Group #14564722 — subgroups

Subgroups — #14564722

Order 2432.

474 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 97 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, complemented
2	C₂×#10597320	C₂	1	1	yes	maximal, characteristic
3	C₂×#10597266	C₂	1	2	yes	maximal, complemented
4	#11926299	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×D₇₆	—	2	2	no	—
8	C₂²×D₇₆	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
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	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	—
15	#10597320	—	2	4	no	—
16	C₁₉×#95	—	2	2	no	—
17	#10597266	—	2	4	no	—
18	C₂³×D₁₉	—	2	2	no	—
19	C₂×Q₈×C₁₉	—	2	1	no	—
20	C₂×Q₈×C₁₉	—	2	2	no	—
21	C₂×Q₈×C₁₉	—	2	2	no	—
22	C₂×Q₈×C₁₉	D₄	1	2	yes	complemented
23	C₂²×C₄×C₁₉	—	2	1	no	—
24	C₂²×C₄×C₁₉	—	2	1	no	—
25	C₂²×C₄×C₁₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×D₇₆	—	2	2	no	—
28	C₂×D₇₆	—	2	2	no	—
29	C₂×D₇₆	—	2	2	no	—
30	C₂×D₇₆	—	2	2	no	—
31	C₂×D₇₆	—	2	8	no	—
32	C₂×D₇₆	—	2	2	no	—
33	C₂×D₇₆	—	4	4	no	—
34	C₂×D₇₆	—	4	4	no	—
35	C₂×D₇₆	D₄	1	4	yes	—
36	C₄⋊C₄×C₁₉	—	2	2	no	—
37	C₄²×C₁₉	—	2	2	no	—
38	#64527	—	2	16	no	—
39	C₂×C₁₉⋊C₈	—	2	2	no	—
40	C₂×C₁₉⋊C₈	—	2	2	no	—
41	C₂×C₁₉⋊C₈	D₄	1	4	yes	complemented
42	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₁₉	—	2	4	no	—
44	C₂²×D₁₉	—	4	2	no	—
45	C₂²×D₁₉	—	4	2	no	—
46	C₂²×D₁₉	—	4	4	no	—
47	C₂²×D₁₉	—	4	4	no	—
48	Q₈×C₁₉	—	2	4	no	—
49	Q₈×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	1	no	—
55	C₂×C₄×C₁₉	—	4	1	no	—
56	C₂×C₄×C₁₉	—	4	2	no	—
57	C₂×C₄×C₁₉	SD₁₆	1	4	yes	complemented
58	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₉	#41	1	1	yes	characteristic
62	D₇₆	—	2	4	no	—
63	D₇₆	—	2	4	no	—
64	D₇₆	—	4	16	no	—
65	D₇₆	—	4	2	no	—
66	D₇₆	—	4	2	no	—
67	C₁₉⋊C₈	—	2	4	no	—
68	#2014	—	19	1	no	maximal, Sylow
69	C₂²×C₁₉	#120	1	1	yes	characteristic
70	C₂²×C₁₉	#121	1	1	yes	characteristic
71	C₂²×C₁₉	#127	1	1	yes	characteristic
72	C₂²×C₁₉	C₂×SD₁₆	1	2	yes	—
73	C₂²×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×D₁₉	—	4	4	no	—
75	C₂×D₁₉	—	4	8	no	—
76	C₂×D₁₉	—	8	4	no	—
77	C₂×D₁₉	—	8	2	no	—
78	C₂×D₁₉	—	8	2	no	—
79	C₄×C₁₉	—	2	4	no	—
80	C₄×C₁₉	—	4	2	no	—
81	C₄×C₁₉	—	8	1	no	—
82	C₄×C₁₉	#120	1	2	yes	—
83	C₄×C₁₉	#121	1	2	yes	—
84	C₂²×SD₁₆	—	19	2	no	—
85	C₂×#127	—	19	1	no	—
86	C₂×#102	—	19	2	no	—
87	#391	D₁₉	1	1	yes	characteristic, complemented
88	#336	—	19	1	no	—
89	C₂×C₁₉	#392	1	1	yes	characteristic
90	C₂×C₁₉	#450	1	2	yes	—
91	C₂×C₁₉	#460	1	2	yes	—
92	C₂×C₁₉	#492	1	1	yes	characteristic
93	C₂×C₁₉	#496	1	1	yes	characteristic
94	D₁₉	—	8	4	no	—
95	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
96	C₂²×D₄	—	19	2	no	—
97	C₂²×D₄	—	38	2	no	—
98	C₂×SD₁₆	—	38	4	no	—
99	C₂×SD₁₆	—	38	4	no	—
100	C₂×SD₁₆	—	38	4	no	—
101	C₂²×C₈	—	19	2	no	—
102	#127	—	38	4	no	—
103	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
104	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
105	#102	—	38	4	no	—
106	#95	—	2	2	no	—
107	C₁₉	#2014	1	1	yes	characteristic, complemented, Sylow
108	C₂⁴	—	38	2	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	D₇₆	1	2	yes	complemented
113	C₂×D₄	—	19	4	no	—
114	C₂×D₄	—	38	8	no	—
115	C₂×D₄	—	38	2	no	—
116	C₂×D₄	—	38	2	no	—
117	C₂×D₄	—	38	2	no	—
118	C₂×D₄	—	38	2	no	—
119	C₂×D₄	—	38	2	no	—
120	C₂×D₄	—	76	4	no	—
121	C₂×D₄	—	76	4	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	C₂²⋊D₁₉	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
126	SD₁₆	—	38	16	no	—
127	C₂×C₈	—	19	4	no	—
128	C₂×C₈	—	38	2	no	—
129	C₂×C₈	—	38	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₂³	—	38	4	no	—
133	C₂³	—	76	2	no	—
134	C₂³	—	76	2	no	—
135	C₂³	—	76	4	no	—
136	C₂³	—	76	4	no	—
137	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
138	Q₈	—	2	4	no	—
139	Q₈	—	4	2	no	—
140	D₄	—	38	4	no	—
141	D₄	—	38	4	no	—
142	D₄	—	76	16	no	—
143	D₄	—	76	2	no	—
144	D₄	—	76	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	#64527	1	4	yes	complemented
153	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
154	C₂×C₄	#64541	1	1	yes	characteristic
155	C₂×C₄	D₄×D₁₉	1	2	yes	—
156	C₂×C₄	D₄×D₁₉	1	2	yes	—
157	C₈	—	38	4	no	—
158	C₂²	—	76	8	no	—
159	C₂²	—	76	4	no	—
160	C₂²	—	152	2	no	—
161	C₂²	—	152	2	no	—
162	C₂²	—	152	4	no	—
163	C₂²	#10597376	1	1	yes	characteristic
164	C₂²	#10597383	1	1	yes	characteristic
165	C₂²	#10597386	1	1	yes	characteristic
166	C₂²	C₂×#64527	1	2	yes	—
167	C₂²	#10597395	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	4	2	no	—
170	C₄	—	8	1	no	—
171	C₄	#10597328	1	2	yes	—
172	C₄	#10597330	1	2	yes	—
173	C₂	—	152	4	no	—
174	C₂	#11926453	1	1	yes	characteristic
175	C₂	#11926694	1	2	yes	—
176	C₂	#11926744	1	2	yes	—
177	C₂	#11926918	1	1	yes	characteristic
178	C₂	#11926934	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#64527

C₂

yes

maximal, complemented

C₂×#10597320

C₂

yes

maximal, characteristic

C₂×#10597266

C₂

yes

maximal, complemented

#11926299

C₂

yes

maximal, characteristic, complemented

C₁₉×#391

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₉

C₂²

yes

characteristic

C₂²×D₇₆

—

C₂²×D₇₆

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

characteristic

C₂×C₄²×C₁₉

C₂²

yes

—

—

—

C₂²×C₁₉⋊C₈

C₂²

yes

—

#10597320

—

C₁₉×#95

—

#10597266

—

C₂³×D₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

D₄

yes

complemented

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

#64527

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

D₄

yes

complemented

C₂³×C₁₉

C₂×D₄

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

Q₈×C₁₉

—

Q₈×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

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

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

C₁₉⋊C₈

—

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

C₂×SD₁₆

yes

—

C₂²×C₁₉

AGL(1,ℤ/8)

yes

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#120

yes

—

C₄×C₁₉

#121

yes

—

C₂²×SD₁₆

—

C₂×#127

—

C₂×#102

—

#391

D₁₉

yes

characteristic, complemented

#336

—

C₂×C₁₉

#392

yes

characteristic

C₂×C₁₉

#450

yes

—

C₂×C₁₉

#460

yes

—

C₂×C₁₉

yes

C₂×C₁₉

yes

D₁₉

—

C₂²×Q₈

C₂×D₁₉

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

100

C₂×SD₁₆

—

101

C₂²×C₈

—

102

#127

—

103

C₂×C₄⋊C₄

C₂×D₁₉

yes

characteristic

104

C₂×C₄²

C₂×D₁₉

yes

characteristic

105

#102

—

106

#95

—

107

C₁₉

#2014

yes

characteristic, complemented, Sylow

108

C₂⁴

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

D₇₆

yes

complemented

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂²⋊D₁₉

yes

—

125

C₂²×C₄

C₂²×D₁₉

yes

characteristic

126

SD₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄²

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂×C₂²⋊D₁₉

yes

characteristic

138

Q₈

—

139

Q₈

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

#64527

yes

complemented

153

C₂×C₄

C₂×D₇₆

yes

characteristic

154

C₂×C₄

#64541

yes

characteristic

155

C₂×C₄

D₄×D₁₉

yes

—

156

C₂×C₄

D₄×D₁₉

yes

—

157

C₈

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

152

—

161

C₂²

—

152

—

162

C₂²

—

152

—

163

C₂²

#10597376

yes

characteristic

164

C₂²

#10597383

yes

characteristic

165

C₂²

#10597386

yes

characteristic

166

C₂²

C₂×#64527

yes

—

167

C₂²

#10597395

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

#10597328

yes

—

172

C₄

#10597330

yes

—

173

C₂

—

152

—

174

C₂

#11926453

yes

characteristic

175

C₂

#11926694

yes

—

176

C₂

#11926744

yes

—

177

C₂

#11926918

yes

characteristic

178

C₂

#11926934

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉
commutator, cosocle	C₂²×C₄×C₁₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting	C₁₉×#391	C₂
socle	C₂³×C₁₉	C₂×D₄
prime radical	C₂×#10597320	C₂
hypercenter	#391	D₁₉
hypocenter	C₁₉	#2014
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉

commutator, cosocle

C₂²×C₄×C₁₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉

Fitting

C₁₉×#391

C₂

socle

C₂³×C₁₉

C₂×D₄

C₂

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
1. C₂²×#64527	C₂	semidirect
3. C₂×#10597266	C₂	semidirect
4. #11926299	C₂	semidirect
5. C₁₉×#391	C₂	semidirect
22. C₂×Q₈×C₁₉	D₄	semidirect
41. C₂×C₁₉⋊C₈	D₄	semidirect
57. C₂×C₄×C₁₉	SD₁₆	semidirect
87. #391	D₁₉	semidirect
107. C₁₉	#2014	semidirect
112. C₂×Q₈	D₇₆	semidirect
152. C₂×C₄	#64527	semidirect

H ≅ G/N

type

1. C₂²×#64527

C₂

semidirect

3. C₂×#10597266

C₂

semidirect

4. #11926299

C₂

semidirect

5. C₁₉×#391

C₂

semidirect

22. C₂×Q₈×C₁₉

D₄

semidirect

41. C₂×C₁₉⋊C₈

D₄

semidirect

57. C₂×C₄×C₁₉

SD₁₆

semidirect

87. #391

D₁₉

semidirect

107. C₁₉

#2014

semidirect

112. C₂×Q₈

D₇₆

semidirect

152. 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×#10597320	2
C₂×#10597266	2
#11926299	2
C₁₉×#391	2
#2014	19

subgroup

index

C₂²×#64527

C₁₉×#391

Subgroups — #14564722

Conjugacy classes of subgroups

Complements

Maximal subgroups