Group #14564725 — subgroups

Subgroups — #14564725

Order 2432.

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

normal subgroups only

This group has 474 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×#64516	C₂	1	2	yes	maximal, complemented
2	C₂×#10597320	C₂	1	1	yes	maximal, characteristic
3	C₂×#10597278	C₂	1	2	yes	maximal, complemented
4	#11926450	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₆	—	2	2	no	—
7	C₂²×D₇₆	C₂²	1	2	yes	—
8	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
12	C₂×#64516	—	2	4	no	—
13	C₂×#64516	—	2	4	no	—
14	C₂×#64516	—	2	4	no	—
15	#10597320	—	2	4	no	—
16	C₁₉⋊#95	—	2	2	no	—
17	#10597278	—	2	4	no	—
18	C₂³×D₁₉	—	2	2	no	—
19	C₂²×C₄×C₁₉	D₄	1	2	yes	—
20	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂²×Dic₁₉	—	2	1	no	—
22	C₂²×Dic₁₉	—	2	1	no	—
23	C₂×D₇₆	—	2	2	no	—
24	C₂×D₇₆	—	2	2	no	—
25	C₂×D₇₆	—	2	2	no	—
26	C₂×D₇₆	—	2	2	no	—
27	C₂×D₇₆	—	2	8	no	—
28	C₂×D₇₆	—	2	2	no	—
29	C₂×D₇₆	—	4	4	no	—
30	C₂×D₇₆	—	4	4	no	—
31	C₂×D₇₆	D₄	1	4	yes	—
32	C₂×Dic₃₈	—	2	2	no	—
33	C₂×Dic₃₈	—	2	2	no	—
34	C₂×Dic₃₈	—	2	1	no	—
35	C₂×Dic₃₈	D₄	1	2	yes	complemented
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	—	2	2	no	—
38	C₂×C₈×C₁₉	D₄	1	4	yes	complemented
39	C₄²×C₁₉	—	2	2	no	—
40	C₄⋊Dic₁₉	—	2	2	no	—
41	#64516	—	2	16	no	—
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	C₂×C₄×C₁₉	—	2	2	no	—
49	C₂×C₄×C₁₉	—	2	2	no	—
50	C₂×C₄×C₁₉	SD₁₆	1	4	yes	complemented
51	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₉	#41	1	1	yes	characteristic
55	C₂×Dic₁₉	—	2	2	no	—
56	C₂×Dic₁₉	—	4	1	no	—
57	C₂×Dic₁₉	—	4	1	no	—
58	C₂×Dic₁₉	—	4	1	no	—
59	C₂×Dic₁₉	—	4	2	no	—
60	D₇₆	—	2	4	no	—
61	D₇₆	—	2	4	no	—
62	D₇₆	—	4	16	no	—
63	D₇₆	—	4	2	no	—
64	D₇₆	—	4	2	no	—
65	Dic₃₈	—	2	4	no	—
66	Dic₃₈	—	4	2	no	—
67	C₈×C₁₉	—	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	#120	1	2	yes	—
81	C₄×C₁₉	#121	1	2	yes	—
82	Dic₁₉	—	4	2	no	—
83	Dic₁₉	—	8	1	no	—
84	C₂²×SD₁₆	—	19	2	no	—
85	C₂×#127	—	19	1	no	—
86	C₂×#102	—	19	2	no	—
87	#391	—	19	1	no	—
88	#336	D₁₉	1	1	yes	characteristic, complemented
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	—	19	1	no	—
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	C₂×D₁₉	1	2	yes	—
102	#127	—	38	4	no	—
103	C₂×C₄⋊C₄	—	19	1	no	—
104	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
105	#102	—	38	4	no	—
106	#95	—	38	2	no	—
107	C₁₉	#2014	1	1	yes	characteristic, complemented, Sylow
108	C₂⁴	—	38	2	no	—
109	C₂×Q₈	—	19	2	no	—
110	C₂×Q₈	—	38	1	no	—
111	C₂×Q₈	—	38	2	no	—
112	C₂×Q₈	—	38	2	no	—
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	—	38	1	no	—
123	C₂²×C₄	—	38	1	no	—
124	C₂²×C₄	D₇₆	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
126	SD₁₆	—	38	16	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	D₇₆	1	4	yes	complemented
130	C₄⋊C₄	—	38	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×D₇₆	1	1	yes	characteristic
138	Q₈	—	38	4	no	—
139	Q₈	—	76	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	—	38	2	no	—
148	C₂×C₄	—	76	1	no	—
149	C₂×C₄	—	76	1	no	—
150	C₂×C₄	—	76	1	no	—
151	C₂×C₄	—	76	2	no	—
152	C₂×C₄	#64516	1	4	yes	complemented
153	C₂×C₄	C₂×D₇₆	1	2	yes	—
154	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
155	C₂×C₄	D₄×D₁₉	1	2	yes	—
156	C₂×C₄	#64545	1	1	yes	characteristic
157	C₈	—	2	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	#10597320	1	1	yes	characteristic
164	C₂²	#10597328	1	1	yes	characteristic
165	C₂²	#10597341	1	1	yes	characteristic
166	C₂²	C₂×#64516	1	2	yes	—
167	C₂²	#10597354	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	76	2	no	—
170	C₄	—	152	1	no	—
171	C₄	#10597328	1	2	yes	—
172	C₄	#10597341	1	2	yes	—
173	C₂	—	152	4	no	—
174	C₂	#11926454	1	1	yes	characteristic
175	C₂	#11926697	1	2	yes	—
176	C₂	#11926745	1	2	yes	—
177	C₂	#11926917	1	1	yes	characteristic
178	C₂	#11926933	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#64516

C₂

yes

maximal, complemented

C₂×#10597320

C₂

yes

maximal, characteristic

C₂×#10597278

C₂

yes

maximal, complemented

#11926450

C₂

yes

maximal, characteristic, complemented

C₁₉×#336

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₆

—

C₂²×D₇₆

C₂²

yes

—

C₂²×Dic₃₈

C₂²

yes

characteristic

C₂²×C₈×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

—

—

—

—

C₁₉⋊#95

—

#10597278

—

C₂³×D₁₉

—

C₂²×C₄×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

D₄

yes

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

D₄

yes

complemented

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

D₄

yes

complemented

C₄²×C₁₉

—

C₄⋊Dic₁₉

—

#64516

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

SD₁₆

yes

complemented

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

#41

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

—

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

#120

yes

—

C₄×C₁₉

#121

yes

—

Dic₁₉

—

Dic₁₉

—

C₂²×SD₁₆

—

—

—

—

D₁₉

yes

characteristic, complemented

C₂×C₁₉

#392

yes

characteristic

C₂×C₁₉

#450

yes

—

C₂×C₁₉

#460

yes

—

C₂×C₁₉

yes

C₂×C₁₉

yes

D₁₉

—

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

100

C₂×SD₁₆

—

101

C₂²×C₈

C₂×D₁₉

yes

—

102

#127

—

103

C₂×C₄⋊C₄

—

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

—

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

D₇₆

yes

—

125

C₂²×C₄

C₂²×D₁₉

yes

characteristic

126

SD₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

D₇₆

yes

complemented

130

C₄⋊C₄

—

131

C₄²

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂×D₇₆

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

#64516

yes

complemented

153

C₂×C₄

C₂×D₇₆

yes

—

154

C₂×C₄

D₄×D₁₉

yes

characteristic

155

C₂×C₄

D₄×D₁₉

yes

—

156

C₂×C₄

#64545

yes

characteristic

157

C₈

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

152

—

161

C₂²

—

152

—

162

C₂²

—

152

—

163

C₂²

#10597320

yes

characteristic

164

C₂²

#10597328

yes

characteristic

165

C₂²

#10597341

yes

characteristic

166

C₂²

C₂×#64516

yes

—

167

C₂²

#10597354

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

152

—

171

C₄

#10597328

yes

—

172

C₄

#10597341

yes

—

173

C₂

—

152

—

174

C₂

#11926454

yes

characteristic

175

C₂

#11926697

yes

—

176

C₂

#11926745

yes

—

177

C₂

#11926917

yes

characteristic

178

C₂

#11926933

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₆
commutator, cosocle	C₂²×C₄×C₁₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting	C₁₉×#336	C₂
socle	C₂³×C₁₉	C₂×D₄
prime radical	C₂×#10597320	C₂
hypercenter	#336	D₁₉
hypocenter	C₁₉	#2014
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₇₆

commutator, cosocle

C₂²×C₄×C₁₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉

Fitting

C₁₉×#336

C₂

socle

C₂³×C₁₉

C₂×D₄

C₂

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
1. C₂²×#64516	C₂	semidirect
3. C₂×#10597278	C₂	semidirect
4. #11926450	C₂	semidirect
5. C₁₉×#336	C₂	semidirect
35. C₂×Dic₃₈	D₄	semidirect
38. C₂×C₈×C₁₉	D₄	semidirect
50. C₂×C₄×C₁₉	SD₁₆	semidirect
88. #336	D₁₉	semidirect
107. C₁₉	#2014	semidirect
129. C₂×C₈	D₇₆	semidirect
152. C₂×C₄	#64516	semidirect

H ≅ G/N

type

1. C₂²×#64516

C₂

semidirect

3. C₂×#10597278

C₂

semidirect

4. #11926450

C₂

semidirect

5. C₁₉×#336

C₂

semidirect

35. C₂×Dic₃₈

D₄

semidirect

38. C₂×C₈×C₁₉

D₄

semidirect

50. C₂×C₄×C₁₉

SD₁₆

semidirect

88. #336

D₁₉

semidirect

107. C₁₉

#2014

semidirect

129. C₂×C₈

D₇₆

semidirect

152. C₂×C₄

#64516

semidirect

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

subgroup	index
C₂²×#64516	2
C₂×#10597320	2
C₂×#10597278	2
#11926450	2
C₁₉×#336	2
#2014	19

subgroup

index

C₂²×#64516

C₁₉×#336

Subgroups — #14564725

Conjugacy classes of subgroups

Complements

Maximal subgroups