Group #18069352 — subgroups

Subgroups — #18069352

Order 4864.

210 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 63 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	#14564616	C₂	1	1	yes	maximal, characteristic
2	#14561464	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1274	C₂	1	1	yes	maximal, characteristic
4	#11927101	C₄	1	1	yes	characteristic
5	#11926597	—	2	1	no	—
6	#11926568	C₄	1	1	yes	characteristic
7	#11926304	—	2	1	no	—
8	#11926304	—	2	1	no	—
9	C₁₉×#431	—	2	1	no	—
10	C₁₉×#376	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
14	#10597372	—	2	1	no	—
15	#10597370	—	4	1	no	—
16	C₂×#64520	—	2	1	no	—
17	#10597316	C₈	1	2	yes	complemented
18	#10597315	—	2	1	no	—
19	C₄×Dic₃₈	—	4	1	no	—
20	C₁₉×C₄⋊C₈	—	4	1	no	—
21	C₁₉×C₂²⋊C₈	—	4	1	no	—
22	C₄×C₈×C₁₉	—	2	2	no	—
23	C₁₉×#95	—	4	1	no	—
24	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
25	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
26	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
27	C₄×C₁₉⋊C₈	—	2	2	no	—
28	C₂²×C₄×C₁₉	—	2	1	no	—
29	C₂²×C₄×C₁₉	—	2	1	no	—
30	C₂²×C₄×C₁₉	—	2	1	no	—
31	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×Dic₃₈	—	4	1	no	—
33	C₂×C₈×C₁₉	—	4	2	no	—
34	C₄²×C₁₉	—	2	1	no	—
35	C₄²×C₁₉	—	2	1	no	—
36	C₄²×C₁₉	—	2	1	no	—
37	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
39	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
40	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
41	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
42	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
43	C₂²⋊Dic₁₉	—	4	1	no	—
44	C₄⋊Dic₁₉	—	2	2	no	—
45	C₄⋊Dic₁₉	—	4	1	no	—
46	Dic₁₉⋊C₄	—	4	2	no	—
47	C₄×Dic₁₉	—	4	1	no	—
48	#64520	—	4	1	no	—
49	#64520	—	4	1	no	—
50	C₂×C₁₉⋊C₈	—	4	2	no	—
51	C₂×C₁₉⋊C₈	—	4	1	no	—
52	#7129	—	19	1	no	maximal, Sylow
53	C₂³×C₁₉	#99	1	1	yes	characteristic
54	C₂×C₄×C₁₉	—	2	1	no	—
55	C₂×C₄×C₁₉	—	2	1	no	—
56	C₂×C₄×C₁₉	—	2	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	—	8	1	no	—
61	C₂×C₄×C₁₉	—	8	1	no	—
62	C₂×C₄×C₁₉	—	8	1	no	—
63	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
65	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
66	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
68	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
69	C₂×Dic₁₉	—	4	2	no	—
70	Dic₃₈	—	8	2	no	—
71	C₈×C₁₉	—	8	2	no	—
72	C₁₉⋊C₈	—	8	2	no	—
73	C₁₉⋊C₈	—	8	1	no	—
74	#1995	—	19	1	no	—
75	#1274	—	19	1	no	—
76	#1274	D₁₉	1	1	yes	characteristic
77	C₂²×C₁₉	—	4	1	no	—
78	C₂²×C₁₉	#323	1	1	yes	characteristic
79	C₂²×C₁₉	#328	1	1	yes	characteristic
80	C₂²×C₁₉	#328	1	1	yes	characteristic
81	C₄×C₁₉	—	4	1	no	—
82	C₄×C₁₉	—	4	1	no	—
83	C₄×C₁₉	—	4	1	no	—
84	C₄×C₁₉	—	4	1	no	—
85	C₄×C₁₉	—	4	1	no	—
86	C₄×C₁₉	—	4	1	no	—
87	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
88	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
89	C₄×C₁₉	#329	1	1	yes	characteristic
90	C₄×C₁₉	#333	1	1	yes	characteristic
91	Dic₁₉	—	8	2	no	—
92	#533	—	19	1	no	—
93	#431	—	2	1	no	—
94	#431	—	38	1	no	—
95	#423	—	19	1	no	—
96	#376	C₂×D₁₉	1	1	yes	characteristic
97	#339	—	38	1	no	—
98	#339	—	38	1	no	—
99	C₂×C₁₉	—	2	1	no	—
100	C₂×C₁₉	—	2	1	no	—
101	C₂×C₁₉	—	2	1	no	—
102	C₂×C₁₉	#1272	1	1	yes	characteristic
103	C₂×M₁₆	—	38	1	no	—
104	C₄⋊Q₈	—	38	1	no	—
105	#125	—	19	2	no	—
106	C₂²⋊Q₈	—	76	1	no	—
107	C₄×Q₈	—	76	1	no	—
108	#117	—	38	1	no	—
109	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
110	C₂×C₄²	D₇₆	1	1	yes	characteristic
111	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
112	C₄⋊C₈	—	4	1	no	—
113	C₄⋊C₈	—	76	1	no	—
114	C₄⋊C₈	—	76	1	no	—
115	C₂²⋊C₈	—	4	1	no	—
116	C₂²⋊C₈	—	76	1	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	38	2	no	—
119	#95	—	4	1	no	—
120	C₁₉	#7129	1	1	yes	characteristic, complemented, Sylow
121	C₂×Q₈	—	76	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
126	M₁₆	—	76	1	no	—
127	M₁₆	—	76	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	76	2	no	—
130	C₂×C₈	—	76	1	no	—
131	C₄⋊C₄	—	38	2	no	—
132	C₄⋊C₄	—	76	2	no	—
133	C₄⋊C₄	—	76	1	no	—
134	C₂²⋊C₄	—	76	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	76	1	no	—
139	C₄²	C₈×D₁₉	1	1	yes	characteristic
140	C₄²	#64515	1	1	yes	characteristic
141	C₄²	#64516	1	1	yes	characteristic
142	C₄²	Dic₇₆	1	1	yes	characteristic
143	C₄²	#64526	1	1	yes	characteristic
144	C₄²	#64528	1	1	yes	characteristic
145	C₂³	#10597263	1	1	yes	characteristic
146	Q₈	—	152	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	—	8	1	no	—
155	C₂×C₄	—	8	1	no	—
156	C₂×C₄	—	76	2	no	—
157	C₂×C₄	#10597262	1	1	yes	characteristic
158	C₂×C₄	#10597267	1	1	yes	characteristic
159	C₂×C₄	#10597273	1	1	yes	characteristic
160	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
161	C₂×C₄	#10597280	1	1	yes	characteristic
162	C₂×C₄	#10597282	1	1	yes	characteristic
163	C₈	—	8	2	no	—
164	C₈	—	152	2	no	—
165	C₈	—	152	1	no	—
166	C₂²	—	4	1	no	—
167	C₂²	#11926267	1	1	yes	characteristic
168	C₂²	#11926280	1	1	yes	characteristic
169	C₂²	#11926282	1	1	yes	characteristic
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	152	2	no	—
177	C₄	#11926274	1	1	yes	characteristic
178	C₄	#11926276	1	1	yes	characteristic
179	C₄	#11926283	1	1	yes	characteristic
180	C₄	#11926292	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	#14561460	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564616

C₂

yes

maximal, characteristic

#14561464

C₂

yes

maximal, characteristic

C₁₉×#1274

C₂

yes

maximal, characteristic

#11927101

C₄

yes

characteristic

#11926597

—

#11926568

C₄

yes

characteristic

#11926304

—

#11926304

—

C₁₉×#431

—

C₁₉×#376

C₂²

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

—

—

—

C₈

yes

complemented

#10597315

—

C₄×Dic₃₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

—

C₁₉×#95

—

C₂²⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×C₈

yes

characteristic

C₄²×C₁₉

M₁₆

yes

characteristic

C₄²×C₁₉

SD₁₆

yes

characteristic

C₄²×C₁₉

SD₁₆

yes

characteristic

C₄²×C₁₉

Q₁₆

yes

characteristic

C₄²×C₁₉

Q₁₆

yes

characteristic

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

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

C₂²⋊C₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

Dic₃₈

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1995

—

#1274

—

#1274

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

Q₈⋊C₈

yes

characteristic

C₄×C₁₉

Q₈⋊C₈

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

Dic₁₉

—

—

—

—

—

C₂×D₁₉

yes

characteristic

#339

—

#339

—

C₂×C₁₉

—

100

C₂×C₁₉

—

101

C₂×C₁₉

—

102

C₂×C₁₉

#1272

yes

characteristic

103

C₂×M₁₆

—

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

107

C₄×Q₈

—

108

#117

—

109

C₂×C₄²

C₄×D₁₉

yes

characteristic

110

C₂×C₄²

D₇₆

yes

characteristic

111

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₂²⋊C₈

—

116

C₂²⋊C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

#95

—

120

C₁₉

#7129

yes

characteristic, complemented, Sylow

121

C₂×Q₈

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

126

M₁₆

—

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₈×D₁₉

yes

characteristic

140

C₄²

#64515

yes

characteristic

141

C₄²

#64516

yes

characteristic

142

C₄²

Dic₇₆

yes

characteristic

143

C₄²

#64526

yes

characteristic

144

C₄²

#64528

yes

characteristic

145

C₂³

#10597263

yes

characteristic

146

Q₈

—

152

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

#10597262

yes

characteristic

158

C₂×C₄

#10597267

yes

characteristic

159

C₂×C₄

#10597273

yes

characteristic

160

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

161

C₂×C₄

#10597280

yes

characteristic

162

C₂×C₄

#10597282

yes

characteristic

163

C₈

—

164

C₈

—

152

—

165

C₈

—

152

—

166

C₂²

—

167

C₂²

#11926267

yes

characteristic

168

C₂²

#11926280

yes

characteristic

169

C₂²

#11926282

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

152

—

177

C₄

#11926274

yes

characteristic

178

C₄

#11926276

yes

characteristic

179

C₄

#11926283

yes

characteristic

180

C₄

#11926292

yes

characteristic

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

#14561460

yes

characteristic

	subgroup	G/H
center	C₄	#11926283
commutator	C₄²×C₁₉	C₂×C₈
Frattini	#376	C₂×D₁₉
Fitting	C₁₉×#1274	C₂
socle	C₂×C₁₉	#1272
prime radical	C₂³×C₁₉	#99
cosocle	C₁₉×#376	C₂²
hypercenter	#1274	D₁₉
hypocenter	C₁₉	#7129
LCS	G ▷ C₄²×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #376 ◁ #1274

subgroup

G/H

center

C₄

#11926283

commutator

C₄²×C₁₉

C₂×C₈

Frattini

#376

C₂×D₁₉

Fitting

C₁₉×#1274

C₂

C₂×C₁₉

C₂³×C₁₉

C₁₉×#376

C₂²

D₁₉

C₁₉

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #376 ◁ #1274

N	H ≅ G/N	type
17. #10597316	C₈	semidirect
120. C₁₉	#7129	semidirect

H ≅ G/N

type

17. #10597316

C₈

semidirect

120. C₁₉

#7129

semidirect

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

subgroup	index
#14564616	2
#14561464	2
C₁₉×#1274	2
#7129	19

subgroup

index

#14564616

#14561464

C₁₉×#1274

#7129

Subgroups — #18069352

Conjugacy classes of subgroups

Complements

Maximal subgroups