Group #18069257 — subgroups

Subgroups — #18069257

Order 4864.

180 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 51 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	#14561575	C₂	1	1	yes	maximal, characteristic
2	#14561510	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2086	C₂	1	1	yes	maximal, characteristic
4	#11926568	—	2	1	no	—
5	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
6	C₂×C₁₉×#107	C₄	1	1	yes	characteristic
7	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₁₉×#389	—	2	1	no	—
9	C₁₉×#340	—	2	1	no	—
10	C₁₉×#340	C₄	1	1	yes	characteristic
11	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
15	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
16	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
17	C₂×#64520	—	4	1	no	—
18	C₁₉×#107	—	4	1	no	—
19	C₁₉×C₄⋊C₈	—	4	1	no	—
20	C₁₉×#95	—	4	1	no	—
21	C₂×C₁₉⋊C₁₆	—	4	1	no	—
22	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
24	C₂²×C₄×C₁₉	—	4	1	no	—
25	C₂²×C₄×C₁₉	—	4	1	no	—
26	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
29	C₂×C₈×C₁₉	—	2	1	no	—
30	C₂×C₈×C₁₉	—	2	1	no	—
31	C₂×C₈×C₁₉	—	4	1	no	—
32	C₂×C₈×C₁₉	—	4	1	no	—
33	C₄⋊C₄×C₁₉	—	2	2	no	—
34	C₄⋊C₄×C₁₉	—	4	1	no	—
35	C₄⋊C₄×C₁₉	—	4	1	no	—
36	C₄⋊C₄×C₁₉	—	4	1	no	—
37	C₄⋊C₄×C₁₉	—	8	1	no	—
38	C₄⋊C₄×C₁₉	—	8	1	no	—
39	C₄²×C₁₉	—	2	1	no	—
40	C₄²×C₁₉	—	2	1	no	—
41	#64520	—	8	1	no	—
42	#64520	—	8	1	no	—
43	C₂×C₁₉⋊C₈	—	4	2	no	—
44	C₁₉⋊C₁₆	—	8	1	no	—
45	#7091	—	19	1	no	maximal, Sylow
46	C₂³×C₁₉	#102	1	1	yes	characteristic
47	C₂×C₄×C₁₉	—	2	1	no	—
48	C₂×C₄×C₁₉	—	2	1	no	—
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	—	4	1	no	—
53	C₂×C₄×C₁₉	—	4	2	no	—
54	C₂×C₄×C₁₉	—	8	1	no	—
55	C₂×C₄×C₁₉	—	8	1	no	—
56	C₂×C₄×C₁₉	—	8	1	no	—
57	C₂×C₄×C₁₉	—	8	1	no	—
58	C₂×C₄×C₁₉	—	8	1	no	—
59	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
60	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
61	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
63	C₈×C₁₉	—	4	1	no	—
64	C₈×C₁₉	—	4	1	no	—
65	C₁₉⋊C₈	—	8	2	no	—
66	#2086	D₁₉	1	1	yes	characteristic
67	#1322	—	19	1	no	—
68	#1292	—	19	1	no	—
69	C₂²×C₁₉	—	2	1	no	—
70	C₂²×C₁₉	—	2	1	no	—
71	C₂²×C₁₉	#327	1	1	yes	characteristic
72	C₂²×C₁₉	#357	1	1	yes	characteristic
73	C₂²×C₁₉	#360	1	1	yes	characteristic
74	C₄×C₁₉	—	2	1	no	—
75	C₄×C₁₉	—	2	1	no	—
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	—	4	1	no	—
78	C₄×C₁₉	—	8	2	no	—
79	C₄×C₁₉	—	16	1	no	—
80	C₄×C₁₉	#352	1	1	yes	characteristic
81	C₄×C₁₉	#356	1	1	yes	characteristic
82	C₂×#107	Dic₁₉	1	1	yes	characteristic
83	#423	—	38	1	no	—
84	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
85	#389	—	2	1	no	—
86	C₂²⋊C₁₆	—	38	1	no	—
87	#340	—	2	1	no	—
88	#340	Dic₁₉	1	1	yes	characteristic
89	C₂×C₁₉	—	2	1	no	—
90	C₂×C₁₉	—	2	1	no	—
91	C₂×C₁₉	#1342	1	1	yes	characteristic
92	C₂×C₁₉	#1346	1	1	yes	characteristic
93	C₂×C₁₉	#1347	1	1	yes	characteristic
94	C₂×M₁₆	—	76	1	no	—
95	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
96	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	2	1	no	—
99	C₂×C₄⋊C₄	—	4	1	no	—
100	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
101	C₂×C₁₆	—	76	1	no	—
102	#107	—	4	1	no	—
103	C₄⋊C₈	—	4	1	no	—
104	C₄⋊C₈	—	38	2	no	—
105	C₄⋊C₈	—	76	1	no	—
106	#95	—	4	1	no	—
107	C₁₉	#7091	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₄	—	4	1	no	—
109	C₂²×C₄	—	4	1	no	—
110	C₂²×C₄	#64525	1	1	yes	characteristic
111	C₂²×C₄	#64526	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
113	M₁₆	—	152	1	no	—
114	M₁₆	—	152	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	76	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	4	1	no	—
124	C₄⋊C₄	—	8	1	no	—
125	C₄⋊C₄	—	8	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₁₆	—	152	1	no	—
129	C₂³	#10597289	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	8	1	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	—	8	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	#10597283	1	1	yes	characteristic
143	C₂×C₄	#10597284	1	1	yes	characteristic
144	C₂×C₄	#10597291	1	1	yes	characteristic
145	C₂×C₄	#10597294	1	1	yes	characteristic
146	C₈	—	4	1	no	—
147	C₈	—	4	1	no	—
148	C₈	—	152	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#11926278	1	1	yes	characteristic
152	C₂²	#11926345	1	1	yes	characteristic
153	C₂²	#11926354	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	4	1	no	—
158	C₄	—	8	2	no	—
159	C₄	—	16	1	no	—
160	C₄	#11926334	1	1	yes	characteristic
161	C₄	#11926343	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#14561629	1	1	yes	characteristic
165	C₂	#14561641	1	1	yes	characteristic
166	C₂	#14561644	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561575

C₂

yes

maximal, characteristic

#14561510

C₂

yes

maximal, characteristic

C₁₉×#2086

C₂

yes

maximal, characteristic

#11926568

—

C₂²⋊(C₁₉⋊C₁₆)

—

C₂×C₁₉×#107

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₁₉×#389

—

C₁₉×#340

—

C₁₉×#340

C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×#64520

—

C₁₉×#107

—

C₁₉×C₄⋊C₈

—

C₁₉×#95

—

C₂×C₁₉⋊C₁₆

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₁₉⋊C₁₆

—

—

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

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

D₁₆

yes

characteristic

C₂×C₄×C₁₉

SD₃₂

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#2086

D₁₉

yes

characteristic

#1322

—

#1292

—

C₂²×C₁₉

—

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

yes

C₄×C₁₉

yes

Dic₁₉

yes

characteristic

#423

—

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

#389

—

C₂²⋊C₁₆

—

#340

—

#340

Dic₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×M₁₆

—

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

101

C₂×C₁₆

—

102

#107

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#95

—

107

C₁₉

#7091

yes

characteristic, complemented, Sylow

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

#64525

yes

characteristic

111

C₂²×C₄

#64526

yes

characteristic

112

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

113

M₁₆

—

152

—

114

M₁₆

—

152

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄²

—

127

C₄²

—

128

C₁₆

—

152

—

129

C₂³

#10597289

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#10597283

yes

characteristic

143

C₂×C₄

#10597284

yes

characteristic

144

C₂×C₄

#10597291

yes

characteristic

145

C₂×C₄

#10597294

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

152

—

149

C₂²

—

150

C₂²

—

151

C₂²

#11926278

yes

characteristic

152

C₂²

#11926345

yes

characteristic

153

C₂²

#11926354

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#11926334

yes

characteristic

161

C₄

#11926343

yes

characteristic

162

C₂

—

163

C₂

—

164

C₂

#14561629

yes

characteristic

165

C₂

#14561641

yes

characteristic

166

C₂

#14561644

yes

characteristic

	subgroup	G/H
center	C₂²	#11926278
commutator	C₂²×C₈×C₁₉	C₂×C₄
Frattini	C₂×C₄⋊C₈	C₂×D₁₉
Fitting	C₁₉×#2086	C₂
socle	C₂²×C₁₉	#327
prime radical	C₂³×C₁₉	#102
cosocle	C₂×C₁₉×C₄⋊C₈	C₂²
hypercenter	#2086	D₁₉
hypocenter	C₁₉	#7091
LCS	G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #2086

subgroup

G/H

center

C₂²

#11926278

commutator

C₂²×C₈×C₁₉

C₂×C₄

Frattini

C₂×C₄⋊C₈

C₂×D₁₉

Fitting

C₁₉×#2086

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₁₉×C₄⋊C₈

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #2086

N	H ≅ G/N	type
107. C₁₉	#7091	semidirect

H ≅ G/N

type

107. C₁₉

#7091

semidirect

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

subgroup	index
#14561575	2
#14561510	2
C₁₉×#2086	2
#7091	19

subgroup

index

#14561575

#14561510

C₁₉×#2086

#7091

Subgroups — #18069257

Conjugacy classes of subgroups

Complements

Maximal subgroups