Group #14564718 — subgroups

Subgroups — #14564718

Order 2432.

406 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#64527	C₂	1	2	yes	maximal
2	C₂×#10597292	C₂	1	2	yes	maximal, complemented
3	#11926446	C₂	1	1	yes	maximal, characteristic
4	#11926299	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
7	C₂²×D₇₆	C₂²	1	1	yes	characteristic
8	C₁₉×C₄⋊Q₈	—	2	4	no	—
9	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
10	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×#64527	—	2	4	no	—
12	C₂×#64527	—	2	4	no	—
13	C₂×#64527	—	2	4	no	—
14	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
15	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	—
17	#10597292	—	2	4	no	—
18	C₂³×D₁₉	—	2	1	no	—
19	C₂×Q₈×C₁₉	—	2	2	no	—
20	C₂×Q₈×C₁₉	—	2	2	no	—
21	C₂×Q₈×C₁₉	—	2	2	no	—
22	C₂×Q₈×C₁₉	—	2	2	no	—
23	C₂×Q₈×C₁₉	—	2	2	no	—
24	C₂×Q₈×C₁₉	D₄	1	4	yes	—
25	C₂²×C₄×C₁₉	—	2	2	no	—
26	C₂²×C₄×C₁₉	D₄	1	2	yes	—
27	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₉	—	2	1	no	—
29	C₂×D₇₆	—	2	2	no	—
30	C₂×D₇₆	—	2	2	no	—
31	C₂×D₇₆	—	2	1	no	—
32	C₂×D₇₆	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₉	—	2	8	no	—
34	C₄²×C₁₉	—	2	2	no	—
35	#64527	—	2	16	no	—
36	(C₂×D₁₉)⋊C₄	—	4	4	no	—
37	C₄⋊Dic₁₉	—	2	2	no	—
38	C₂×C₁₉⋊C₈	—	2	2	no	—
39	C₂×C₁₉⋊C₈	—	2	2	no	—
40	C₂×C₁₉⋊C₈	D₄	1	4	yes	—
41	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₁₉	—	2	2	no	—
43	C₂²×D₁₉	—	4	1	no	—
44	C₂²×D₁₉	—	4	1	no	—
45	C₂²×D₁₉	—	4	2	no	—
46	C₂²×D₁₉	—	4	2	no	—
47	Q₈×C₁₉	—	2	4	no	—
48	Q₈×C₁₉	—	2	4	no	—
49	Q₈×C₁₉	—	4	2	no	—
50	Q₈×C₁₉	—	4	2	no	—
51	C₂×C₄×C₁₉	—	2	2	no	—
52	C₂×C₄×C₁₉	—	2	2	no	—
53	C₂×C₄×C₁₉	—	2	4	no	—
54	C₂×C₄×C₁₉	—	4	2	no	—
55	C₂×C₄×C₁₉	—	4	2	no	—
56	C₂×C₄×C₁₉	SD₁₆	1	4	yes	complemented
57	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₉	#41	1	1	yes	characteristic
61	C₂×Dic₁₉	—	4	1	no	—
62	C₂×Dic₁₉	—	4	2	no	—
63	D₇₆	—	2	4	no	—
64	D₇₆	—	4	2	no	—
65	C₁₉⋊C₈	—	2	4	no	—
66	#2013	—	19	1	no	maximal, Sylow
67	C₂²×C₁₉	#120	1	1	yes	characteristic
68	C₂²×C₁₉	#121	1	1	yes	characteristic
69	C₂²×C₁₉	#127	1	1	yes	characteristic
70	C₂²×C₁₉	C₂×SD₁₆	1	2	yes	—
71	C₂²×C₁₉	#137	1	2	yes	—
72	C₂×D₁₉	—	4	2	no	—
73	C₂×D₁₉	—	4	4	no	—
74	C₂×D₁₉	—	8	2	no	—
75	C₂×D₁₉	—	8	1	no	—
76	C₂×D₁₉	—	8	1	no	—
77	C₄×C₁₉	—	2	4	no	—
78	C₄×C₁₉	—	4	4	no	—
79	C₄×C₁₉	#120	1	2	yes	—
80	C₄×C₁₉	#121	1	2	yes	—
81	Dic₁₉	—	8	1	no	—
82	C₂²×SD₁₆	—	19	2	no	—
83	C₂×C₄⋊Q₈	D₁₉	1	1	yes	characteristic, complemented
84	C₂×#103	—	19	2	no	—
85	#390	—	19	1	no	—
86	#336	—	19	1	no	—
87	C₂×C₁₉	#392	1	1	yes	characteristic
88	C₂×C₁₉	#448	1	2	yes	—
89	C₂×C₁₉	#461	1	2	yes	—
90	C₂×C₁₉	#492	1	1	yes	characteristic
91	C₂×C₁₉	#497	1	1	yes	characteristic
92	D₁₉	—	8	2	no	—
93	C₂²×Q₈	C₂×D₁₉	1	2	yes	—
94	C₂²×D₄	—	19	1	no	—
95	C₂×SD₁₆	—	38	4	no	—
96	C₂×SD₁₆	—	38	4	no	—
97	C₂×SD₁₆	—	38	4	no	—
98	C₂²×C₈	—	19	2	no	—
99	C₄⋊Q₈	—	2	4	no	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	19	1	no	—
102	C₂×C₂²⋊C₄	—	38	2	no	—
103	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
104	#103	—	38	4	no	—
105	C₁₉	#2013	1	1	yes	characteristic, complemented, Sylow
106	C₂⁴	—	38	1	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	C₂²⋊D₁₉	1	4	yes	—
113	C₂×D₄	—	19	2	no	—
114	C₂×D₄	—	38	2	no	—
115	C₂×D₄	—	38	1	no	—
116	C₂×D₄	—	38	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	38	1	no	—
119	C₂²×C₄	C₂²⋊D₁₉	1	2	yes	—
120	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
121	SD₁₆	—	38	16	no	—
122	C₂×C₈	—	19	4	no	—
123	C₂×C₈	—	38	2	no	—
124	C₂×C₈	—	38	2	no	—
125	C₄⋊C₄	—	2	8	no	—
126	C₄⋊C₄	—	38	2	no	—
127	C₂²⋊C₄	—	76	4	no	—
128	C₄²	—	2	2	no	—
129	C₂³	—	38	2	no	—
130	C₂³	—	76	1	no	—
131	C₂³	—	76	2	no	—
132	C₂³	—	76	2	no	—
133	C₂³	—	76	1	no	—
134	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	2	4	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	4	2	no	—
139	D₄	—	38	4	no	—
140	D₄	—	76	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	76	1	no	—
147	C₂×C₄	—	76	2	no	—
148	C₂×C₄	#64527	1	4	yes	complemented
149	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
150	C₂×C₄	D₄×D₁₉	1	2	yes	—
151	C₂×C₄	#64543	1	1	yes	characteristic
152	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
153	C₈	—	38	4	no	—
154	C₂²	—	76	2	no	—
155	C₂²	—	76	4	no	—
156	C₂²	—	152	1	no	—
157	C₂²	—	152	1	no	—
158	C₂²	—	152	2	no	—
159	C₂²	#10597384	1	1	yes	characteristic
160	C₂²	#10597386	1	1	yes	characteristic
161	C₂²	C₂×#64527	1	2	yes	—
162	C₂²	#10597388	1	2	yes	—
163	C₂²	#10597399	1	1	yes	characteristic
164	C₄	—	2	4	no	—
165	C₄	—	4	4	no	—
166	C₄	—	152	1	no	—
167	C₄	#10597383	1	2	yes	—
168	C₄	#10597385	1	2	yes	—
169	C₂	—	152	2	no	—
170	C₂	#11926452	1	1	yes	characteristic
171	C₂	#11926683	1	2	yes	—
172	C₂	#11926746	1	2	yes	—
173	C₂	#11926918	1	1	yes	characteristic
174	C₂	#11926936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#64527

C₂

yes

maximal

C₂×#10597292

C₂

yes

maximal, complemented

#11926446

C₂

yes

maximal, characteristic

#11926299

C₂

yes

maximal, characteristic, complemented

C₂×C₁₉×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₉

C₂²

yes

—

C₂²×D₇₆

C₂²

yes

characteristic

C₁₉×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂²

yes

—

—

—

C₂×(C₂×D₁₉)⋊C₄

—

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂²×C₁₉⋊C₈

C₂²

yes

—

#10597292

—

C₂³×D₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

#64527

—

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

D₄

yes

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

SD₁₆

yes

complemented

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

#41

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

C₁₉⋊C₈

—

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

C₂×SD₁₆

yes

—

C₂²×C₁₉

#137

yes

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#120

yes

—

C₄×C₁₉

#121

yes

—

Dic₁₉

—

C₂²×SD₁₆

—

C₂×C₄⋊Q₈

D₁₉

yes

characteristic, complemented

C₂×#103

—

#390

—

#336

—

C₂×C₁₉

#392

yes

characteristic

C₂×C₁₉

#448

yes

—

C₂×C₁₉

#461

yes

—

C₂×C₁₉

yes

C₂×C₁₉

yes

D₁₉

—

C₂²×Q₈

C₂×D₁₉

yes

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₄²

C₂×D₁₉

yes

characteristic

104

#103

—

105

C₁₉

#2013

yes

characteristic, complemented, Sylow

106

C₂⁴

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

C₂²⋊D₁₉

yes

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊D₁₉

yes

—

120

C₂²×C₄

C₂²×D₁₉

yes

characteristic

121

SD₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂×C₂²⋊D₁₉

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#64527

yes

complemented

149

C₂×C₄

D₄×D₁₉

yes

characteristic

150

C₂×C₄

D₄×D₁₉

yes

—

151

C₂×C₄

#64543

yes

characteristic

152

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

153

C₈

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

152

—

157

C₂²

—

152

—

158

C₂²

—

152

—

159

C₂²

#10597384

yes

characteristic

160

C₂²

#10597386

yes

characteristic

161

C₂²

C₂×#64527

yes

—

162

C₂²

#10597388

yes

—

163

C₂²

#10597399

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

152

—

167

C₄

#10597383

yes

—

168

C₄

#10597385

yes

—

169

C₂

—

152

—

170

C₂

#11926452

yes

characteristic

171

C₂

#11926683

yes

—

172

C₂

#11926746

yes

—

173

C₂

#11926918

yes

characteristic

174

C₂

#11926936

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉
commutator, cosocle	C₂²×C₄×C₁₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×C₄⋊Q₈	C₂
socle	C₂³×C₁₉	C₂×D₄
prime radical	C₂²×D₇₆	C₂²
hypercenter	C₂×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	#2013
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉

commutator, cosocle

C₂²×C₄×C₁₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉

Fitting

C₂×C₁₉×C₄⋊Q₈

C₂

socle

C₂³×C₁₉

C₂×D₄

prime radical

C₂²×D₇₆

C₂²

hypercenter

C₂×C₄⋊Q₈

D₁₉

hypocenter

C₁₉

#2013

LCS

G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×#10597292	C₂	semidirect
4. #11926299	C₂	semidirect
5. C₂×C₁₉×C₄⋊Q₈	C₂	semidirect
56. C₂×C₄×C₁₉	SD₁₆	semidirect
83. C₂×C₄⋊Q₈	D₁₉	semidirect
105. C₁₉	#2013	semidirect
148. C₂×C₄	#64527	semidirect

H ≅ G/N

type

2. C₂×#10597292

C₂

semidirect

4. #11926299

C₂

semidirect

5. C₂×C₁₉×C₄⋊Q₈

C₂

semidirect

56. C₂×C₄×C₁₉

SD₁₆

semidirect

83. C₂×C₄⋊Q₈

D₁₉

semidirect

105. C₁₉

#2013

semidirect

148. C₂×C₄

#64527

semidirect

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

subgroup	index
C₂²×#64527	2
C₂×#10597292	2
#11926446	2
#11926299	2
C₂×C₁₉×C₄⋊Q₈	2
#2013	19

subgroup

index

C₂²×#64527

C₂×C₁₉×C₄⋊Q₈

Subgroups — #14564718

Conjugacy classes of subgroups

Complements

Maximal subgroups