Group #14564856 — subgroups

Subgroups — #14564856

Order 2432.

306 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 71 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×#10597294	C₂	1	2	yes	maximal, complemented
2	#11926545	C₂	1	2	yes	maximal, complemented
3	#11926413	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926313	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#537	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₁₉	C₂²	1	2	yes	—
7	C₁₉×C₂²⋊Q₈	—	2	2	no	—
8	C₁₉×#121	—	2	2	no	—
9	C₁₉×#121	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₉⋊C₄	—	2	2	no	—
13	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
14	C₂×#64520	C₂²	1	2	yes	—
15	#10597294	—	2	4	no	—
16	#10597292	—	2	2	no	—
17	#10597289	—	2	2	no	—
18	#41×C₁₉	—	2	2	no	—
19	#41×C₁₉	—	2	2	no	—
20	#41×C₁₉	—	2	2	no	—
21	#41×C₁₉	—	2	2	no	—
22	C₂×Q₈×C₁₉	D₄	1	2	yes	—
23	C₂×D₄×C₁₉	—	2	2	no	—
24	C₂×D₄×C₁₉	—	2	2	no	—
25	C₂×D₄×C₁₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₉	—	2	2	no	—
27	C₂²×C₄×C₁₉	D₄	1	2	yes	—
28	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₉	—	2	1	no	—
30	C₂²×Dic₁₉	—	2	1	no	—
31	C₄⋊C₄×C₁₉	—	2	2	no	—
32	C₄⋊C₄×C₁₉	—	2	2	no	—
33	C₂²⋊C₄×C₁₉	—	2	4	no	—
34	C₄⋊Dic₁₉	—	2	2	no	—
35	Dic₁₉⋊C₄	—	4	4	no	—
36	C₄×Dic₁₉	—	2	1	no	—
37	C₄×Dic₁₉	D₄	1	2	yes	complemented
38	#64520	—	2	2	no	—
39	#64520	—	2	2	no	—
40	C₂×C₁₉⋊C₈	—	2	2	no	—
41	C₂³×C₁₉	—	2	2	no	—
42	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₉	—	2	2	no	—
44	Q₈×C₁₉	—	2	2	no	—
45	D₄×C₁₉	—	2	2	no	—
46	D₄×C₁₉	—	2	2	no	—
47	D₄×C₁₉	—	4	2	no	—
48	D₄×C₁₉	—	4	2	no	—
49	D₄×C₁₉	—	8	2	no	—
50	C₂×C₄×C₁₉	—	2	2	no	—
51	C₂×C₄×C₁₉	—	2	2	no	—
52	C₂×C₄×C₁₉	—	4	2	no	—
53	C₂×C₄×C₁₉	—	4	2	no	—
54	C₂×C₄×C₁₉	—	4	2	no	—
55	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₉	#41	1	2	yes	—
58	C₂×C₄×C₁₉	#41	1	1	yes	characteristic
59	C₂×Dic₁₉	—	2	2	no	—
60	C₂×Dic₁₉	—	2	2	no	—
61	C₂×Dic₁₉	—	2	2	no	—
62	C₂×Dic₁₉	—	4	2	no	—
63	C₂×Dic₁₉	—	4	1	no	—
64	C₁₉⋊C₈	—	4	2	no	—
65	#2039	—	19	1	no	maximal, Sylow
66	C₂²×C₁₉	—	2	1	no	—
67	C₂²×C₁₉	—	2	1	no	—
68	C₂²×C₁₉	—	4	2	no	—
69	C₂²×C₁₉	—	4	2	no	—
70	C₂²×C₁₉	—	4	2	no	—
71	C₂²×C₁₉	#120	1	1	yes	characteristic
72	C₂²×C₁₉	#121	1	1	yes	characteristic
73	C₂²×C₁₉	#124	1	1	yes	characteristic
74	C₄×C₁₉	—	4	2	no	—
75	C₄×C₁₉	—	4	2	no	—
76	C₄×C₁₉	#121	1	2	yes	—
77	C₄×C₁₉	#123	1	2	yes	—
78	Dic₁₉	—	2	4	no	—
79	Dic₁₉	—	8	1	no	—
80	#537	D₁₉	1	1	yes	characteristic
81	C₂×#104	—	19	2	no	—
82	#417	—	19	2	no	—
83	#384	—	19	1	no	—
84	#343	—	19	1	no	—
85	C₂×C₁₉	—	2	1	no	—
86	C₂×C₁₉	—	2	1	no	—
87	C₂×C₁₉	—	8	2	no	—
88	C₂×C₁₉	#394	1	1	yes	characteristic
89	C₂×C₁₉	#453	1	1	yes	characteristic
90	C₂×C₁₉	#456	1	1	yes	characteristic
91	C₂×#41	C₂×D₁₉	1	2	yes	—
92	C₂×M₁₆	—	19	2	no	—
93	C₂²⋊Q₈	—	2	2	no	—
94	#121	—	2	2	no	—
95	#121	—	2	2	no	—
96	C₂×C₄⋊C₄	—	19	1	no	—
97	C₂×C₄⋊C₄	—	38	2	no	—
98	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
99	C₂×C₄²	—	19	1	no	—
100	#104	—	38	4	no	—
101	#103	—	38	2	no	—
102	#102	—	38	2	no	—
103	C₁₉	#2039	1	1	yes	characteristic, complemented, Sylow
104	#41	—	2	2	no	—
105	#41	—	2	2	no	—
106	#41	—	2	2	no	—
107	#41	—	2	2	no	—
108	C₂×Q₈	C₂²⋊D₁₉	1	2	yes	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	C₂²⋊D₁₉	1	2	yes	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	38	1	no	—
114	C₂²×C₄	—	38	1	no	—
115	C₂²×C₄	C₂²⋊D₁₉	1	2	yes	—
116	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
117	M₁₆	—	38	2	no	—
118	M₁₆	—	38	2	no	—
119	C₂×C₈	—	38	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	38	2	no	—
123	C₄⋊C₄	—	76	4	no	—
124	C₂²⋊C₄	—	2	4	no	—
125	C₄²	—	19	2	no	—
126	C₄²	—	38	1	no	—
127	C₂³	—	2	2	no	—
128	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
129	Q₈	—	2	2	no	—
130	Q₈	—	2	2	no	—
131	D₄	—	2	2	no	—
132	D₄	—	2	2	no	—
133	D₄	—	4	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	8	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	38	2	no	—
142	C₂×C₄	—	38	2	no	—
143	C₂×C₄	—	38	2	no	—
144	C₂×C₄	—	76	2	no	—
145	C₂×C₄	—	76	1	no	—
146	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
147	C₂×C₄	#64543	1	1	yes	characteristic
148	C₂×C₄	#64543	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
150	C₈	—	76	2	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	2	no	—
156	C₂²	#10597382	1	1	yes	characteristic
157	C₂²	#10597384	1	1	yes	characteristic
158	C₂²	#10597399	1	1	yes	characteristic
159	C₄	—	4	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	38	4	no	—
162	C₄	—	152	1	no	—
163	C₄	#10597381	1	2	yes	—
164	C₄	#10597385	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	8	2	no	—
168	C₂	#11926458	1	1	yes	characteristic
169	C₂	#11926712	1	1	yes	characteristic
170	C₂	#11926727	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10597294

C₂

yes

maximal, complemented

#11926545

C₂

yes

maximal, complemented

#11926413

C₂

yes

maximal, characteristic, complemented

#11926313

C₂

yes

maximal, characteristic, complemented

C₁₉×#537

C₂

yes

maximal, characteristic

C₂×#41×C₁₉

C₂²

yes

—

C₁₉×C₂²⋊Q₈

—

C₁₉×#121

—

C₁₉×#121

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×Dic₁₉⋊C₄

—

C₂×C₄×Dic₁₉

C₂²

yes

characteristic

C₂×#64520

C₂²

yes

—

#10597294

—

#10597292

—

#10597289

—

#41×C₁₉

—

#41×C₁₉

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

D₄

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

C₄×Dic₁₉

D₄

yes

complemented

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

#41

yes

—

C₂×C₄×C₁₉

#41

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₁₉⋊C₈

—

#2039

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#121

yes

—

C₄×C₁₉

#123

yes

—

Dic₁₉

—

Dic₁₉

—

#537

D₁₉

yes

—

—

—

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×#41

C₂×D₁₉

yes

—

C₂×M₁₆

—

C₂²⋊Q₈

—

#121

—

#121

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₉

yes

characteristic

C₂×C₄²

—

100

#104

—

101

#103

—

102

#102

—

103

C₁₉

#2039

yes

characteristic, complemented, Sylow

104

#41

—

105

#41

—

106

#41

—

107

#41

—

108

C₂×Q₈

C₂²⋊D₁₉

yes

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂²⋊D₁₉

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂²⋊D₁₉

yes

—

116

C₂²×C₄

C₂²×D₁₉

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₂³

—

128

C₂³

C₂×C₂²⋊D₁₉

yes

characteristic

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

D₄×D₁₉

yes

characteristic

147

C₂×C₄

#64543

yes

characteristic

148

C₂×C₄

#64543

yes

—

149

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

150

C₈

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

#10597382

yes

characteristic

157

C₂²

#10597384

yes

characteristic

158

C₂²

#10597399

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

152

—

163

C₄

#10597381

yes

—

164

C₄

#10597385

yes

—

165

C₂

—

166

C₂

—

167

C₂

—

168

C₂

#11926458

yes

characteristic

169

C₂

#11926712

yes

characteristic

170

C₂

#11926727

yes

characteristic

	subgroup	G/H
center	C₂²	#10597399
commutator, cosocle	C₂²×C₄×C₁₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting, prime radical	C₁₉×#537	C₂
socle	C₂²×C₁₉	#120
hypercenter	#537	D₁₉
hypocenter	C₁₉	#2039
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #537

subgroup

G/H

center

C₂²

#10597399

commutator, cosocle

C₂²×C₄×C₁₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉

Fitting, prime radical

C₁₉×#537

C₂

C₂²×C₁₉

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #537

N	H ≅ G/N	type
1. C₂×#10597294	C₂	semidirect
2. #11926545	C₂	semidirect
3. #11926413	C₂	semidirect
4. #11926313	C₂	semidirect
37. C₄×Dic₁₉	D₄	semidirect
103. C₁₉	#2039	semidirect

H ≅ G/N

type

1. C₂×#10597294

C₂

semidirect

2. #11926545

C₂

semidirect

3. #11926413

C₂

semidirect

4. #11926313

C₂

semidirect

37. C₄×Dic₁₉

D₄

semidirect

103. C₁₉

#2039

semidirect

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

subgroup	index
C₂×#10597294	2
#11926545	2
#11926413	2
#11926313	2
C₁₉×#537	2
#2039	19

subgroup

index

C₁₉×#537

Subgroups — #14564856

Conjugacy classes of subgroups

Complements

Maximal subgroups