Group #14280568 — subgroups

Subgroups — #14280568

Order 1920.

392 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 145 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	#11752337	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#11746751	C₂	1	4	yes	maximal, complemented
4	#10603811	—	3	1	no	maximal, Hall
5	C₂×C₃×C₈×D₅	C₄	1	2	yes	—
6	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
7	#91233	C₄	1	4	yes	—
8	#91232	C₄	1	4	yes	—
9	C₄×C₃⋊#170	C₂²	1	4	yes	—
10	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₃×C₈×Dic₅	C₂²	1	2	yes	—
12	C₃×C₄×#168	C₄	1	1	yes	characteristic
13	#69726	—	5	1	no	maximal, Hall
14	#65678	—	3	2	no	—
15	C₄×C₈×D₅	D₃	1	1	yes	characteristic
16	#170⋊C₈	—	3	4	no	—
17	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
18	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃⋊#170	C₂×C₄	1	8	yes	—
20	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
23	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
24	C₂×C₃×#168	C₂×C₄	1	2	yes	—
25	C₃×C₈×D₅	—	2	4	no	—
26	C₂×C₃×C₄×C₈	—	5	1	no	—
27	#4925	—	5	2	no	—
28	#4454	—	5	4	no	—
29	C₂×C₈×D₅	Dic₃	1	2	yes	—
30	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
31	#4004	—	3	4	no	—
32	#4003	—	3	4	no	—
33	C₄×#170	—	3	4	no	—
34	C₄×C₅×C₈	Dic₃	1	1	yes	characteristic
35	C₈×Dic₅	C₂×D₃	1	2	yes	—
36	C₄×#168	Dic₃	1	1	yes	characteristic
37	#1452	—	15	1	no	Sylow
38	C₂²×C₃×D₅	C₄²	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
42	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
43	C₃×C₄×D₅	—	2	4	no	—
44	C₃×C₄×D₅	M₁₆	1	4	yes	—
45	C₃⋊#170	—	2	8	no	—
46	C₃×C₅×C₈	—	2	2	no	—
47	C₃×#168	—	2	2	no	—
48	C₂²×C₃×C₈	—	5	2	no	—
49	C₂×C₃×C₄²	—	5	1	no	—
50	C₃×C₄×C₈	—	5	2	no	—
51	C₃×C₄×C₈	—	5	1	no	—
52	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
53	#830	—	5	4	no	—
54	C₄⋊(C₃⋊C₈)	—	5	4	no	—
55	C₄×C₃⋊C₈	—	5	4	no	—
56	C₂×C₄×D₅	C₄×D₃	1	2	yes	—
57	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
58	C₂×#170	—	3	8	no	—
59	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
60	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
61	C₄×Dic₅	C₄×D₃	1	2	yes	—
62	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
63	C₂×#168	C₂×Dic₃	1	2	yes	—
64	C₈×D₅	—	2	4	no	—
65	#431	—	15	2	no	—
66	C₂×C₄×C₈	—	5	1	no	—
67	#322	—	15	4	no	—
68	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₃×D₅	—	2	2	no	—
70	C₂×C₃×D₅	#97	1	2	yes	—
71	C₃×C₄×C₅	—	2	2	no	—
72	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
73	C₃×Dic₅	#97	1	2	yes	—
74	C₃×Dic₅	#131	1	4	yes	—
75	C₂²×C₃×C₄	—	5	1	no	—
76	C₂²×C₃×C₄	—	5	2	no	—
77	C₂×C₃×C₈	—	5	2	no	—
78	C₂×C₃×C₈	—	10	4	no	—
79	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
80	C₃×C₄²	—	5	2	no	—
81	C₃×C₄²	—	5	1	no	—
82	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₂×C₃⋊C₈	—	5	8	no	—
84	C₂²×D₅	C₄×Dic₃	1	1	yes	characteristic
85	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
86	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
87	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
88	C₂×Dic₅	C₂×C₄×D₃	1	2	yes	—
89	C₄×D₅	—	2	4	no	—
90	C₄×D₅	#208	1	4	yes	—
91	#170	—	6	8	no	—
92	C₅×C₈	—	2	2	no	—
93	#168	—	2	2	no	—
94	C₂²×C₈	—	5	2	no	—
95	C₂×C₄²	—	5	1	no	—
96	C₄⋊C₈	—	15	4	no	—
97	C₂²⋊C₈	—	15	4	no	—
98	C₄×C₈	—	5	2	no	—
99	C₄×C₈	—	5	1	no	—
100	C₄×C₈	—	15	4	no	—
101	C₄×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
102	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
103	C₂×C₃×C₅	#405	1	2	yes	—
104	C₃×D₅	—	2	2	no	—
105	C₂³×C₃	—	5	1	no	—
106	C₂×C₃×C₄	—	5	1	no	—
107	C₂×C₃×C₄	—	5	2	no	—
108	C₂×C₃×C₄	—	5	4	no	—
109	C₂×C₃×C₄	—	10	4	no	—
110	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
111	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
112	C₃×C₈	—	2	2	no	—
113	C₃×C₈	—	10	2	no	—
114	C₃⋊C₈	—	10	8	no	—
115	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
116	C₂×D₅	—	2	2	no	—
117	C₂×D₅	#803	1	2	yes	—
118	C₄×C₅	—	2	2	no	—
119	C₄×C₅	C₂×#208	1	2	yes	—
120	Dic₅	#803	1	2	yes	—
121	Dic₅	#901	1	4	yes	—
122	C₂²×C₄	—	5	1	no	—
123	C₂²×C₄	—	5	2	no	—
124	C₂×C₈	—	5	2	no	—
125	C₂×C₈	—	10	4	no	—
126	C₂×C₈	—	15	8	no	—
127	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
128	C₄²	—	5	2	no	—
129	C₄²	—	5	1	no	—
130	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
131	C₃×C₅	#1452	1	1	yes	characteristic, complemented, Hall
132	C₂²×C₃	—	5	2	no	—
133	C₂²×C₃	—	10	2	no	—
134	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₃×C₄	—	2	2	no	—
136	C₃×C₄	—	5	2	no	—
137	C₃×C₄	—	5	4	no	—
138	C₃×C₄	D₅⋊M₁₆	1	2	yes	—
139	C₂×C₅	C₂×#803	1	1	yes	characteristic
140	C₂×C₅	#5101	1	2	yes	—
141	D₅	—	2	2	no	—
142	C₂³	—	5	1	no	—
143	C₂×C₄	—	5	2	no	—
144	C₂×C₄	—	5	1	no	—
145	C₂×C₄	—	5	4	no	—
146	C₂×C₄	—	10	4	no	—
147	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
148	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	10	2	no	—
151	C₈	—	30	8	no	—
152	C₂×C₃	—	10	2	no	—
153	C₂×C₃	#65676	1	1	yes	characteristic
154	C₂×C₃	#65718	1	2	yes	—
155	C₅	#69726	1	1	yes	characteristic, complemented, Sylow
156	C₂²	—	5	2	no	—
157	C₂²	—	10	2	no	—
158	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	5	2	no	—
161	C₄	—	5	4	no	—
162	C₄	D₅⋊#208	1	2	yes	—
163	C₃	#10603811	1	1	yes	characteristic, complemented, Sylow
164	C₂	—	10	2	no	—
165	C₂	#11752335	1	1	yes	characteristic
166	C₂	#11752377	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11752337

C₂

yes

maximal

C₃×C₄×C₈×D₅

C₂

yes

maximal, characteristic

#11746751

C₂

yes

maximal, complemented

#10603811

—

maximal, Hall

C₂×C₃×C₈×D₅

C₄

yes

—

C₃×C₄²×D₅

C₂²

yes

characteristic

#91233

C₄

yes

—

#91232

C₄

yes

—

C₄×C₃⋊#170

C₂²

yes

—

C₃×C₄×C₅×C₈

C₄

yes

characteristic

C₃×C₈×Dic₅

C₂²

yes

—

C₃×C₄×#168

C₄

yes

—

—

C₄×C₈×D₅

D₃

yes

characteristic

#170⋊C₈

—

C₂×C₃×C₄×D₅

C₂×C₄

yes

—

C₂×C₃×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₃⋊#170

C₂×C₄

yes

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

—

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₃×C₄×Dic₅

C₂×C₄

yes

—

C₃×C₄×Dic₅

C₂³

yes

characteristic

C₂×C₃×#168

C₂×C₄

yes

—

C₃×C₈×D₅

—

C₂×C₃×C₄×C₈

—

#4925

—

#4454

—

C₂×C₈×D₅

Dic₃

yes

—

C₄²×D₅

C₂×D₃

yes

—

—

—

C₄×C₅×C₈

Dic₃

yes

characteristic

C₈×Dic₅

C₂×D₃

yes

—

C₄×#168

Dic₃

yes

characteristic

#1452

—

Sylow

C₂²×C₃×D₅

C₄²

yes

characteristic

C₂×C₃×C₄×C₅

C₄²

yes

—

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×Dic₅

C₄²

yes

characteristic

C₂×C₃×Dic₅

C₂²×C₄

yes

—

C₃×C₄×D₅

—

C₃×C₄×D₅

M₁₆

yes

—

C₃⋊#170

—

C₃×C₅×C₈

—

C₃×#168

—

C₂²×C₃×C₈

—

C₂×C₃×C₄²

—

C₃×C₄×C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

#830

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

C₂×C₄×D₅

C₄×D₃

yes

—

C₂×C₄×D₅

C₂×Dic₃

yes

characteristic

C₂×#170

—

C₂×C₅×C₈

C₂×Dic₃

yes

—

C₄²×C₅

C₂×Dic₃

yes

characteristic

C₄×Dic₅

C₄×D₃

yes

—

C₄×Dic₅

C₂²×D₃

yes

characteristic

C₂×#168

C₂×Dic₃

yes

—

C₈×D₅

—

#431

—

C₂×C₄×C₈

—

#322

—

C₂²×C₃×C₅

C₂×C₄²

yes

characteristic

C₂×C₃×D₅

—

C₂×C₃×D₅

#97

yes

—

C₃×C₄×C₅

—

C₃×C₄×C₅

C₂×M₁₆

yes

—

C₃×Dic₅

#97

yes

—

C₃×Dic₅

#131

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₃⋊C₈

—

C₂²×D₅

C₄×Dic₃

yes

characteristic

C₂×C₄×C₅

C₄×Dic₃

yes

—

C₂×C₄×C₅

C₂²×Dic₃

yes

characteristic

C₂×Dic₅

C₄×Dic₃

yes

characteristic

C₂×Dic₅

C₂×C₄×D₃

yes

—

C₄×D₅

—

C₄×D₅

#208

yes

—

#170

—

C₅×C₈

—

#168

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

ΓL(1,𝔽₁₆)

yes

characteristic

102

C₂×C₃×C₅

C₂×#97

yes

characteristic

103

C₂×C₃×C₅

#405

yes

—

104

C₃×D₅

—

105

C₂³×C₃

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

C₄×AGL(1,𝔽₅)

yes

—

111

C₂×C₃×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

112

C₃×C₈

—

113

C₃×C₈

—

114

C₃⋊C₈

—

115

C₂²×C₅

C₂×C₄×Dic₃

yes

characteristic

116

C₂×D₅

—

117

C₂×D₅

#803

yes

—

118

C₄×C₅

—

119

C₄×C₅

C₂×#208

yes

—

120

Dic₅

#803

yes

—

121

Dic₅

#901

yes

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂×ΓL(1,𝔽₁₆)

yes

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

131

C₃×C₅

#1452

yes

characteristic, complemented, Hall

132

C₂²×C₃

—

133

C₂²×C₃

—

134

C₂²×C₃

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

135

C₃×C₄

—

136

C₃×C₄

—

137

C₃×C₄

—

138

C₃×C₄

D₅⋊M₁₆

yes

—

139

C₂×C₅

C₂×#803

yes

characteristic

140

C₂×C₅

#5101

yes

—

141

D₅

—

142

C₂³

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₄×ΓL(1,𝔽₁₆)

yes

—

148

C₂×C₄

C₂²×ΓL(1,𝔽₁₆)

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₂×C₃

—

153

C₂×C₃

#65676

yes

characteristic

154

C₂×C₃

#65718

yes

—

155

C₅

#69726

yes

characteristic, complemented, Sylow

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₄×ΓL(1,𝔽₁₆)

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

D₅⋊#208

yes

—

163

C₃

#10603811

yes

characteristic, complemented, Sylow

164

C₂

—

165

C₂

#11752335

yes

characteristic

166

C₂

#11752377

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₃×C₄×C₅×C₈	C₄
prime radical	C₂²×C₃×D₅	C₄²
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₄×C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1452
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×ΓL(1,𝔽₁₆)

commutator, socle

C₂²×C₃×C₅

C₂×C₄²

Fitting

C₃×C₄×C₅×C₈

C₄

prime radical

C₂²×C₃×D₅

C₄²

cosocle

C₃×C₄×Dic₅

C₂³

hypercenter

C₄×C₈

ΓL(1,𝔽₁₆)

hypocenter

C₃×C₅

#1452

LCS

G ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈

N	H ≅ G/N	type
3. #11746751	C₂	semidirect
131. C₃×C₅	#1452	semidirect
155. C₅	#69726	semidirect
163. C₃	#10603811	semidirect

H ≅ G/N

type

3. #11746751

C₂

semidirect

131. C₃×C₅

#1452

semidirect

155. C₅

#69726

semidirect

163. C₃

#10603811

semidirect

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

subgroup	index
#11752337	2
C₃×C₄×C₈×D₅	2
#11746751	2
#10603811	3
#69726	5

subgroup

index

C₃×C₄×C₈×D₅

Subgroups — #14280568

Conjugacy classes of subgroups

Complements

Maximal subgroups