Group #18069342 — subgroups

Subgroups — #18069342

Order 4864.

182 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 67 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	#14561715	C₂	1	1	yes	maximal, characteristic
2	#14561480	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1946	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597269	—	2	1	no	—
5	#11926597	—	2	1	no	—
6	C₂×C₁₉×#108	C₄	1	1	yes	characteristic
7	C₁₉×#427	C₄	1	1	yes	characteristic
8	C₁₉×#424	—	2	1	no	—
9	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
10	C₁₉×#339	—	2	1	no	—
11	C₂×M₁₆×C₁₉	—	2	1	no	—
12	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
14	C₁₉×#117	—	2	1	no	—
15	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
16	C₁₉×#108	—	4	1	no	—
17	C₁₉×#107	—	2	1	no	—
18	C₁₉×#106	C₈	1	2	yes	complemented
19	C₁₉×C₄⋊C₈	—	4	1	no	—
20	C₁₉×C₄⋊C₈	—	4	1	no	—
21	C₁₉×#97	—	4	1	no	—
22	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
23	#10597269	—	4	1	no	—
24	#10597269	—	4	1	no	—
25	C₂×C₁₉⋊C₁₆	—	2	2	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	M₁₆×C₁₉	—	4	1	no	—
31	M₁₆×C₁₉	—	4	1	no	—
32	C₂×C₈×C₁₉	—	2	1	no	—
33	C₂×C₈×C₁₉	—	2	1	no	—
34	C₂×C₈×C₁₉	—	2	1	no	—
35	C₂×C₈×C₁₉	—	2	1	no	—
36	C₂×C₈×C₁₉	—	4	1	no	—
37	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
40	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
41	C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂²⋊C₄×C₁₉	—	4	1	no	—
43	C₄²×C₁₉	—	2	1	no	—
44	C₄²×C₁₉	—	4	1	no	—
45	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
46	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
47	C₂×C₁₉⋊C₈	—	4	2	no	—
48	C₁₉⋊C₁₆	—	4	2	no	—
49	#7125	—	19	1	no	maximal, Sylow
50	C₂³×C₁₉	#104	1	1	yes	characteristic
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	4	2	no	—
54	C₂×C₄×C₁₉	—	8	1	no	—
55	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
58	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
59	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
60	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
61	C₈×C₁₉	—	2	1	no	—
62	C₈×C₁₉	—	2	1	no	—
63	C₈×C₁₉	—	2	1	no	—
64	C₈×C₁₉	—	2	1	no	—
65	C₈×C₁₉	—	8	1	no	—
66	C₁₉⋊C₈	—	8	2	no	—
67	#1946	D₁₉	1	1	yes	characteristic
68	#1375	—	19	1	no	—
69	#1281	—	19	1	no	—
70	C₂²×C₁₉	—	4	1	no	—
71	C₂²×C₁₉	D₄⋊C₈	1	1	yes	characteristic
72	C₂²×C₁₉	#328	1	1	yes	characteristic
73	C₂²×C₁₉	#330	1	1	yes	characteristic
74	C₄×C₁₉	—	4	1	no	—
75	C₄×C₁₉	—	4	1	no	—
76	C₄×C₁₉	—	8	2	no	—
77	C₄×C₁₉	#324	1	1	yes	characteristic
78	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
79	C₄×C₁₉	#327	1	1	yes	characteristic
80	C₄×C₁₉	#332	1	1	yes	characteristic
81	C₂×M₃₂	—	38	1	no	—
82	#431	—	38	1	no	—
83	C₂×#108	Dic₁₉	1	1	yes	characteristic
84	#427	Dic₁₉	1	1	yes	characteristic
85	#424	—	2	1	no	—
86	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
87	#339	—	2	1	no	—
88	C₂×C₁₉	—	2	1	no	—
89	C₂×C₁₉	—	2	1	no	—
90	C₂×C₁₉	—	2	1	no	—
91	C₂×C₁₉	#1271	1	1	yes	characteristic
92	C₂×M₁₆	—	2	1	no	—
93	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
94	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
95	#117	—	2	1	no	—
96	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
97	M₃₂	—	76	1	no	—
98	M₃₂	—	76	1	no	—
99	C₂×C₁₆	—	38	2	no	—
100	#108	—	4	1	no	—
101	#107	—	2	1	no	—
102	#106	C₁₉⋊C₈	1	2	yes	complemented
103	C₄⋊C₈	—	4	1	no	—
104	C₄⋊C₈	—	4	1	no	—
105	C₄⋊C₈	—	76	1	no	—
106	C₂²⋊C₈	—	38	2	no	—
107	#97	—	4	1	no	—
108	C₄×C₈	—	76	1	no	—
109	C₁₉	#7125	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
112	M₁₆	—	4	1	no	—
113	M₁₆	—	4	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	76	2	no	—
120	C₂×C₈	C₂×C₁₉⋊C₈	1	1	yes	characteristic
121	C₂×C₈	#64520	1	1	yes	characteristic
122	C₂×C₈	#64525	1	1	yes	characteristic
123	C₂×C₈	#64526	1	1	yes	characteristic
124	C₄⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	4	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	4	1	no	—
128	C₄²	#64527	1	1	yes	characteristic
129	C₄²	#64528	1	1	yes	characteristic
130	C₁₆	—	76	2	no	—
131	C₂³	#10597294	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
137	C₂×C₄	#10597289	1	1	yes	characteristic
138	C₂×C₄	#10597291	1	1	yes	characteristic
139	C₂×C₄	#10597292	1	1	yes	characteristic
140	C₂×C₄	#10597293	1	1	yes	characteristic
141	C₂×C₄	#10597294	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	1	no	—
147	C₈	—	152	2	no	—
148	C₂²	—	4	1	no	—
149	C₂²	D₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
150	C₂²	#11926281	1	1	yes	characteristic
151	C₂²	#11926286	1	1	yes	characteristic
152	C₄	—	4	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	8	2	no	—
155	C₄	#11926270	1	1	yes	characteristic
156	C₄	Q₈⋊(C₁₉⋊C₈)	1	1	yes	characteristic
157	C₄	#11926278	1	1	yes	characteristic
158	C₄	#11926290	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	#14561458	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561715

C₂

yes

maximal, characteristic

#14561480

C₂

yes

maximal, characteristic

C₁₉×#1946

C₂

yes

maximal, characteristic

C₂×#10597269

—

#11926597

—

C₂×C₁₉×#108

C₄

yes

characteristic

C₁₉×#427

C₄

yes

characteristic

C₁₉×#424

—

C₂×C₁₉×#97

C₂²

yes

characteristic

C₁₉×#339

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₁₉×#117

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₁₉×#108

—

C₁₉×#107

—

C₁₉×#106

C₈

yes

complemented

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×#97

—

C₂²⋊(C₁₉⋊C₈)

—

#10597269

—

#10597269

—

C₂×C₁₉⋊C₁₆

—

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

D₈

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

SD₁₆

yes

characteristic

C₄²×C₁₉

Q₁₆

yes

characteristic

C₂×C₁₉⋊C₈

—

C₁₉⋊C₁₆

—

—

C₂³×C₁₉

yes

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

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#1946

D₁₉

yes

characteristic

#1375

—

#1281

—

C₂²×C₁₉

—

C₂²×C₁₉

D₄⋊C₈

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#324

yes

characteristic

C₄×C₁₉

Q₈⋊C₈

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×M₃₂

—

#431

—

C₂×#108

Dic₁₉

yes

characteristic

#427

Dic₁₉

yes

characteristic

#424

—

C₂×#97

C₂×D₁₉

yes

characteristic

#339

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1271

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

#117

—

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

100

#108

—

101

#107

—

102

#106

C₁₉⋊C₈

yes

complemented

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₂²⋊C₈

—

107

#97

—

108

C₄×C₈

—

109

C₁₉

#7125

yes

characteristic, complemented, Sylow

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₂×C₁₉⋊C₈

yes

characteristic

121

C₂×C₈

#64520

yes

characteristic

122

C₂×C₈

#64525

yes

characteristic

123

C₂×C₈

#64526

yes

characteristic

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₄²

—

128

C₄²

#64527

yes

characteristic

129

C₄²

#64528

yes

characteristic

130

C₁₆

—

131

C₂³

#10597294

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

137

C₂×C₄

#10597289

yes

characteristic

138

C₂×C₄

#10597291

yes

characteristic

139

C₂×C₄

#10597292

yes

characteristic

140

C₂×C₄

#10597293

yes

characteristic

141

C₂×C₄

#10597294

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

152

—

148

C₂²

—

149

C₂²

D₄⋊(C₁₉⋊C₈)

yes

characteristic

150

C₂²

#11926281

yes

characteristic

151

C₂²

#11926286

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#11926270

yes

characteristic

156

C₄

Q₈⋊(C₁₉⋊C₈)

yes

characteristic

157

C₄

#11926278

yes

characteristic

158

C₄

#11926290

yes

characteristic

159

C₂

—

160

C₂

—

161

C₂

—

162

C₂

#14561458

yes

characteristic

	subgroup	G/H
center	C₄	#11926278
commutator	C₂×C₈×C₁₉	C₂×C₈
Frattini	C₂×#97	C₂×D₁₉
Fitting	C₁₉×#1946	C₂
socle	C₂×C₁₉	#1271
prime radical	C₂³×C₁₉	#104
cosocle	C₂×C₁₉×#97	C₂²
hypercenter	#1946	D₁₉
hypocenter	C₁₉	#7125
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

subgroup

G/H

center

C₄

#11926278

commutator

C₂×C₈×C₁₉

C₂×C₈

Frattini

C₂×#97

C₂×D₁₉

Fitting

C₁₉×#1946

C₂

C₂×C₁₉

C₂³×C₁₉

C₂×C₁₉×#97

C₂²

D₁₉

C₁₉

G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

N	H ≅ G/N	type
18. C₁₉×#106	C₈	semidirect
102. #106	C₁₉⋊C₈	semidirect
109. C₁₉	#7125	semidirect

H ≅ G/N

type

18. C₁₉×#106

C₈

semidirect

102. #106

C₁₉⋊C₈

semidirect

109. C₁₉

#7125

semidirect

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

subgroup	index
#14561715	2
#14561480	2
C₁₉×#1946	2
#7125	19

subgroup

index

#14561715

#14561480

C₁₉×#1946

#7125

Subgroups — #18069342

Conjugacy classes of subgroups

Complements

Maximal subgroups