characteristic
79	C₄×C₁₉	—	2	1	no	—
80	C₄×C₁₉	—	2	1	no	—
81	C₄×C₁₉	—	4	1	no	—
82	C₄×C₁₉	—	4	1	no	—
83	C₄×C₁₉	—	8	1	no	—
84	C₄×C₁₉	—	8	1	no	—
85	Dic₁₉	—	8	2	no	—
86	C₂×#125	—	19	1	no	—
87	C₂×#103	—	38	1	no	—
88	C₂×#101	—	38	1	no	—
89	#391	C₂×D₁₉	1	1	yes	characteristic
90	#340	—	38	1	no	—
91	#336	—	19	1	no	—
92	#324	—	2	1	no	—
93	C₂×C₁₉	—	2	1	no	—
94	C₂×C₁₉	—	2	1	no	—
95	C₂×C₁₉	#1351	1	1	yes	characteristic
96	C₂×C₁₉	#1413	1	1	yes	characteristic
97	C₂×C₁₉	#1414	1	1	yes	characteristic
98	C₂²×Q₈	D₇₆	1	1	yes	characteristic
99	C₂×M₁₆	—	76	1	no	—
100	C₂²×C₈	—	38	1	no	—
101	#125	—	76	2	no	—
102	C₂×C₄⋊C₄	—	38	1	no	—
103	C₂×C₄⋊C₄	—	76	1	no	—
104	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
105	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
106	#103	—	76	2	no	—
107	#101	—	38	2	no	—
108	#101	—	76	1	no	—
109	C₂²⋊C₈	—	4	1	no	—
110	#95	—	4	1	no	—
111	C₁₉	#7429	1	1	yes	characteristic, complemented, Sylow
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	4	1	no	—
114	C₂×Q₈	—	4	1	no	—
115	C₂×Q₈	#64516	1	1	yes	characteristic
116	C₂×Q₈	Dic₇₆	1	1	yes	characteristic
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	76	1	no	—
121	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
122	M₁₆	—	76	2	no	—
123	M₁₆	—	152	1	no	—
124	C₂×C₈	—	8	1	no	—
125	C₂×C₈	—	76	1	no	—
126	C₂×C₈	—	76	1	no	—
127	C₂×C₈	—	76	1	no	—
128	C₂×C₈	—	152	1	no	—
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	38	2	no	—
131	C₄⋊C₄	—	76	1	no	—
132	C₄⋊C₄	—	76	2	no	—
133	C₄⋊C₄	—	76	2	no	—
134	C₄²	—	4	1	no	—
135	C₂³	#10597263	1	1	yes	characteristic
136	Q₈	—	4	1	no	—
137	Q₈	—	4	1	no	—
138	Q₈	—	8	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	76	2	no	—
151	C₂×C₄	—	152	1	no	—
152	C₂×C₄	—	152	1	no	—
153	C₂×C₄	#10597273	1	1	yes	characteristic
154	C₂×C₄	#10597282	1	1	yes	characteristic
155	C₈	—	16	1	no	—
156	C₈	—	152	1	no	—
157	C₈	—	152	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#11926282	1	1	yes	characteristic
161	C₂²	#11926332	1	1	yes	characteristic
162	C₂²	#11926338	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	8	1	no	—
168	C₄	—	8	1	no	—
169	C₄	—	152	2	no	—
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	#14561657	1	1	yes	characteristic
173	C₂	#14561807	1	1	yes	characteristic
174	C₂	#14561809	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564786

C₂

yes

maximal, characteristic

#14561762

C₂

yes

maximal, characteristic

C₁₉×#1319

C₂

yes

maximal, characteristic

C₂×#10597316

C₄

yes

—

—

—

C₄

yes

characteristic

C₁₉×#391

C₂²

yes

characteristic

C₁₉×#324

—

C₂²×Q₈×C₁₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×Dic₁₉⋊C₄

—

C₂×#64520

—

C₂²×C₁₉⋊C₈

—

#10597316

—

C₁₉×C₂²⋊C₈

—

C₁₉×#95

—

#10597293

—

#10597293

—

#10597292

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

SD₁₆

yes

characteristic

C₂×Q₈×C₁₉

Q₁₆

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×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

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2026

—

#1394

—

#1319

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#328

yes

characteristic

C₂²×C₁₉

AΣL(1,𝔽₁₆)

yes

characteristic

C₂²×C₁₉

#354

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

Dic₁₉

—

—

—

—

C₂×D₁₉

yes

—

—

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×Q₈

D₇₆

yes

characteristic

C₂×M₁₆

—

100

C₂²×C₈

—

101

#125

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

characteristic

105

C₂×C₄²

C₄×D₁₉

yes

characteristic

106

#103

—

107

#101

—

108

#101

—

109

C₂²⋊C₈

—

110

#95

—

111

C₁₉

#7429

yes

characteristic, complemented, Sylow

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

#64516

yes

characteristic

116

C₂×Q₈

Dic₇₆

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

122

M₁₆

—

123

M₁₆

—

152

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

152

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

—

135

C₂³

#10597263

yes

characteristic

136

Q₈

—

137

Q₈

—

138

Q₈

—

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

—

150

C₂×C₄

—

151

C₂×C₄

—

152

—

152

C₂×C₄

—

152

—

153

C₂×C₄

#10597273

yes

characteristic

154

C₂×C₄

#10597282

yes

characteristic

155

C₈

—

156

C₈

—

152

—

157

C₈

—

152

—

158

C₂²

—

159

C₂²

—

160

C₂²

#11926282

yes

characteristic

161

C₂²

#11926332

yes

characteristic

162

C₂²

#11926338

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

152

—

170

C₂

—

171

C₂

—

172

C₂

#14561657

yes

characteristic

173

C₂

#14561807

yes

characteristic

174

C₂

#14561809

yes

characteristic

	subgroup	G/H
center	C₂²	#11926338
commutator	C₂×C₄²×C₁₉	C₂×C₄
Frattini	#391	C₂×D₁₉
Fitting	C₁₉×#1319	C₂
socle	C₂²×C₁₉	#354
prime radical	C₂³×C₁₉	#99
cosocle	C₁₉×#391	C₂²
hypercenter	#1319	D₁₉
hypocenter	C₁₉	#7429
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ #1319

subgroup

G/H

center

C₂²

#11926338

commutator

C₂×C₄²×C₁₉

C₂×C₄

Frattini

#391

C₂×D₁₉

Fitting

C₁₉×#1319

C₂

C₂²×C₁₉

C₂³×C₁₉

C₁₉×#391

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ #1319

N	H ≅ G/N	type
111. C₁₉	#7429	semidirect

H ≅ G/N

type

111. C₁₉

#7429

semidirect

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

subgroup	index
#14564786	2
#14561762	2
C₁₉×#1319	2
#7429	19

subgroup

index

#14564786

#14561762

C₁₉×#1319

#7429

Subgroups — #18070128

Conjugacy classes of subgroups

Complements

Maximal subgroups