Group #18070083 — subgroups

Subgroups — #18070083

Order 4864.

198 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 37 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	#14564922	C₂	1	1	yes	maximal, characteristic
2	C₁₉⋊#1309	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1327	C₂	1	1	yes	maximal, characteristic, complemented
4	#11927344	C₄	1	1	yes	characteristic
5	#11926568	—	2	1	no	—
6	C₂×#10597262	—	2	1	no	—
7	#11926526	—	2	1	no	—
8	C₁₉⋊#338	C₄	1	1	yes	characteristic
9	C₁₉×#423	C₂²	1	1	yes	characteristic
10	C₁₉×#349	—	2	1	no	—
11	C₂×#64541	—	2	1	no	—
12	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
13	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
15	C₂×#64520	—	2	1	no	—
16	C₂×#64520	—	2	2	no	—
17	#10597355	—	4	1	no	—
18	#10597354	—	4	1	no	—
19	#10597350	—	4	1	no	—
20	C₁₉×M₃₂	—	4	1	no	—
21	C₁₉×C₄⋊C₈	—	4	1	no	—
22	#10597281	—	4	1	no	—
23	#10597280	—	4	1	no	—
24	#10597262	—	4	1	no	—
25	#10597262	—	4	1	no	—
26	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
27	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
28	C₂²×C₄×C₁₉	—	2	1	no	—
29	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₂²⋊D₁₉	—	4	1	no	—
31	#64541	—	4	1	no	—
32	#64541	—	4	1	no	—
33	#64541	—	4	1	no	—
34	#64541	—	4	1	no	—
35	C₂×D₇₆	—	2	1	no	—
36	C₂×C₄×D₁₉	—	4	1	no	—
37	C₂×Dic₃₈	—	2	1	no	—
38	M₁₆×C₁₉	—	4	1	no	—
39	M₁₆×C₁₉	—	4	1	no	—
40	C₂×C₈×C₁₉	—	2	1	no	—
41	C₂×C₈×C₁₉	—	2	1	no	—
42	C₄²×C₁₉	—	2	1	no	—
43	C₄²×C₁₉	D₈	1	1	yes	characteristic
44	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
45	#64520	—	4	1	no	—
46	#64520	—	4	1	no	—
47	#64520	—	8	2	no	—
48	C₂×C₁₉⋊C₈	—	4	2	no	—
49	C₂×C₁₉⋊C₈	—	4	1	no	—
50	Dic₇₆	—	4	1	no	—
51	D₁₅₂	—	4	1	no	—
52	#64516	—	4	2	no	—
53	C₁₆×C₁₉	—	8	1	no	—
54	#7414	—	19	1	no	maximal, Sylow
55	C₂³×C₁₉	#104	1	1	yes	characteristic
56	C₂²×D₁₉	—	4	1	no	—
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	1	no	—
61	C₂×C₄×C₁₉	—	4	1	no	—
62	C₂×C₄×C₁₉	—	4	1	no	—
63	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
64	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
65	C₂²⋊D₁₉	—	8	1	no	—
66	C₂²⋊D₁₉	—	8	1	no	—
67	C₂×Dic₁₉	—	4	1	no	—
68	D₇₆	—	4	1	no	—
69	D₇₆	—	4	1	no	—
70	C₄×D₁₉	—	8	1	no	—
71	C₄×D₁₉	—	8	1	no	—
72	Dic₃₈	—	4	1	no	—
73	Dic₃₈	—	4	1	no	—
74	C₈×C₁₉	—	4	1	no	—
75	C₈×C₁₉	—	4	1	no	—
76	C₁₉⋊C₈	—	8	1	no	—
77	C₁₉⋊C₈	—	8	2	no	—
78	#2050	—	19	1	no	—
79	#1327	D₁₉	1	1	yes	characteristic, complemented
80	#1309	—	19	1	no	—
81	C₂²×C₁₉	—	2	1	no	—
82	C₂²×C₁₉	—	4	1	no	—
83	C₂²×C₁₉	#327	1	1	yes	characteristic
84	C₂×D₁₉	—	8	1	no	—
85	C₂×D₁₉	—	16	1	no	—
86	C₄×C₁₉	—	2	1	no	—
87	C₄×C₁₉	—	2	1	no	—
88	C₄×C₁₉	—	2	1	no	—
89	C₄×C₁₉	—	4	1	no	—
90	C₄×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
91	C₄×C₁₉	#355	1	1	yes	characteristic
92	Dic₁₉	—	8	1	no	—
93	#575	—	19	1	no	—
94	#423	—	38	1	no	—
95	#423	C₂×D₁₉	1	1	yes	characteristic
96	C₂×#104	—	38	1	no	—
97	#413	—	38	1	no	—
98	#349	—	2	1	no	—
99	#338	—	19	1	no	—
100	C₂×C₁₉	—	2	1	no	—
101	C₂×C₁₉	—	4	1	no	—
102	C₂×C₁₉	#1345	1	1	yes	characteristic
103	D₁₉	—	16	1	no	—
104	C₂×#41	—	38	1	no	—
105	#137	—	76	1	no	—
106	AGL(1,ℤ/8)	—	76	1	no	—
107	#135	—	76	1	no	—
108	C₂×M₁₆	—	38	1	no	—
109	C₂×M₁₆	—	38	2	no	—
110	C₂×M₁₆	C₄×D₁₉	1	1	yes	characteristic
111	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
112	C₂×C₄²	D₇₆	1	1	yes	characteristic
113	M₃₂	—	4	1	no	—
114	C₄⋊C₈	—	4	1	no	—
115	C₄⋊C₈	—	76	1	no	—
116	C₄⋊C₈	—	76	1	no	—
117	#104	—	76	1	no	—
118	#104	—	76	1	no	—
119	#101	—	76	1	no	—
120	#100	—	76	1	no	—
121	C₁₉	#7414	1	1	yes	characteristic, complemented, Sylow
122	#41	—	76	1	no	—
123	#41	—	76	1	no	—
124	#41	—	76	1	no	—
125	#41	—	76	1	no	—
126	C₂×Q₈	—	38	1	no	—
127	C₂×D₄	—	38	1	no	—
128	C₂×D₄	—	76	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	76	1	no	—
131	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
132	Q₁₆	—	76	1	no	—
133	SD₁₆	—	76	2	no	—
134	D₈	—	76	1	no	—
135	M₁₆	—	4	1	no	—
136	M₁₆	—	4	1	no	—
137	M₁₆	—	76	1	no	—
138	M₁₆	—	76	1	no	—
139	M₁₆	—	152	2	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	76	2	no	—
143	C₂×C₈	—	76	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	#64516	1	1	yes	characteristic
146	C₄²	D₁₅₂	1	1	yes	characteristic
147	C₁₆	—	8	1	no	—
148	C₂³	—	76	1	no	—
149	C₂³	#10597282	1	1	yes	characteristic
150	Q₈	—	76	1	no	—
151	Q₈	—	76	1	no	—
152	D₄	—	76	1	no	—
153	D₄	—	76	1	no	—
154	D₄	—	152	1	no	—
155	D₄	—	152	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	76	1	no	—
163	C₂×C₄	—	152	1	no	—
164	C₂×C₄	—	152	1	no	—
165	C₂×C₄	#10597263	1	1	yes	characteristic
166	C₂×C₄	#10597278	1	1	yes	characteristic
167	C₈	—	4	1	no	—
168	C₈	—	4	1	no	—
169	C₈	—	152	1	no	—
170	C₈	—	152	2	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	4	1	no	—
173	C₂²	—	304	1	no	—
174	C₂²	—	152	1	no	—
175	C₂²	#11926279	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	4	1	no	—
180	C₄	—	152	1	no	—
181	C₄	#11926330	1	1	yes	characteristic
182	C₄	#11926340	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	4	1	no	—
185	C₂	—	304	1	no	—
186	C₂	#14561639	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564922

C₂

yes

maximal, characteristic

C₁₉⋊#1309

C₂

yes

maximal, characteristic, complemented

C₁₉×#1327

C₂

yes

maximal, characteristic, complemented

#11927344

C₄

yes

—

—

—

C₁₉⋊#338

C₄

yes

characteristic

C₁₉×#423

C₂²

yes

characteristic

C₁₉×#349

—

C₂×#64541

—

C₂×M₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

—

—

—

—

—

C₁₉×M₃₂

—

C₁₉×C₄⋊C₈

—

—

—

—

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₂²⋊D₁₉

—

—

—

—

—

C₂×D₇₆

—

C₂×C₄×D₁₉

—

C₂×Dic₃₈

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

D₈

yes

characteristic

C₄²×C₁₉

SD₁₆

yes

—

—

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

Dic₇₆

—

D₁₅₂

—

#64516

—

C₁₆×C₁₉

—

—

C₂³×C₁₉

yes

C₂²×D₁₉

—

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

C₂²⋊D₁₉

—

C₂²⋊D₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

C₄×D₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2050

—

#1327

D₁₉

yes

characteristic, complemented

#1309

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#327

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

AΣL(1,𝔽₁₆)

yes

characteristic

C₄×C₁₉

#355

yes

characteristic

Dic₁₉

—

#575

—

#423

—

#423

C₂×D₁₉

yes

—

—

—

—

100

C₂×C₁₉

—

101

C₂×C₁₉

—

102

C₂×C₁₉

#1345

yes

characteristic

103

D₁₉

—

104

C₂×#41

—

105

#137

—

106

AGL(1,ℤ/8)

—

107

#135

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

110

C₂×M₁₆

C₄×D₁₉

yes

characteristic

111

C₂×M₁₆

C₂²⋊D₁₉

yes

characteristic

112

C₂×C₄²

D₇₆

yes

characteristic

113

M₃₂

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

#104

—

118

#104

—

119

#101

—

120

#100

—

121

C₁₉

#7414

yes

characteristic, complemented, Sylow

122

#41

—

123

#41

—

124

#41

—

125

#41

—

126

C₂×Q₈

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

132

Q₁₆

—

133

SD₁₆

—

134

D₈

—

135

M₁₆

—

136

M₁₆

—

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

152

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₄²

—

145

C₄²

#64516

yes

characteristic

146

C₄²

D₁₅₂

yes

characteristic

147

C₁₆

—

148

C₂³

—

149

C₂³

#10597282

yes

characteristic

150

Q₈

—

151

Q₈

—

152

D₄

—

153

D₄

—

154

D₄

—

152

—

155

D₄

—

152

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

152

—

164

C₂×C₄

—

152

—

165

C₂×C₄

#10597263

yes

characteristic

166

C₂×C₄

#10597278

yes

characteristic

167

C₈

—

168

C₈

—

169

C₈

—

152

—

170

C₈

—

152

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

304

—

174

C₂²

—

152

—

175

C₂²

#11926279

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

152

—

181

C₄

#11926330

yes

characteristic

182

C₄

#11926340

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

—

304

—

186

C₂

#14561639

yes

characteristic

	subgroup	G/H
center	C₂	#14561639
commutator	C₂×M₁₆×C₁₉	C₂×C₄
Frattini	#423	C₂×D₁₉
Fitting	C₁₉×#1327	C₂
socle	C₂×C₁₉	#1345
prime radical	#11927344	C₄
cosocle	C₁₉×#423	C₂²
hypercenter	#1327	D₁₉
hypocenter	C₁₉	#7414
LCS	G ▷ C₂×M₁₆×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ #1327

subgroup

G/H

center

C₂

#14561639

commutator

C₂×M₁₆×C₁₉

C₂×C₄

Frattini

#423

C₂×D₁₉

Fitting

C₁₉×#1327

C₂

C₂×C₁₉

C₄

C₁₉×#423

C₂²

D₁₉

C₁₉

G ▷ C₂×M₁₆×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ #1327

N	H ≅ G/N	type
2. C₁₉⋊#1309	C₂	semidirect
3. C₁₉×#1327	C₂	semidirect
79. #1327	D₁₉	semidirect
121. C₁₉	#7414	semidirect

H ≅ G/N

type

2. C₁₉⋊#1309

C₂

semidirect

3. C₁₉×#1327

C₂

semidirect

79. #1327

D₁₉

semidirect

121. C₁₉

#7414

semidirect

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

subgroup	index
#14564922	2
C₁₉⋊#1309	2
C₁₉×#1327	2
#7414	19

subgroup

index

#14564922

C₁₉⋊#1309

C₁₉×#1327

#7414

Subgroups — #18070083

Conjugacy classes of subgroups

Complements

Maximal subgroups