Group #14564396 — subgroups

Subgroups — #14564396

Order 2432.

256 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂×#10597265	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
4	#11926306	C₂	1	2	yes	maximal
5	C₁₉×#384	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
7	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
8	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₉	—	2	1	no	—
13	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	—
14	C₁₉⋊#95	—	2	1	no	—
15	#10597265	—	2	4	no	—
16	C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	complemented
17	C₂²×C₄×C₁₉	—	2	2	no	—
18	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂²×Dic₁₉	—	2	1	no	—
22	C₂²×Dic₁₉	—	2	1	no	—
23	C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₄⋊C₄×C₁₉	—	4	2	no	—
26	C₄⋊C₄×C₁₉	—	4	2	no	—
27	C₄²×C₁₉	C₂×C₄	1	2	yes	—
28	C₄²×C₁₉	Q₈	1	2	yes	—
29	C₄⋊Dic₁₉	—	2	1	no	—
30	C₄⋊Dic₁₉	—	2	1	no	—
31	C₄⋊Dic₁₉	D₄	1	2	yes	—
32	C₄⋊Dic₁₉	Q₈	1	2	yes	—
33	C₄×Dic₁₉	—	4	2	no	—
34	C₂×C₁₉⋊C₈	—	2	2	no	—
35	C₂×C₁₉⋊C₈	—	2	2	no	—
36	C₂×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
37	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉	—	2	1	no	—
39	C₂×C₄×C₁₉	—	2	1	no	—
40	C₂×C₄×C₁₉	—	4	2	no	—
41	C₂×C₄×C₁₉	—	4	2	no	—
42	C₂×C₄×C₁₉	—	4	2	no	—
43	C₂×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₉	SD₁₆	1	2	yes	—
46	C₂×C₄×C₁₉	SD₁₆	1	2	yes	—
47	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₉	#41	1	1	yes	characteristic
53	C₂×Dic₁₉	—	2	2	no	—
54	C₂×Dic₁₉	—	2	2	no	—
55	C₂×Dic₁₉	—	4	1	no	—
56	C₂×Dic₁₉	—	4	1	no	—
57	C₂×Dic₁₉	—	4	1	no	—
58	C₂×Dic₁₉	—	4	1	no	—
59	C₁₉⋊C₈	—	2	4	no	—
60	#1948	—	19	1	no	maximal, Sylow
61	C₂²×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₉	#121	1	1	yes	characteristic
63	C₂²×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂²×C₁₉	#137	1	1	yes	characteristic
68	C₄×C₁₉	—	2	2	no	—
69	C₄×C₁₉	—	8	2	no	—
70	C₄×C₁₉	#106	1	4	yes	complemented
71	C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
72	C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
73	C₄×C₁₉	#125	1	1	yes	characteristic
74	C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
75	Dic₁₉	—	4	2	no	—
76	Dic₁₉	—	4	2	no	—
77	C₂×#106	—	19	2	no	—
78	C₂×C₄⋊C₈	—	19	1	no	—
79	#384	D₁₉	1	1	yes	characteristic
80	C₄×C₄⋊C₄	—	19	1	no	—
81	#340	—	19	2	no	—
82	C₂×C₁₉	#389	1	1	yes	characteristic
83	C₂×C₁₉	C₂×#106	1	1	yes	characteristic
84	C₂×C₁₉	#428	1	1	yes	characteristic
85	C₂×C₁₉	#460	1	1	yes	characteristic
86	C₂×C₁₉	#461	1	1	yes	characteristic
87	C₂×C₁₉	#475	1	1	yes	characteristic
88	C₂×C₁₉	#476	1	1	yes	characteristic
89	C₂²×C₈	—	19	2	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	19	1	no	—
93	C₂×C₄⋊C₄	—	19	1	no	—
94	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
95	C₂×C₄²	—	38	1	no	—
96	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
97	#106	—	38	4	no	—
98	C₄⋊C₈	—	19	4	no	—
99	#95	—	38	1	no	—
100	C₁₉	#1948	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	38	1	no	—
103	C₂²×C₄	—	38	1	no	—
104	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
107	C₂×C₈	—	19	4	no	—
108	C₂×C₈	—	38	2	no	—
109	C₂×C₈	—	38	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	19	2	no	—
115	C₄⋊C₄	—	19	2	no	—
116	C₄⋊C₄	—	38	1	no	—
117	C₄⋊C₄	—	38	1	no	—
118	C₄²	—	76	2	no	—
119	C₄²	Dic₃₈	1	2	yes	—
120	C₄²	C₄×D₁₉	1	2	yes	—
121	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	38	2	no	—
128	C₂×C₄	—	38	2	no	—
129	C₂×C₄	—	76	1	no	—
130	C₂×C₄	—	76	1	no	—
131	C₂×C₄	—	76	1	no	—
132	C₂×C₄	—	76	1	no	—
133	C₂×C₄	Dic₁₉⋊C₄	1	2	yes	—
134	C₂×C₄	Dic₁₉⋊C₄	1	2	yes	—
135	C₂×C₄	#64526	1	2	yes	—
136	C₂×C₄	#64527	1	2	yes	—
137	C₂×C₄	C₂×Dic₃₈	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
139	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
140	C₂×C₄	#64543	1	1	yes	characteristic
141	C₂×C₄	Q₈×D₁₉	1	1	yes	characteristic
142	C₂×C₄	#64545	1	1	yes	characteristic
143	C₈	—	38	4	no	—
144	C₂²	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
145	C₂²	#10597378	1	1	yes	characteristic
146	C₂²	C₂×#64526	1	1	yes	characteristic
147	C₂²	#10597384	1	1	yes	characteristic
148	C₂²	C₂×#64527	1	1	yes	characteristic
149	C₂²	#10597388	1	1	yes	characteristic
150	C₂²	#10597392	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	8	2	no	—
153	C₄	—	76	2	no	—
154	C₄	—	76	2	no	—
155	C₄	#10597265	1	4	yes	complemented
156	C₄	#10597333	1	1	yes	characteristic
157	C₄	#10597334	1	1	yes	characteristic
158	C₄	#10597336	1	1	yes	characteristic
159	C₄	#10597339	1	1	yes	characteristic
160	C₂	#11926440	1	1	yes	characteristic
161	C₂	C₂×#10597265	1	1	yes	characteristic
162	C₂	#11926584	1	1	yes	characteristic
163	C₂	#11926741	1	1	yes	characteristic
164	C₂	#11926746	1	1	yes	characteristic
165	C₂	#11926826	1	1	yes	characteristic
166	C₂	#11926831	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10597265

C₂

yes

maximal

C₂×C₄⋊(C₁₉⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₉

C₂

yes

maximal, characteristic

#11926306

C₂

yes

maximal

C₁₉×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₉

—

C₂²×C₁₉⋊C₈

C₂²

yes

—

C₁₉⋊#95

—

#10597265

—

C₄⋊(C₁₉⋊C₈)

C₄

yes

complemented

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×C₄

yes

—

C₄²×C₁₉

Q₈

yes

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

D₄

yes

—

C₄⋊Dic₁₉

Q₈

yes

—

C₄×Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

C₂×C₄

yes

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₉

SD₁₆

yes

—

C₂×C₄×C₁₉

SD₁₆

yes

—

C₂×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₁₉⋊C₈

—

#1948

—

maximal, Sylow

C₂²×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉

#121

yes

characteristic

C₂²×C₁₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉

C₂×SD₁₆

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#106

yes

complemented

C₄×C₁₉

C₄×D₄

yes

characteristic

C₄×C₁₉

C₄×Q₈

yes

characteristic

C₄×C₁₉

#125

yes

characteristic

C₄×C₁₉

C₄⋊Q₈

yes

characteristic

Dic₁₉

—

Dic₁₉

—

C₂×#106

—

C₂×C₄⋊C₈

—

#384

D₁₉

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₉

yes

characteristic

#106

—

C₄⋊C₈

—

#95

—

100

C₁₉

#1948

yes

characteristic, complemented, Sylow

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

105

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

106

C₂²×C₄

C₂²×D₁₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

Dic₃₈

yes

—

120

C₄²

C₄×D₁₉

yes

—

121

C₂³

C₂×C₂²⋊D₁₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

Dic₁₉⋊C₄

yes

—

134

C₂×C₄

Dic₁₉⋊C₄

yes

—

135

C₂×C₄

#64526

yes

—

136

C₂×C₄

#64527

yes

—

137

C₂×C₄

C₂×Dic₃₈

yes

characteristic

138

C₂×C₄

C₂×C₄×D₁₉

yes

characteristic

139

C₂×C₄

D₄×D₁₉

yes

characteristic

140

C₂×C₄

#64543

yes

characteristic

141

C₂×C₄

Q₈×D₁₉

yes

characteristic

142

C₂×C₄

#64545

yes

characteristic

143

C₈

—

144

C₂²

C₂×Dic₁₉⋊C₄

yes

characteristic

145

C₂²

#10597378

yes

characteristic

146

C₂²

C₂×#64526

yes

characteristic

147

C₂²

#10597384

yes

characteristic

148

C₂²

C₂×#64527

yes

characteristic

149

C₂²

#10597388

yes

characteristic

150

C₂²

#10597392

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#10597265

yes

complemented

156

C₄

#10597333

yes

characteristic

157

C₄

#10597334

yes

characteristic

158

C₄

#10597336

yes

characteristic

159

C₄

#10597339

yes

characteristic

160

C₂

#11926440

yes

characteristic

161

C₂

C₂×#10597265

yes

characteristic

162

C₂

#11926584

yes

characteristic

163

C₂

#11926741

yes

characteristic

164

C₂

#11926746

yes

characteristic

165

C₂

#11926826

yes

characteristic

166

C₂

#11926831

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting	C₁₉×#384	C₂
socle, prime radical	C₂³×C₁₉	C₂×D₄
cosocle	C₂²×C₄×C₁₉	C₂³
hypercenter	#384	D₁₉
hypocenter	C₁₉	#1948
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉

commutator

C₂×C₄×C₁₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉

Fitting

C₁₉×#384

C₂

socle, prime radical

C₂³×C₁₉

C₂×D₄

cosocle

C₂²×C₄×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
16. C₄⋊(C₁₉⋊C₈)	C₄	semidirect
70. C₄×C₁₉	#106	semidirect
100. C₁₉	#1948	semidirect
155. C₄	#10597265	semidirect

H ≅ G/N

type

16. C₄⋊(C₁₉⋊C₈)

C₄

semidirect

70. C₄×C₁₉

#106

semidirect

100. C₁₉

#1948

semidirect

155. C₄

#10597265

semidirect

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

subgroup	index
C₂×#10597265	2
C₂×C₄⋊(C₁₉⋊C₈)	2
C₄×C₄⋊Dic₁₉	2
#11926306	2
C₁₉×#384	2
#1948	19

subgroup

index

C₂×#10597265

C₂×C₄⋊(C₁₉⋊C₈)

C₄×C₄⋊Dic₁₉

#11926306

C₁₉×#384

#1948

Subgroups — #14564396

Conjugacy classes of subgroups

Complements

Maximal subgroups