Group #18076216 — subgroups

Subgroups — #18076216

Order 4864.

342 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 155 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	#14563787	C₂	1	1	yes	maximal, characteristic
2	#14563752	C₂	1	1	yes	maximal, characteristic
3	#14561470	C₂	1	4	yes	maximal
4	C₁₉×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	#11926299	—	2	2	no	—
9	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
10	C₄³×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#336	C₄	1	2	yes	—
12	C₁₉×#336	C₄	1	2	yes	—
13	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₉	D₄	1	2	yes	—
17	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
18	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
19	C₂²×C₁₉⋊C₈	—	2	2	no	—
20	C₂²×C₁₉⋊C₈	—	2	2	no	—
21	C₄×C₈×C₁₉	—	2	2	no	—
22	C₄×C₈×C₁₉	—	2	2	no	—
23	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
24	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
25	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
26	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₁₉	—	2	4	no	—
32	C₂×C₈×C₁₉	—	2	4	no	—
33	C₂×C₈×C₁₉	—	2	4	no	—
34	C₄²×C₁₉	—	2	4	no	—
35	C₄²×C₁₉	—	2	2	no	—
36	C₄²×C₁₉	—	2	2	no	—
37	C₄²×C₁₉	—	2	2	no	—
38	C₄²×C₁₉	—	2	2	no	—
39	C₄²×C₁₉	#41	1	2	yes	—
40	C₄²×C₁₉	#41	1	2	yes	—
41	C₂×C₁₉⋊C₈	—	4	2	no	—
42	C₂×C₁₉⋊C₈	—	4	2	no	—
43	C₂×C₁₉⋊C₈	—	4	4	no	—
44	C₂×C₁₉⋊C₈	—	4	2	no	—
45	C₂×C₁₉⋊C₈	—	4	2	no	—
46	#8789	—	19	1	no	maximal, Sylow
47	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉	—	2	2	no	—
49	C₂×C₄×C₁₉	—	2	2	no	—
50	C₂×C₄×C₁₉	—	2	2	no	—
51	C₂×C₄×C₁₉	—	2	2	no	—
52	C₂×C₄×C₁₉	—	2	2	no	—
53	C₂×C₄×C₁₉	—	2	2	no	—
54	C₂×C₄×C₁₉	#108	1	4	yes	—
55	C₂×C₄×C₁₉	#117	1	2	yes	—
56	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
58	C₂×C₄×C₁₉	#125	1	2	yes	—
59	C₂×C₄×C₁₉	#131	1	2	yes	—
60	C₂×C₄×C₁₉	#131	1	2	yes	—
61	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₂×C₄×C₁₉	#137	1	2	yes	—
63	C₈×C₁₉	—	4	4	no	—
64	C₁₉⋊C₈	—	8	2	no	—
65	C₁₉⋊C₈	—	8	2	no	—
66	#1842	—	19	1	no	—
67	#1831	—	19	1	no	—
68	#1744	D₁₉	1	1	yes	characteristic
69	#1277	—	19	4	no	—
70	C₂²×C₁₉	#382	1	1	yes	characteristic
71	C₂²×C₁₉	#424	1	1	yes	characteristic
72	C₂²×C₁₉	#424	1	1	yes	characteristic
73	C₂²×C₁₉	#428	1	2	yes	—
74	C₂²×C₁₉	C₂×#108	1	2	yes	—
75	C₄×C₁₉	—	2	4	no	—
76	C₄×C₁₉	—	2	2	no	—
77	C₄×C₁₉	—	4	2	no	—
78	C₄×C₁₉	#433	1	4	yes	—
79	C₄×C₁₉	#489	1	2	yes	—
80	C₄×C₁₉	#490	1	1	yes	characteristic
81	C₄×C₁₉	#491	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	19	2	no	—
83	C₂×C₄⋊C₈	—	19	2	no	—
84	C₂×C₄⋊C₈	—	38	2	no	—
85	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
86	C₄³	C₂×D₁₉	1	1	yes	characteristic
87	#336	—	38	2	no	—
88	#336	Dic₁₉	1	2	yes	—
89	#336	Dic₁₉	1	2	yes	—
90	C₂×C₁₉	#1573	1	2	yes	—
91	C₂×C₁₉	#1574	1	2	yes	—
92	C₂×C₁₉	#1832	1	1	yes	characteristic
93	C₂×C₁₉	#1836	1	1	yes	characteristic
94	C₂×C₁₉	#1837	1	1	yes	characteristic
95	C₂²×C₈	—	38	2	no	—
96	C₂²×C₈	—	38	2	no	—
97	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
98	C₂×C₄²	Dic₃₈	1	2	yes	—
99	C₂×C₄²	D₇₆	1	2	yes	—
100	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
101	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
102	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
103	C₄⋊C₈	—	38	4	no	—
104	C₄⋊C₈	—	38	4	no	—
105	C₄⋊C₈	—	76	4	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	2	2	no	—
108	C₁₉	#8789	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
110	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
111	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	76	2	no	—
118	C₂×C₈	—	76	2	no	—
119	C₂×C₈	—	76	4	no	—
120	C₂×C₈	—	76	2	no	—
121	C₂×C₈	—	76	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	#64541	1	2	yes	—
128	C₄²	#64541	1	2	yes	—
129	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	#10597276	1	4	yes	—
137	C₂×C₄	#10597316	1	2	yes	—
138	C₂×C₄	#10597321	1	1	yes	characteristic
139	C₂×C₄	#10597321	1	1	yes	characteristic
140	C₂×C₄	#10597354	1	2	yes	—
141	C₂×C₄	#10597355	1	2	yes	—
142	C₂×C₄	#10597372	1	2	yes	—
143	C₂×C₄	#10597394	1	2	yes	—
144	C₂×C₄	#10597394	1	2	yes	—
145	C₈	—	4	4	no	—
146	C₈	—	152	2	no	—
147	C₈	—	152	2	no	—
148	C₂²	#11926410	1	1	yes	characteristic
149	C₂²	#11926571	1	1	yes	characteristic
150	C₂²	#11926571	1	1	yes	characteristic
151	C₂²	#11926586	1	2	yes	—
152	C₂²	C₂×#10597276	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	#11926605	1	4	yes	—
157	C₄	#11926904	1	2	yes	—
158	C₄	#11926909	1	1	yes	characteristic
159	C₄	#11926913	1	1	yes	characteristic
160	C₂	#14562420	1	2	yes	—
161	C₂	#14562424	1	2	yes	—
162	C₂	#14563755	1	1	yes	characteristic
163	C₂	#14563769	1	1	yes	characteristic
164	C₂	#14563773	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563787

C₂

yes

maximal, characteristic

#14563752

C₂

yes

maximal, characteristic

#14561470

C₂

yes

maximal

C₁₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₉⋊C₈)

—

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

#11926299

—

C₂×C₄×C₈×C₁₉

C₂²

yes

—

C₄³×C₁₉

C₂²

yes

characteristic

C₁₉×#336

C₄

yes

—

C₁₉×#336

C₄

yes

—

C₂²×C₈×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

Q₈

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂²×C₁₉⋊C₈

—

C₂²×C₁₉⋊C₈

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂²×C₄×C₁₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

#41

yes

—

C₄²×C₁₉

#41

yes

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#8789

—

maximal, Sylow

C₂³×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#108

yes

—

C₂×C₄×C₁₉

#117

yes

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#131

yes

—

C₂×C₄×C₁₉

#131

yes

—

C₂×C₄×C₁₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1842

—

#1831

—

#1744

D₁₉

yes

characteristic

#1277

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

—

C₂²×C₁₉

C₂×#108

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#433

yes

—

C₄×C₁₉

#489

yes

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₁₉

yes

—

C₄³

C₂×D₁₉

yes

characteristic

#336

—

#336

Dic₁₉

yes

—

#336

Dic₁₉

yes

—

C₂×C₁₉

#1573

yes

—

C₂×C₁₉

#1574

yes

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₁₉

yes

—

C₂×C₄²

Dic₃₈

yes

—

C₂×C₄²

D₇₆

yes

—

100

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

101

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

102

C₂×C₄²

C₂²×D₁₉

yes

characteristic

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₁₉

#8789

yes

characteristic, complemented, Sylow

109

C₂²×C₄

C₄⋊Dic₁₉

yes

—

110

C₂²×C₄

C₄⋊Dic₁₉

yes

—

111

C₂²×C₄

C₂×Dic₃₈

yes

characteristic

112

C₂²×C₄

C₂×D₇₆

yes

characteristic

113

C₂²×C₄

C₂²×Dic₁₉

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#64541

yes

—

128

C₄²

#64541

yes

—

129

C₂³

C₂×C₄⋊Dic₁₉

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#10597276

yes

—

137

C₂×C₄

#10597316

yes

—

138

C₂×C₄

#10597321

yes

characteristic

139

C₂×C₄

#10597321

yes

characteristic

140

C₂×C₄

#10597354

yes

—

141

C₂×C₄

#10597355

yes

—

142

C₂×C₄

#10597372

yes

—

143

C₂×C₄

#10597394

yes

—

144

C₂×C₄

#10597394

yes

—

145

C₈

—

146

C₈

—

152

—

147

C₈

—

152

—

148

C₂²

#11926410

yes

characteristic

149

C₂²

#11926571

yes

characteristic

150

C₂²

#11926571

yes

characteristic

151

C₂²

#11926586

yes

—

152

C₂²

C₂×#10597276

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#11926605

yes

—

157

C₄

#11926904

yes

—

158

C₄

#11926909

yes

characteristic

159

C₄

#11926913

yes

characteristic

160

C₂

#14562420

yes

—

161

C₂

#14562424

yes

—

162

C₂

#14563755

yes

characteristic

163

C₂

#14563769

yes

characteristic

164

C₂

#14563773

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1744	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1744	D₁₉
hypocenter	C₁₉	#8789
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₇₆

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1744

C₂

socle, prime radical

C₂³×C₁₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
108. C₁₉	#8789	semidirect

H ≅ G/N

type

108. C₁₉

#8789

semidirect

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

subgroup	index
#14563787	2
#14563752	2
#14561470	2
C₁₉×#1744	2
#8789	19

subgroup

index

C₁₉×#1744

Subgroups — #18076216

Conjugacy classes of subgroups

Complements

Maximal subgroups