Group #14512458 — subgroups

Subgroups — #14512458

Order 1920.

214 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 42 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	C₈×D₅×A₄	C₂	1	1	yes	maximal, characteristic
2	A₄⋊#674	C₂	1	2	yes	maximal, complemented
3	#10604706	—	3	1	no	maximal, Hall
4	C₄×D₅×A₄	C₄	1	1	yes	characteristic
5	C₅×C₈×A₄	C₄	1	1	yes	characteristic
6	#91219	—	4	1	no	maximal
7	A₄×#168	C₂²	1	1	yes	characteristic
8	#73104	—	5	1	no	maximal, Hall
9	C₂²×C₈×D₅	D₃	1	1	yes	characteristic
10	C₂×#3990	—	3	1	no	—
11	C₂×#3990	—	3	1	no	—
12	C₂²⋊#674	—	3	2	no	—
13	#64887	—	3	2	no	—
14	C₂×D₅×A₄	C₈	1	1	yes	characteristic
15	C₄×C₅×A₄	C₂×C₄	1	1	yes	characteristic
16	A₄×Dic₅	C₈	1	1	yes	characteristic
17	C₃×C₈×D₅	—	4	1	no	—
18	C₃⋊#674	—	4	2	no	—
19	C₂×C₈×A₄	—	5	1	no	—
20	A₄⋊C₁₆	—	5	2	no	—
21	C₂²×C₄×D₅	Dic₃	1	1	yes	characteristic
22	C₂²×C₅×C₈	Dic₃	1	1	yes	characteristic
23	C₂²×#168	C₂×D₃	1	1	yes	characteristic
24	C₂×C₈×D₅	—	3	1	no	—
25	C₂×C₈×D₅	—	3	1	no	—
26	C₂×C₈×D₅	—	3	1	no	—
27	C₂×C₈×D₅	—	3	1	no	—
28	C₂×#674	—	3	2	no	—
29	C₂×#674	—	3	2	no	—
30	#3990	—	6	1	no	—
31	#3990	—	6	1	no	—
32	#3990	—	6	1	no	—
33	#3990	—	6	1	no	—
34	#2112	—	15	1	no	Sylow
35	C₂×C₅×A₄	C₂×C₈	1	1	yes	characteristic
36	D₅×A₄	—	2	1	no	—
37	C₃×C₄×D₅	—	4	1	no	—
38	C₃×C₅×C₈	—	4	1	no	—
39	C₃×#168	—	4	1	no	—
40	C₂×C₄×A₄	—	5	1	no	—
41	C₈×A₄	—	5	1	no	—
42	C₈×A₄	AGL(1,𝔽₅)	1	1	yes	characteristic
43	#812	—	20	1	no	—
44	C₂³×D₅	C₃⋊C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
46	C₂²×Dic₅	C₃⋊C₈	1	1	yes	characteristic
47	C₂×C₄×D₅	—	3	1	no	—
48	C₂×C₄×D₅	—	3	1	no	—
49	C₂×C₄×D₅	—	3	1	no	—
50	C₂×C₄×D₅	—	3	1	no	—
51	C₂×C₅×C₈	—	3	1	no	—
52	C₂×C₅×C₈	—	3	1	no	—
53	C₂×#168	—	3	1	no	—
54	C₂×#168	—	3	1	no	—
55	C₈×D₅	—	3	1	no	—
56	C₈×D₅	—	3	1	no	—
57	C₈×D₅	—	3	1	no	—
58	C₈×D₅	—	6	1	no	—
59	C₈×D₅	S₄	1	1	yes	characteristic
60	#674	—	6	2	no	—
61	#674	—	6	2	no	—
62	C₂³×C₈	—	5	1	no	—
63	C₂×M₃₂	—	15	1	no	—
64	C₂×M₃₂	—	15	1	no	—
65	C₂²⋊C₁₆	—	15	2	no	—
66	C₂²⋊C₁₆	—	15	2	no	—
67	C₂×C₃×D₅	—	4	1	no	—
68	C₅×A₄	M₃₂	1	1	yes	characteristic, complemented
69	C₃×C₄×C₅	—	4	1	no	—
70	C₃×Dic₅	—	4	1	no	—
71	C₂²×A₄	—	5	1	no	—
72	C₄×A₄	—	5	1	no	—
73	C₄×A₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₂×C₃×C₈	—	20	1	no	—
75	C₃⋊C₁₆	—	20	2	no	—
76	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
77	C₂²×D₅	—	2	1	no	—
78	C₂²×D₅	—	3	1	no	—
79	C₂²×D₅	—	3	1	no	—
80	C₂²×D₅	—	6	1	no	—
81	C₂×C₄×C₅	—	3	1	no	—
82	C₂×C₄×C₅	—	3	1	no	—
83	C₂×Dic₅	—	3	1	no	—
84	C₂×Dic₅	—	3	1	no	—
85	C₄×D₅	—	3	1	no	—
86	C₄×D₅	—	3	1	no	—
87	C₄×D₅	—	3	1	no	—
88	C₄×D₅	—	6	1	no	—
89	C₄×D₅	SL(2,ℤ/4)	1	1	yes	characteristic
90	C₅×C₈	—	3	1	no	—
91	C₅×C₈	SL(2,ℤ/4)	1	1	yes	characteristic
92	#168	—	3	1	no	—
93	#168	C₂×S₄	1	1	yes	characteristic
94	C₂³×C₄	—	5	1	no	—
95	C₂²×C₈	—	5	1	no	—
96	C₂²×C₈	—	15	1	no	—
97	C₂²×C₈	—	15	1	no	—
98	C₂²×C₈	—	15	1	no	—
99	C₂²×C₈	—	15	1	no	—
100	C₂²×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
101	M₃₂	—	30	1	no	—
102	M₃₂	—	30	1	no	—
103	M₃₂	—	30	1	no	—
104	M₃₂	—	30	1	no	—
105	C₂×C₁₆	—	15	2	no	—
106	C₂×C₁₆	—	15	2	no	—
107	C₂×C₃×C₅	—	4	1	no	—
108	C₃×D₅	—	8	1	no	—
109	C₂×A₄	—	10	1	no	—
110	C₂×A₄	D₅⋊C₈	1	1	yes	characteristic
111	C₂×C₃×C₄	—	20	1	no	—
112	C₃×C₈	—	4	1	no	—
113	C₃×C₈	—	20	1	no	—
114	C₂²×C₅	—	3	1	no	—
115	C₂²×C₅	—	3	1	no	—
116	C₂²×C₅	#812	1	1	yes	characteristic, complemented
117	C₂×D₅	—	3	2	no	—
118	C₂×D₅	—	3	1	no	—
119	C₂×D₅	—	6	1	no	—
120	C₂×D₅	—	6	1	no	—
121	C₂×D₅	—	6	1	no	—
122	C₂×D₅	A₄⋊C₈	1	1	yes	characteristic
123	C₄×C₅	—	3	1	no	—
124	C₄×C₅	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
125	Dic₅	—	3	1	no	—
126	Dic₅	A₄⋊C₈	1	1	yes	characteristic
127	C₂⁴	—	5	1	no	—
128	C₂²×C₄	—	5	1	no	—
129	C₂²×C₄	—	15	1	no	—
130	C₂²×C₄	—	15	1	no	—
131	C₂²×C₄	—	15	1	no	—
132	C₂²×C₄	—	15	1	no	—
133	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
134	C₂×C₈	—	3	1	no	—
135	C₂×C₈	—	3	1	no	—
136	C₂×C₈	—	5	1	no	—
137	C₂×C₈	—	15	1	no	—
138	C₂×C₈	—	15	1	no	—
139	C₂×C₈	—	15	1	no	—
140	C₂×C₈	—	15	1	no	—
141	C₂×C₈	—	15	1	no	—
142	C₂×C₈	—	30	1	no	—
143	C₁₆	—	30	2	no	—
144	C₁₆	—	30	2	no	—
145	C₃×C₅	—	4	1	no	Hall
146	C₂²×C₃	—	20	1	no	—
147	A₄	#3990	1	1	yes	characteristic, complemented
148	C₃×C₄	—	4	1	no	—
149	C₃×C₄	—	20	1	no	—
150	C₂×C₅	—	3	1	no	—
151	C₂×C₅	—	3	1	no	—
152	C₂×C₅	C₂×A₄⋊C₈	1	1	yes	characteristic
153	D₅	—	2	1	no	—
154	D₅	—	6	1	no	—
155	C₂³	—	10	1	no	—
156	C₂³	—	15	1	no	—
157	C₂³	—	15	1	no	—
158	C₂³	—	30	1	no	—
159	C₂³	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
160	C₂×C₄	—	3	1	no	—
161	C₂×C₄	—	3	1	no	—
162	C₂×C₄	—	5	1	no	—
163	C₂×C₄	—	15	1	no	—
164	C₂×C₄	—	15	1	no	—
165	C₂×C₄	—	15	1	no	—
166	C₂×C₄	—	15	1	no	—
167	C₂×C₄	—	15	1	no	—
168	C₂×C₄	—	30	1	no	—
169	C₈	—	3	1	no	—
170	C₈	—	5	1	no	—
171	C₈	—	15	1	no	—
172	C₈	A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₂×C₃	—	4	1	no	—
174	C₂×C₃	—	40	1	no	—
175	C₅	#73104	1	1	yes	characteristic, complemented, Sylow
176	C₂²	—	3	1	no	—
177	C₂²	—	3	1	no	—
178	C₂²	—	5	1	no	—
179	C₂²	—	15	2	no	—
180	C₂²	—	15	1	no	—
181	C₂²	—	30	1	no	—
182	C₂²	—	30	1	no	—
183	C₂²	—	30	1	no	—
184	C₂²	#91219	1	1	yes	characteristic, complemented
185	C₄	—	3	1	no	—
186	C₄	—	5	1	no	—
187	C₄	—	15	1	no	—
188	C₄	C₂×A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
189	C₃	—	4	1	no	Sylow
190	C₂	—	3	1	no	—
191	C₂	—	3	1	no	—
192	C₂	—	10	1	no	—
193	C₂	—	30	1	no	—
194	C₂	D₅⋊(A₄⋊C₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×D₅×A₄

C₂

yes

maximal, characteristic

A₄⋊#674

C₂

yes

maximal, complemented

#10604706

—

maximal, Hall

C₄×D₅×A₄

C₄

yes

characteristic

C₅×C₈×A₄

C₄

yes

—

C₂²

yes

characteristic

#73104

—

maximal, Hall

C₂²×C₈×D₅

D₃

yes

characteristic

C₂×#3990

—

C₂×#3990

—

C₂²⋊#674

—

#64887

—

C₂×D₅×A₄

C₈

yes

characteristic

C₄×C₅×A₄

C₂×C₄

yes

characteristic

A₄×Dic₅

C₈

yes

characteristic

C₃×C₈×D₅

—

C₃⋊#674

—

C₂×C₈×A₄

—

A₄⋊C₁₆

—

C₂²×C₄×D₅

Dic₃

yes

characteristic

C₂²×C₅×C₈

Dic₃

yes

characteristic

C₂²×#168

C₂×D₃

yes

characteristic

C₂×C₈×D₅

—

C₂×C₈×D₅

—

C₂×C₈×D₅

—

C₂×C₈×D₅

—

—

—

—

—

—

—

—

C₂×C₅×A₄

C₂×C₈

yes

characteristic

D₅×A₄

—

C₃×C₄×D₅

—

C₃×C₅×C₈

—

C₃×#168

—

C₂×C₄×A₄

—

C₈×A₄

—

C₈×A₄

AGL(1,𝔽₅)

yes

characteristic

#812

—

C₂³×D₅

C₃⋊C₈

yes

characteristic

C₂²×C₄×C₅

C₂×Dic₃

yes

characteristic

C₂²×Dic₅

C₃⋊C₈

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×#168

—

C₂×#168

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

S₄

yes

characteristic

#674

—

#674

—

C₂³×C₈

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₂×C₃×D₅

—

C₅×A₄

M₃₂

yes

characteristic, complemented

C₃×C₄×C₅

—

C₃×Dic₅

—

C₂²×A₄

—

C₄×A₄

—

C₄×A₄

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₃×C₈

—

C₃⋊C₁₆

—

C₂³×C₅

C₂×C₃⋊C₈

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

SL(2,ℤ/4)

yes

characteristic

C₅×C₈

—

C₅×C₈

yes

—

C₂×S₄

yes

characteristic

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

ΓL(1,𝔽₁₆)

yes

characteristic

101

M₃₂

—

102

M₃₂

—

103

M₃₂

—

104

M₃₂

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₃×C₅

—

108

C₃×D₅

—

109

C₂×A₄

—

110

C₂×A₄

D₅⋊C₈

yes

characteristic

111

C₂×C₃×C₄

—

112

C₃×C₈

—

113

C₃×C₈

—

114

C₂²×C₅

—

115

C₂²×C₅

—

116

C₂²×C₅

#812

yes

characteristic, complemented

117

C₂×D₅

—

118

C₂×D₅

—

119

C₂×D₅

—

120

C₂×D₅

—

121

C₂×D₅

—

122

C₂×D₅

A₄⋊C₈

yes

characteristic

123

C₄×C₅

—

124

C₄×C₅

C₂×SL(2,ℤ/4)

yes

characteristic

125

Dic₅

—

126

Dic₅

A₄⋊C₈

yes

characteristic

127

C₂⁴

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₁₆

—

144

C₁₆

—

145

C₃×C₅

—

Hall

146

C₂²×C₃

—

147

A₄

#3990

yes

characteristic, complemented

148

C₃×C₄

—

149

C₃×C₄

—

150

C₂×C₅

—

151

C₂×C₅

—

152

C₂×C₅

C₂×A₄⋊C₈

yes

characteristic

153

D₅

—

154

D₅

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

D₅⋊(C₃⋊C₈)

yes

characteristic

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

A₄⋊AGL(1,𝔽₅)

yes

characteristic

173

C₂×C₃

—

174

C₂×C₃

—

175

C₅

#73104

yes

characteristic, complemented, Sylow

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

#91219

yes

characteristic, complemented

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

C₂×A₄⋊AGL(1,𝔽₅)

yes

characteristic

189

C₃

—

Sylow

190

C₂

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

D₅⋊(A₄⋊C₈)

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×A₄⋊AGL(1,𝔽₅)
commutator	C₂×C₅×A₄	C₂×C₈
Fitting	C₂²×C₅×C₈	Dic₃
socle	C₂³×C₅	C₂×C₃⋊C₈
prime radical	C₂×D₅×A₄	C₈
cosocle	A₄×#168	C₂²
hypercenter	C₈	A₄⋊AGL(1,𝔽₅)
hypocenter	C₅×A₄	M₃₂
LCS	G ▷ C₂×C₅×A₄ ▷ C₅×A₄
UCS	C₁ ◁ C₄ ◁ C₈

subgroup

G/H

center, Frattini

C₄

C₂×A₄⋊AGL(1,𝔽₅)

commutator

C₂×C₅×A₄

C₂×C₈

Fitting

C₂²×C₅×C₈

Dic₃

socle

C₂³×C₅

C₂×C₃⋊C₈

prime radical

C₂×D₅×A₄

C₈

C₂²

C₈

C₅×A₄

M₃₂

LCS

G ▷ C₂×C₅×A₄ ▷ C₅×A₄

UCS

C₁ ◁ C₄ ◁ C₈

N	H ≅ G/N	type
2. A₄⋊#674	C₂	semidirect
68. C₅×A₄	M₃₂	semidirect
116. C₂²×C₅	#812	semidirect
147. A₄	#3990	semidirect
175. C₅	#73104	semidirect
184. C₂²	#91219	semidirect

H ≅ G/N

type

2. A₄⋊#674

C₂

semidirect

68. C₅×A₄

M₃₂

semidirect

116. C₂²×C₅

#812

semidirect

147. A₄

#3990

semidirect

175. C₅

#73104

semidirect

184. C₂²

#91219

semidirect

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

subgroup	index
C₈×D₅×A₄	2
A₄⋊#674	2
#10604706	3
#91219	4
#73104	5

subgroup

index

C₈×D₅×A₄

Subgroups — #14512458

Conjugacy classes of subgroups

Complements

Maximal subgroups