Group #18069131 — subgroups

Subgroups — #18069131

Order 4864.

292 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 131 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	#14563752	C₂	1	2	yes	maximal
2	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
4	#11926299	—	2	2	no	—
5	#11926299	C₄	1	4	yes	—
6	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
7	C₂×C₁₉×C₄⋊C₈	C₄	1	2	yes	—
8	C₄³×C₁₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₉	—	2	2	no	—
10	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
12	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₉	D₄	1	2	yes	—
14	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
15	C₂²×C₁₉⋊C₈	—	2	4	no	—
16	C₁₉×C₄⋊C₈	—	4	2	no	—
17	C₁₉×C₄⋊C₈	—	4	2	no	—
18	C₁₉×C₄⋊C₈	—	4	2	no	—
19	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
20	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
21	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
22	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
23	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
25	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₉	—	4	2	no	—
27	C₂×C₈×C₁₉	—	4	4	no	—
28	C₄²×C₁₉	—	2	1	no	—
29	C₄²×C₁₉	—	2	1	no	—
30	C₄²×C₁₉	—	2	2	no	—
31	C₄²×C₁₉	—	4	2	no	—
32	C₄²×C₁₉	—	4	1	no	—
33	C₄²×C₁₉	D₈	1	2	yes	—
34	C₄²×C₁₉	SD₁₆	1	2	yes	—
35	C₄²×C₁₉	SD₁₆	1	2	yes	—
36	C₄²×C₁₉	Q₁₆	1	2	yes	—
37	C₂×C₁₉⋊C₈	—	4	4	no	—
38	C₂×C₁₉⋊C₈	—	4	4	no	—
39	C₂×C₁₉⋊C₈	—	4	4	no	—
40	#7040	—	19	1	no	maximal, Sylow
41	C₂³×C₁₉	#95	1	1	yes	characteristic
42	C₂×C₄×C₁₉	—	2	2	no	—
43	C₂×C₄×C₁₉	—	2	1	no	—
44	C₂×C₄×C₁₉	—	2	2	no	—
45	C₂×C₄×C₁₉	—	2	1	no	—
46	C₂×C₄×C₁₉	—	2	2	no	—
47	C₂×C₄×C₁₉	—	2	2	no	—
48	C₂×C₄×C₁₉	—	2	1	no	—
49	C₂×C₄×C₁₉	—	2	2	no	—
50	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
51	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
52	C₂×C₄×C₁₉	#102	1	2	yes	—
53	C₂×C₄×C₁₉	#102	1	2	yes	—
54	C₂×C₄×C₁₉	#103	1	2	yes	—
55	C₂×C₄×C₁₉	#103	1	2	yes	—
56	C₂×C₄×C₁₉	#106	1	2	yes	—
57	C₂×C₄×C₁₉	#107	1	2	yes	—
58	C₂×C₄×C₁₉	#108	1	2	yes	—
59	C₈×C₁₉	—	8	2	no	—
60	C₁₉⋊C₈	—	8	4	no	—
61	#1842	D₁₉	1	1	yes	characteristic
62	#1831	—	19	2	no	—
63	C₂²×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
64	C₂²×C₁₉	#340	1	2	yes	—
65	C₂²×C₁₉	#341	1	1	yes	characteristic
66	C₂²×C₁₉	#343	1	1	yes	characteristic
67	C₂²×C₁₉	#344	1	2	yes	—
68	C₄×C₁₉	—	2	2	no	—
69	C₄×C₁₉	—	2	2	no	—
70	C₄×C₁₉	—	2	2	no	—
71	C₄×C₁₉	—	2	2	no	—
72	C₄×C₁₉	—	2	2	no	—
73	C₄×C₁₉	—	4	1	no	—
74	C₄×C₁₉	#331	1	1	yes	characteristic
75	C₄×C₁₉	#332	1	2	yes	—
76	C₄×C₁₉	#333	1	1	yes	characteristic
77	C₂×C₄⋊C₈	—	2	2	no	—
78	C₂×C₄⋊C₈	—	38	2	no	—
79	C₂×C₄⋊C₈	Dic₁₉	1	2	yes	—
80	C₄³	C₂×D₁₉	1	1	yes	characteristic
81	#336	—	19	4	no	—
82	#336	—	38	2	no	—
83	C₂×C₁₉	#1286	1	2	yes	—
84	C₂×C₁₉	#1290	1	1	yes	characteristic
85	C₂×C₁₉	#1296	1	1	yes	characteristic
86	C₂×C₁₉	#1301	1	2	yes	—
87	C₂×C₁₉	#1307	1	1	yes	characteristic
88	C₂²×C₈	—	2	2	no	—
89	C₂²×C₈	—	38	4	no	—
90	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
91	C₂×C₄²	C₄×D₁₉	1	2	yes	—
92	C₂×C₄²	D₇₆	1	1	yes	characteristic
93	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
95	C₄⋊C₈	—	4	2	no	—
96	C₄⋊C₈	—	4	2	no	—
97	C₄⋊C₈	—	4	2	no	—
98	C₄⋊C₈	—	76	2	no	—
99	C₄⋊C₈	—	76	2	no	—
100	C₁₉	#7040	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
102	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
103	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
104	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
105	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
106	C₂×C₈	—	4	4	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	76	4	no	—
109	C₂×C₈	—	76	4	no	—
110	C₂×C₈	—	76	4	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	4	2	no	—
115	C₄²	—	4	1	no	—
116	C₄²	#64525	1	2	yes	—
117	C₄²	#64526	1	2	yes	—
118	C₄²	#64527	1	2	yes	—
119	C₄²	#64528	1	2	yes	—
120	C₂³	C₁₉⋊#95	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#10597264	1	2	yes	—
130	C₂×C₄	#10597265	1	2	yes	—
131	C₂×C₄	#10597266	1	2	yes	—
132	C₂×C₄	#10597267	1	2	yes	—
133	C₂×C₄	#10597276	1	2	yes	—
134	C₂×C₄	#10597289	1	2	yes	—
135	C₂×C₄	#10597290	1	1	yes	characteristic
136	C₂×C₄	#10597292	1	2	yes	—
137	C₂×C₄	#10597293	1	1	yes	characteristic
138	C₈	—	8	2	no	—
139	C₈	—	152	4	no	—
140	C₂²	#11926301	1	1	yes	characteristic
141	C₂²	#11926306	1	2	yes	—
142	C₂²	#11926309	1	1	yes	characteristic
143	C₂²	#11926313	1	1	yes	characteristic
144	C₂²	#11926316	1	2	yes	—
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	1	no	—
151	C₄	#11926288	1	1	yes	characteristic
152	C₄	#11926290	1	2	yes	—
153	C₄	#11926293	1	1	yes	characteristic
154	C₂	#14561495	1	2	yes	—
155	C₂	#14561505	1	1	yes	characteristic
156	C₂	#14561518	1	1	yes	characteristic
157	C₂	#14561532	1	2	yes	—
158	C₂	#14561546	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563752

C₂

yes

maximal

C₁₉×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₉⋊C₈)

—

#11926299

—

#11926299

C₄

yes

—

C₂×C₁₉×C₄⋊C₈

—

C₂×C₁₉×C₄⋊C₈

C₄

yes

—

C₄³×C₁₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

Q₈

yes

characteristic

C₂²×C₁₉⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂²×C₄×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

D₈

yes

—

C₄²×C₁₉

SD₁₆

yes

—

C₄²×C₁₉

SD₁₆

yes

—

C₄²×C₁₉

Q₁₆

yes

—

C₂×C₁₉⋊C₈

—

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

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#102

yes

—

C₂×C₄×C₁₉

#103

yes

—

C₂×C₄×C₁₉

#103

yes

—

C₂×C₄×C₁₉

#106

yes

—

C₂×C₄×C₁₉

#107

yes

—

C₂×C₄×C₁₉

#108

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#1842

D₁₉

yes

characteristic

#1831

—

C₂²×C₁₉

H(3,ℤ/4)

yes

characteristic

C₂²×C₁₉

#340

yes

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#331

yes

characteristic

C₄×C₁₉

#332

yes

—

C₄×C₁₉

#333

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

Dic₁₉

yes

—

C₄³

C₂×D₁₉

yes

characteristic

#336

—

#336

—

C₂×C₁₉

#1286

yes

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

—

C₂×C₁₉

#1307

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

Dic₃₈

yes

characteristic

C₂×C₄²

C₄×D₁₉

yes

—

C₂×C₄²

D₇₆

yes

characteristic

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

C₂×C₄²

C₂²⋊D₁₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₁₉

#7040

yes

characteristic, complemented, Sylow

101

C₂²×C₄

C₄×Dic₁₉

yes

characteristic

102

C₂²×C₄

Dic₁₉⋊C₄

yes

—

103

C₂²×C₄

C₄⋊Dic₁₉

yes

characteristic

104

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

—

105

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

#64525

yes

—

117

C₄²

#64526

yes

—

118

C₄²

#64527

yes

—

119

C₄²

#64528

yes

—

120

C₂³

C₁₉⋊#95

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#10597264

yes

—

130

C₂×C₄

#10597265

yes

—

131

C₂×C₄

#10597266

yes

—

132

C₂×C₄

#10597267

yes

—

133

C₂×C₄

#10597276

yes

—

134

C₂×C₄

#10597289

yes

—

135

C₂×C₄

#10597290

yes

characteristic

136

C₂×C₄

#10597292

yes

—

137

C₂×C₄

#10597293

yes

characteristic

138

C₈

—

139

C₈

—

152

—

140

C₂²

#11926301

yes

characteristic

141

C₂²

#11926306

yes

—

142

C₂²

#11926309

yes

characteristic

143

C₂²

#11926313

yes

characteristic

144

C₂²

#11926316

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#11926288

yes

characteristic

152

C₄

#11926290

yes

—

153

C₄

#11926293

yes

characteristic

154

C₂

#14561495

yes

—

155

C₂

#14561505

yes

characteristic

156

C₂

#14561518

yes

characteristic

157

C₂

#14561532

yes

—

158

C₂

#14561546

yes

characteristic

	subgroup	G/H
center	C₂³	C₁₉⋊#95
commutator	C₂²×C₄×C₁₉	C₄²
Frattini	C₄³	C₂×D₁₉
Fitting	C₁₉×#1842	C₂
socle, prime radical	C₂³×C₁₉	#95
cosocle	C₄³×C₁₉	C₂²
hypercenter	#1842	D₁₉
hypocenter	C₁₉	#7040
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₁₉⋊#95

commutator

C₂²×C₄×C₁₉

C₄²

Frattini

C₄³

C₂×D₁₉

Fitting

C₁₉×#1842

C₂

socle, prime radical

C₂³×C₁₉

#95

cosocle

C₄³×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
100. C₁₉	#7040	semidirect

H ≅ G/N

type

100. C₁₉

#7040

semidirect

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

subgroup	index
#14563752	2
C₁₉×#1842	2
#7040	19

subgroup

index

#14563752

C₁₉×#1842

#7040

Subgroups — #18069131

Conjugacy classes of subgroups

Complements

Maximal subgroups