Group #18082307 — subgroups

Subgroups — #18082307

Order 4864.

558 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 207 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	#14564597	C₂	1	4	yes	maximal
2	#14563795	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597275	—	2	2	no	—
6	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
7	#11926414	C₂²	1	2	yes	—
8	#11926306	—	2	8	no	—
9	C₈×C₁₉⋊C₈	—	2	2	no	—
10	C₂²×Q₁₆×C₁₉	—	2	2	no	—
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₁₉×#494	—	2	2	no	—
13	C₁₉×#494	C₄	1	4	yes	complemented
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₁₉	—	2	2	no	—
16	C₂×Q₁₆×C₁₉	—	2	8	no	—
17	C₂×Q₁₆×C₁₉	—	4	4	no	—
18	C₂×Q₁₆×C₁₉	—	4	4	no	—
19	C₂²×C₈×C₁₉	D₄	1	2	yes	—
20	C₁₉×C₄⋊Q₈	—	2	2	no	—
21	C₁₉×C₄⋊Q₈	—	2	4	no	—
22	C₁₉×C₄⋊Q₈	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
24	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₉	—	2	4	no	—
26	C₂×C₄⋊Dic₁₉	—	2	2	no	—
27	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
28	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₉	D₄	1	2	yes	—
30	#10597275	—	4	4	no	—
31	C₄×C₁₉⋊C₈	—	2	2	no	—
32	C₄×C₁₉⋊C₈	—	2	2	no	—
33	C₂×Q₈×C₁₉	—	2	4	no	—
34	C₂×Q₈×C₁₉	—	4	2	no	—
35	C₂×Q₈×C₁₉	—	4	4	no	—
36	C₂×Q₈×C₁₉	—	4	4	no	—
37	C₂²×C₄×C₁₉	—	4	2	no	—
38	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₁₉	—	4	2	no	—
41	Q₁₆×C₁₉	—	4	16	no	—
42	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
43	C₂×C₈×C₁₉	D₈	1	4	yes	—
44	C₂×C₈×C₁₉	SD₁₆	1	4	yes	—
45	C₄⋊C₄×C₁₉	—	2	8	no	—
46	C₄⋊C₄×C₁₉	—	4	4	no	—
47	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₉	#41	1	2	yes	—
50	C₄⋊Dic₁₉	—	4	4	no	—
51	C₄⋊Dic₁₉	—	4	8	no	—
52	C₂×C₁₉⋊C₈	—	2	4	no	—
53	C₂×C₁₉⋊C₈	—	2	4	no	—
54	C₂×C₁₉⋊C₈	Q₁₆	1	8	yes	complemented
55	#9825	—	19	1	no	maximal, Sylow
56	C₂³×C₁₉	#127	1	1	yes	characteristic
57	Q₈×C₁₉	—	4	8	no	—
58	Q₈×C₁₉	—	8	4	no	—
59	C₂×C₄×C₁₉	—	4	4	no	—
60	C₂×C₄×C₁₉	—	8	2	no	—
61	C₂×C₄×C₁₉	—	8	2	no	—
62	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₁₉	#121	1	2	yes	—
65	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
68	C₂×C₄×C₁₉	C₂×Q₁₆	1	4	yes	—
69	C₂×C₄×C₁₉	#135	1	4	yes	—
70	C₂×Dic₁₉	—	8	2	no	—
71	C₂×Dic₁₉	—	8	4	no	—
72	C₈×C₁₉	#102	1	8	yes	—
73	C₁₉⋊C₈	—	2	8	no	—
74	C₂×#494	D₁₉	1	1	yes	characteristic
75	#1990	—	19	4	no	—
76	#1845	—	19	1	no	—
77	C₂×C₈²	—	19	1	no	—
78	C₂²×C₁₉	#390	1	1	yes	characteristic
79	C₂²×C₁₉	#492	1	1	yes	characteristic
80	C₂²×C₁₉	#493	1	1	yes	characteristic
81	C₂²×C₁₉	#494	1	2	yes	—
82	C₂²×C₁₉	#495	1	2	yes	—
83	C₄×C₁₉	—	8	4	no	—
84	C₄×C₁₉	C₂×#102	1	2	yes	—
85	C₄×C₁₉	C₄×Q₁₆	1	4	yes	—
86	C₄×C₁₉	#467	1	4	yes	—
87	C₄×C₁₉	#486	1	2	yes	—
88	Dic₁₉	—	16	2	no	—
89	C₂²×Q₁₆	—	2	2	no	—
90	C₂×C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
91	#494	—	2	2	no	—
92	#494	Dic₁₉	1	4	yes	complemented
93	C₂×#107	—	38	2	no	—
94	C₂×C₄×C₈	—	19	2	no	—
95	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
96	#384	—	19	2	no	—
97	#340	—	38	8	no	—
98	C₈²	—	38	2	no	—
99	C₂×C₁₉	#1707	1	2	yes	—
100	C₂×C₁₉	#1709	1	1	yes	characteristic
101	C₂×C₁₉	#1713	1	1	yes	characteristic
102	C₂×C₁₉	#1957	1	1	yes	characteristic
103	C₂×C₁₉	#1969	1	2	yes	—
104	C₂²×Q₈	—	2	2	no	—
105	C₂×Q₁₆	—	2	8	no	—
106	C₂×Q₁₆	—	4	4	no	—
107	C₂×Q₁₆	—	4	4	no	—
108	C₂²×C₈	—	19	4	no	—
109	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
110	C₄⋊Q₈	—	2	4	no	—
111	C₄⋊Q₈	—	2	2	no	—
112	C₄⋊Q₈	C₂×Dic₁₉	1	4	yes	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	38	4	no	—
115	C₂×C₄⋊C₄	—	38	2	no	—
116	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
117	#107	—	76	4	no	—
118	C₄×C₈	—	38	2	no	—
119	C₄×C₈	—	38	2	no	—
120	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
121	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
122	C₁₉	#9825	1	1	yes	characteristic, complemented, Sylow
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂×Q₈	—	4	4	no	—
126	C₂×Q₈	—	4	4	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	76	2	no	—
129	C₂²×C₄	D₄×D₁₉	1	2	yes	—
130	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
131	Q₁₆	—	4	16	no	—
132	C₂×C₈	—	19	8	no	—
133	C₂×C₈	—	38	4	no	—
134	C₂×C₈	—	38	4	no	—
135	C₂×C₈	#64525	1	4	yes	—
136	C₂×C₈	#64526	1	4	yes	—
137	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
138	C₄⋊C₄	—	2	8	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	76	4	no	—
141	C₄⋊C₄	—	76	8	no	—
142	C₄²	#64543	1	2	yes	—
143	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
144	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
145	C₂³	#10597386	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	Q₈	—	8	4	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	152	2	no	—
152	C₂×C₄	—	152	4	no	—
153	C₂×C₄	Q₁₆×D₁₉	1	4	yes	—
154	C₂×C₄	#10597365	1	4	yes	—
155	C₂×C₄	C₂×#64525	1	2	yes	—
156	C₂×C₄	C₂×#64526	1	2	yes	—
157	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
158	C₂×C₄	#10597382	1	1	yes	characteristic
159	C₂×C₄	#10597384	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
161	C₈	—	38	8	no	—
162	C₈	#10597289	1	8	yes	—
163	C₂²	#11926447	1	1	yes	characteristic
164	C₂²	#11926916	1	1	yes	characteristic
165	C₂²	#11926920	1	1	yes	characteristic
166	C₂²	#11926922	1	2	yes	—
167	C₂²	#11926929	1	2	yes	—
168	C₄	—	8	4	no	—
169	C₄	—	304	2	no	—
170	C₄	C₂×#10597289	1	2	yes	—
171	C₄	Q₁₆×Dic₁₉	1	4	yes	—
172	C₄	#11926782	1	4	yes	—
173	C₄	#11926890	1	2	yes	—
174	C₂	#14563229	1	2	yes	—
175	C₂	#14563242	1	1	yes	characteristic
176	C₂	#14563251	1	1	yes	characteristic
177	C₂	#14564442	1	1	yes	characteristic
178	C₂	#14564497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14564597

C₂

yes

maximal

#14563795

C₂

yes

maximal, characteristic

C₂×C₈×C₁₉⋊C₈

C₂

yes

maximal, characteristic

C₂×C₁₉×#494

C₂

yes

maximal, characteristic

C₂×#10597275

—

C₂×C₄×C₁₉⋊C₈

C₂²

yes

—

#11926414

C₂²

yes

—

#11926306

—

C₈×C₁₉⋊C₈

—

C₂²×Q₁₆×C₁₉

—

C₂×C₁₉×C₄⋊Q₈

C₂²

yes

—

C₁₉×#494

—

C₁₉×#494

C₄

yes

complemented

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₂²×Q₈×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂²×C₁₉⋊C₈

D₄

yes

—

C₄×C₈×C₁₉

C₂×C₄

yes

—

C₄×C₈×C₁₉

D₄

yes

—

#10597275

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×Dic₁₉

—

Q₁₆×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

#41

yes

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

Q₁₆

yes

—

C₂³×C₁₉

yes

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#124

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₂×C₄×C₁₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₉

C₂×Q₁₆

yes

—

C₂×C₄×C₁₉

#135

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

#102

yes

—

C₁₉⋊C₈

—

C₂×#494

D₁₉

yes

characteristic

#1990

—

#1845

—

C₂×C₈²

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

—

C₂²×C₁₉

#495

yes

—

C₄×C₁₉

—

C₄×C₁₉

C₂×#102

yes

—

C₄×C₁₉

C₄×Q₁₆

yes

—

C₄×C₁₉

#467

yes

—

C₄×C₁₉

#486

yes

—

Dic₁₉

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₁₉

yes

—

#494

—

#494

Dic₁₉

yes

complemented

C₂×#107

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

#384

—

#340

—

C₈²

—

C₂×C₁₉

#1707

yes

—

100

C₂×C₁₉

#1709

yes

characteristic

101

C₂×C₁₉

#1713

yes

characteristic

102

C₂×C₁₉

#1957

yes

characteristic

103

C₂×C₁₉

#1969

yes

—

104

C₂²×Q₈

—

105

C₂×Q₁₆

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂²×C₈

—

109

C₂²×C₈

C₂²⋊D₁₉

yes

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

C₂×Dic₁₉

yes

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²×D₁₉

yes

characteristic

117

#107

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₂×Dic₁₉

yes

—

121

C₄×C₈

C₂²⋊D₁₉

yes

—

122

C₁₉

#9825

yes

characteristic, complemented, Sylow

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

D₄×D₁₉

yes

—

130

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

131

Q₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

#64525

yes

—

136

C₂×C₈

#64526

yes

—

137

C₂×C₈

C₂²⋊Dic₁₉

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

#64543

yes

—

143

C₄²

C₂²×Dic₁₉

yes

characteristic

144

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

145

C₂³

#10597386

yes

characteristic

146

Q₈

—

147

Q₈

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

—

152

C₂×C₄

—

152

—

153

C₂×C₄

Q₁₆×D₁₉

yes

—

154

C₂×C₄

#10597365

yes

—

155

C₂×C₄

C₂×#64525

yes

—

156

C₂×C₄

C₂×#64526

yes

—

157

C₂×C₄

D₄×Dic₁₉

yes

—

158

C₂×C₄

#10597382

yes

characteristic

159

C₂×C₄

#10597384

yes

—

160

C₂×C₄

C₂×C₂²⋊Dic₁₉

yes

characteristic

161

C₈

—

162

C₈

#10597289

yes

—

163

C₂²

#11926447

yes

characteristic

164

C₂²

#11926916

yes

characteristic

165

C₂²

#11926920

yes

characteristic

166

C₂²

#11926922

yes

—

167

C₂²

#11926929

yes

—

168

C₄

—

169

C₄

—

304

—

170

C₄

C₂×#10597289

yes

—

171

C₄

Q₁₆×Dic₁₉

yes

—

172

C₄

#11926782

yes

—

173

C₄

#11926890

yes

—

174

C₂

#14563229

yes

—

175

C₂

#14563242

yes

characteristic

176

C₂

#14563251

yes

characteristic

177

C₂

#14564442

yes

characteristic

178

C₂

#14564497

yes

—

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#494	C₂
socle, prime radical	C₂³×C₁₉	#127
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#494	D₁₉
hypocenter	C₁₉	#9825
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂³

#10597386

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₂×C₁₉×#494

C₂

socle, prime radical

C₂³×C₁₉

#127

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
13. C₁₉×#494	C₄	semidirect
54. C₂×C₁₉⋊C₈	Q₁₆	semidirect
92. #494	Dic₁₉	semidirect
122. C₁₉	#9825	semidirect

H ≅ G/N

type

13. C₁₉×#494

C₄

semidirect

54. C₂×C₁₉⋊C₈

Q₁₆

semidirect

92. #494

Dic₁₉

semidirect

122. C₁₉

#9825

semidirect

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

subgroup	index
#14564597	2
#14563795	2
C₂×C₈×C₁₉⋊C₈	2
C₂×C₁₉×#494	2
#9825	19

subgroup

index

#14564597

#14563795

C₂×C₈×C₁₉⋊C₈

C₂×C₁₉×#494

#9825

Subgroups — #18082307

Conjugacy classes of subgroups

Complements

Maximal subgroups