Group #14561987 — subgroups

Subgroups — #14561987

Order 2432.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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	#11926601	C₂	1	1	yes	maximal, characteristic, complemented
2	#11926597	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926268	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#124	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₁₉	C₄	1	1	yes	characteristic
6	C₂²×D₄×C₁₉	C₄	1	1	yes	characteristic
7	C₁₉×#124	—	2	1	no	—
8	C₁₉×#124	—	2	1	no	—
9	C₁₉×#124	—	2	1	no	—
10	C₁₉×#124	—	2	1	no	—
11	C₂×C₂²⋊C₄×C₁₉	C₄	1	2	yes	—
12	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
15	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
16	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
17	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
18	#10597260	—	2	1	no	—
19	C₄×C₁₉⋊C₈	—	2	1	no	—
20	C₂⁴×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×Q₈×C₁₉	—	2	1	no	—
22	C₂×Q₈×C₁₉	—	2	1	no	—
23	C₂×Q₈×C₁₉	—	4	2	no	—
24	C₂×D₄×C₁₉	—	2	1	no	—
25	C₂×D₄×C₁₉	—	2	1	no	—
26	C₂×D₄×C₁₉	—	4	2	no	—
27	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²⋊C₄×C₁₉	—	2	2	no	—
32	C₂²⋊C₄×C₁₉	—	2	2	no	—
33	C₂²⋊C₄×C₁₉	—	2	2	no	—
34	C₂²⋊C₄×C₁₉	—	2	2	no	—
35	C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₁₉⋊C₈	—	2	2	no	—
40	C₂×C₁₉⋊C₈	—	2	2	no	—
41	C₂³×C₁₉	—	2	2	no	—
42	C₂³×C₁₉	—	2	4	no	—
43	C₂³×C₁₉	—	4	2	no	—
44	C₂³×C₁₉	—	4	2	no	—
45	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
46	Q₈×C₁₉	—	4	2	no	—
47	Q₈×C₁₉	—	4	2	no	—
48	D₄×C₁₉	—	4	2	no	—
49	D₄×C₁₉	—	4	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	—	4	1	no	—
54	C₂×C₄×C₁₉	—	4	2	no	—
55	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₁₉⋊C₈	—	4	2	no	—
62	C₁₉⋊C₈	—	4	2	no	—
63	#1469	—	19	1	no	maximal, Sylow
64	C₂²×C₁₉	—	2	1	no	—
65	C₂²×C₁₉	—	2	1	no	—
66	C₂²×C₁₉	—	4	2	no	—
67	C₂²×C₁₉	—	4	2	no	—
68	C₂²×C₁₉	—	4	2	no	—
69	C₂²×C₁₉	—	8	4	no	—
70	C₂²×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₁₉	#131	1	2	yes	—
72	C₄×C₁₉	—	2	1	no	—
73	C₄×C₁₉	—	2	1	no	—
74	C₄×C₁₉	—	2	1	no	—
75	C₄×C₁₉	—	2	1	no	—
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	—	4	2	no	—
78	C₄×C₁₉	#100	1	2	yes	—
79	C₂×#124	D₁₉	1	1	yes	characteristic
80	#432	—	19	1	no	—
81	#431	—	19	1	no	—
82	#323	—	19	4	no	—
83	C₂×C₁₉	—	2	2	no	—
84	C₂×C₁₉	—	8	2	no	—
85	C₂×C₁₉	#408	1	1	yes	characteristic
86	C₂×C₁₉	#410	1	1	yes	characteristic
87	C₂×C₁₉	C₂×#100	1	1	yes	characteristic
88	C₂²×Q₈	Dic₁₉	1	1	yes	characteristic
89	C₂²×D₄	Dic₁₉	1	1	yes	characteristic
90	#124	—	2	1	no	—
91	#124	—	2	1	no	—
92	#124	—	2	1	no	—
93	#124	—	2	1	no	—
94	C₂×C₂²⋊C₄	Dic₁₉	1	2	yes	—
95	C₂×C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
96	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
97	C₄⋊C₈	—	38	1	no	—
98	C₄⋊C₈	—	38	1	no	—
99	C₂²⋊C₈	—	19	2	no	—
100	C₂²⋊C₈	—	19	2	no	—
101	#97	—	38	1	no	—
102	C₄×C₈	—	38	1	no	—
103	C₁₉	#1469	1	1	yes	characteristic, complemented, Sylow
104	C₂⁴	C₂×Dic₁₉	1	2	yes	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	4	2	no	—
111	C₂²×C₄	C₂×Dic₁₉	1	1	yes	characteristic
112	C₂²×C₄	C₂×Dic₁₉	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₉	1	2	yes	—
114	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
115	C₂×C₈	—	38	2	no	—
116	C₂×C₈	—	38	2	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
122	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
123	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
124	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	—	2	4	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	4	2	no	—
129	C₂³	C₂²×Dic₁₉	1	1	yes	characteristic
130	Q₈	—	4	2	no	—
131	Q₈	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
145	C₈	—	76	2	no	—
146	C₈	—	76	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	8	4	no	—
153	C₂²	#10597394	1	2	yes	—
154	C₂²	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	4	2	no	—
161	C₄	#10597290	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	—	8	2	no	—
164	C₂	#11926511	1	1	yes	characteristic
165	C₂	#11926518	1	1	yes	characteristic
166	C₂	C₂×#10597290	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11926601

C₂

yes

maximal, characteristic, complemented

#11926597

C₂

yes

maximal, characteristic, complemented

#11926268

C₂

yes

maximal, complemented

C₂×C₁₉×#124

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₉

C₄

yes

characteristic

C₂²×D₄×C₁₉

C₄

yes

characteristic

C₁₉×#124

—

C₁₉×#124

—

C₁₉×#124

—

C₁₉×#124

—

C₂×C₂²⋊C₄×C₁₉

C₄

yes

—

C₂×C₂²⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂²⋊(C₁₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₉⋊C₈)

C₂²

yes

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

#10597260

—

C₄×C₁₉⋊C₈

—

C₂⁴×C₁₉

C₂×C₄

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂²×C₄

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1469

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉

#131

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#100

yes

—

C₂×#124

D₁₉

yes

—

—

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×Q₈

Dic₁₉

yes

characteristic

C₂²×D₄

Dic₁₉

yes

—

—

—

—

C₂×C₂²⋊C₄

Dic₁₉

yes

—

C₂×C₂²⋊C₄

C₂×D₁₉

yes

—

C₂×C₄²

C₂×D₁₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

101

#97

—

102

C₄×C₈

—

103

C₁₉

#1469

yes

characteristic, complemented, Sylow

104

C₂⁴

C₂×Dic₁₉

yes

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂²×C₄

C₂×Dic₁₉

yes

characteristic

112

C₂²×C₄

C₂×Dic₁₉

yes

characteristic

113

C₂²×C₄

C₂×Dic₁₉

yes

—

114

C₂²×C₄

C₂²×D₁₉

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₂²⋊D₁₉

yes

characteristic

122

C₄²

C₂²⋊D₁₉

yes

characteristic

123

C₄²

C₂²⋊D₁₉

yes

characteristic

124

C₄²

C₂²⋊D₁₉

yes

characteristic

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂²×Dic₁₉

yes

characteristic

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊Dic₁₉

yes

characteristic

140

C₂×C₄

C₂²⋊Dic₁₉

yes

characteristic

141

C₂×C₄

C₂²⋊Dic₁₉

yes

characteristic

142

C₂×C₄

C₂²⋊Dic₁₉

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#10597394

yes

—

154

C₂²

C₂×C₂²⋊Dic₁₉

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#10597290

yes

—

162

C₂

—

163

C₂

—

164

C₂

#11926511

yes

characteristic

165

C₂

#11926518

yes

characteristic

166

C₂

C₂×#10597290

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₉
commutator	C₂³×C₁₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#124	C₂
socle	C₂²×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₉	C₄
cosocle	C₂²×C₄×C₁₉	C₂³
hypercenter	C₂×#124	D₁₉
hypocenter	C₁₉	#1469
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₉

commutator

C₂³×C₁₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉

Fitting

C₂×C₁₉×#124

C₂

socle

C₂²×C₁₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₁₉

C₄

cosocle

C₂²×C₄×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #11926601	C₂	semidirect
2. #11926597	C₂	semidirect
3. #11926268	C₂	semidirect
103. C₁₉	#1469	semidirect

H ≅ G/N

type

1. #11926601

C₂

semidirect

2. #11926597

C₂

semidirect

3. #11926268

C₂

semidirect

103. C₁₉

#1469

semidirect

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

subgroup	index
#11926601	2
#11926597	2
#11926268	2
C₂×C₁₉×#124	2
#1469	19

subgroup

index

C₂×C₁₉×#124

Subgroups — #14561987

Conjugacy classes of subgroups

Complements

Maximal subgroups