Group #14512465 — subgroups

Subgroups — D₅⋊#4628 #14512465

Order 1920.

198 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 40 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	D₅×#131⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#11751558	C₂	1	2	yes	maximal, complemented
3	D₅⋊#350	—	3	1	no	maximal, Hall
4	D₅×#41⋊C₃	C₄	1	1	yes	characteristic
5	C₅×#131⋊C₃	C₄	1	1	yes	characteristic
6	D₅⋊(C₃⋊C₁₆)	—	4	1	no	maximal
7	#4094⋊C₃	C₂²	1	1	yes	characteristic
8	C₂×#4628	—	5	1	no	maximal, Hall
9	D₅×#131	D₃	1	1	yes	characteristic
10	D₅⋊M₃₂	—	3	1	no	—
11	C₂×D₅⋊C₁₆	—	3	1	no	—
12	#64932	—	3	2	no	—
13	#64898	—	3	2	no	—
14	C₅×#41⋊C₃	C₂×C₄	1	1	yes	characteristic
15	D₅×SL(2,𝔽₃)	C₈	1	1	yes	characteristic
16	C₅⋊(#41⋊C₃)	C₈	1	1	yes	characteristic
17	C₃×C₈×D₅	—	4	1	no	—
18	C₃⋊#674	—	4	2	no	—
19	C₂×#131⋊C₃	—	5	1	no	—
20	#4628	—	5	2	no	—
21	D₅×#41	Dic₃	1	1	yes	characteristic
22	C₅×#131	Dic₃	1	1	yes	characteristic
23	#4094	C₂×D₃	1	1	yes	characteristic
24	#4053	—	3	1	no	—
25	D₅×M₁₆	—	3	1	no	—
26	#4047	—	3	1	no	—
27	C₂×C₈×D₅	—	3	1	no	—
28	#3998	—	3	2	no	—
29	C₂×#674	—	3	2	no	—
30	#3990	—	6	1	no	—
31	D₅⋊C₁₆	—	6	1	no	—
32	D₅⋊C₁₆	—	6	1	no	—
33	D₅⋊C₁₆	—	6	1	no	—
34	C₂×#350	—	15	1	no	Sylow
35	C₃×C₄×D₅	—	4	1	no	—
36	C₅×SL(2,𝔽₃)	C₂×C₈	1	1	yes	characteristic
37	C₃×C₅×C₈	—	4	1	no	—
38	C₃×#168	—	4	1	no	—
39	C₂×#41⋊C₃	—	5	1	no	—
40	#131⋊C₃	—	5	1	no	—
41	#131⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₂×C₃⋊C₁₆	—	20	1	no	—
43	C₅×#41	C₂×Dic₃	1	1	yes	characteristic
44	#713	C₃⋊C₈	1	1	yes	characteristic
45	Q₈×D₅	C₃⋊C₈	1	1	yes	characteristic
46	#711	—	3	1	no	—
47	D₄×D₅	—	3	1	no	—
48	#709	—	3	1	no	—
49	C₂×C₄×D₅	—	3	1	no	—
50	C₅×M₁₆	—	3	1	no	—
51	C₂×C₅×C₈	—	3	1	no	—
52	#681	—	3	1	no	—
53	C₂×#168	—	3	1	no	—
54	#676	—	6	1	no	—
55	C₈×D₅	—	3	1	no	—
56	C₈×D₅	—	3	1	no	—
57	C₈×D₅	—	3	1	no	—
58	C₈×D₅	S₄	1	1	yes	characteristic
59	#674	—	6	2	no	—
60	#674	—	6	2	no	—
61	C₂×#131	—	5	1	no	—
62	C₂×M₃₂	—	15	1	no	—
63	C₂²×C₁₆	—	15	1	no	—
64	#350	—	15	2	no	—
65	#350	—	15	2	no	—
66	C₂×C₃×D₅	—	4	1	no	—
67	C₃×C₄×C₅	—	4	1	no	—
68	C₃×Dic₅	—	4	1	no	—
69	#41⋊C₃	—	5	1	no	—
70	#41⋊C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
71	C₂×SL(2,𝔽₃)	—	5	1	no	—
72	C₂×C₃×C₈	—	20	1	no	—
73	C₃⋊C₁₆	—	20	2	no	—
74	C₂²×D₅	—	3	1	no	—
75	C₅×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
76	C₅×D₄	—	3	1	no	—
77	C₂×C₄×C₅	—	3	1	no	—
78	C₂²⋊D₅	—	6	1	no	—
79	C₂×Dic₅	—	3	1	no	—
80	D₂₀	—	3	1	no	—
81	C₄×D₅	—	3	1	no	—
82	C₄×D₅	—	3	1	no	—
83	C₄×D₅	—	3	1	no	—
84	C₄×D₅	SL(2,ℤ/4)	1	1	yes	characteristic
85	Dic₁₀	—	3	1	no	—
86	C₅×C₈	—	3	1	no	—
87	C₅×C₈	SL(2,ℤ/4)	1	1	yes	characteristic
88	#168	—	3	1	no	—
89	#168	C₂×S₄	1	1	yes	characteristic
90	C₂×#41	—	5	1	no	—
91	#131	—	5	1	no	—
92	#131	—	15	1	no	—
93	#131	—	15	1	no	—
94	#131	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
95	C₂×M₁₆	—	15	1	no	—
96	C₂²×C₈	—	15	1	no	—
97	M₃₂	—	15	2	no	—
98	M₃₂	—	30	1	no	—
99	C₂×C₁₆	—	15	2	no	—
100	C₂×C₁₆	—	30	1	no	—
101	C₂×C₁₆	—	30	1	no	—
102	C₂×C₁₆	—	30	1	no	—
103	C₂×C₃×C₅	—	4	1	no	—
104	C₃×D₅	—	4	2	no	—
105	C₂×C₃×C₄	—	20	1	no	—
106	SL(2,𝔽₃)	D₅⋊C₈	1	1	yes	characteristic
107	C₃×C₈	—	4	1	no	—
108	C₃×C₈	—	20	1	no	—
109	C₂²×C₅	—	3	1	no	—
110	C₂×D₅	—	3	1	no	—
111	C₂×D₅	—	6	2	no	—
112	C₂×D₅	A₄⋊C₈	1	1	yes	characteristic
113	C₄×C₅	—	3	1	no	—
114	C₄×C₅	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
115	Dic₅	—	3	1	no	—
116	Dic₅	A₄⋊C₈	1	1	yes	characteristic
117	#41	—	5	1	no	—
118	#41	—	15	1	no	—
119	#41	—	15	1	no	—
120	#41	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
121	C₂×Q₈	—	5	1	no	—
122	C₂×D₄	—	15	1	no	—
123	C₂²×C₄	—	15	1	no	—
124	M₁₆	—	3	1	no	—
125	M₁₆	—	15	1	no	—
126	M₁₆	—	30	1	no	—
127	C₂×C₈	—	3	1	no	—
128	C₂×C₈	—	5	1	no	—
129	C₂×C₈	—	15	1	no	—
130	C₂×C₈	—	15	1	no	—
131	C₂×C₈	—	15	1	no	—
132	C₂×C₈	—	15	1	no	—
133	C₁₆	—	30	2	no	—
134	C₁₆	—	30	2	no	—
135	C₃×C₅	—	4	1	no	Hall
136	C₂²×C₃	—	20	1	no	—
137	C₃×C₄	—	4	1	no	—
138	C₃×C₄	—	20	1	no	—
139	C₂×C₅	—	6	1	no	—
140	C₂×C₅	C₂×A₄⋊C₈	1	1	yes	characteristic
141	D₅	—	6	1	no	—
142	D₅	#4628	1	2	yes	complemented
143	C₂³	—	15	1	no	—
144	Q₈	—	15	1	no	—
145	Q₈	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
146	D₄	—	3	1	no	—
147	D₄	—	15	1	no	—
148	D₄	—	30	1	no	—
149	C₂×C₄	—	3	1	no	—
150	C₂×C₄	—	5	1	no	—
151	C₂×C₄	—	15	1	no	—
152	C₂×C₄	—	15	1	no	—
153	C₂×C₄	—	15	1	no	—
154	C₂×C₄	—	15	1	no	—
155	C₈	—	3	1	no	—
156	C₈	—	5	1	no	—
157	C₈	—	15	1	no	—
158	C₈	A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₂×C₃	—	4	1	no	—
160	C₂×C₃	—	20	2	no	—
161	C₅	C₂×#4628	1	1	yes	characteristic, complemented, Sylow
162	C₂²	—	3	1	no	—
163	C₂²	—	5	1	no	—
164	C₂²	—	15	1	no	—
165	C₂²	—	30	2	no	—
166	C₄	—	3	1	no	—
167	C₄	—	5	1	no	—
168	C₄	—	15	1	no	—
169	C₄	C₂×A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₃	—	4	1	no	Sylow
171	C₂	—	5	2	no	—
172	C₂	—	6	1	no	—
173	C₂	—	30	1	no	—
174	C₂	D₅⋊(A₄⋊C₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅×#131⋊C₃

C₂

yes

maximal, characteristic

#11751558

C₂

yes

maximal, complemented

D₅⋊#350

—

maximal, Hall

D₅×#41⋊C₃

C₄

yes

characteristic

C₅×#131⋊C₃

C₄

yes

characteristic

D₅⋊(C₃⋊C₁₆)

—

maximal

#4094⋊C₃

C₂²

yes

—

D₃

yes

characteristic

D₅⋊M₃₂

—

C₂×D₅⋊C₁₆

—

#64932

—

#64898

—

C₅×#41⋊C₃

C₂×C₄

yes

characteristic

D₅×SL(2,𝔽₃)

C₈

yes

characteristic

C₅⋊(#41⋊C₃)

C₈

yes

characteristic

C₃×C₈×D₅

—

C₃⋊#674

—

C₂×#131⋊C₃

—

#4628

—

D₅×#41

Dic₃

yes

characteristic

C₅×#131

Dic₃

yes

characteristic

#4094

C₂×D₃

yes

characteristic

#4053

—

D₅×M₁₆

—

#4047

—

C₂×C₈×D₅

—

#3998

—

C₂×#674

—

#3990

—

D₅⋊C₁₆

—

D₅⋊C₁₆

—

D₅⋊C₁₆

—

C₂×#350

—

Sylow

C₃×C₄×D₅

—

C₅×SL(2,𝔽₃)

C₂×C₈

yes

characteristic

C₃×C₅×C₈

—

C₃×#168

—

C₂×#41⋊C₃

—

—

yes

C₂×C₃⋊C₁₆

—

C₅×#41

C₂×Dic₃

yes

characteristic

#713

C₃⋊C₈

yes

characteristic

Q₈×D₅

C₃⋊C₈

yes

characteristic

#711

—

D₄×D₅

—

#709

—

C₂×C₄×D₅

—

C₅×M₁₆

—

C₂×C₅×C₈

—

#681

—

C₂×#168

—

#676

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

S₄

yes

—

—

—

C₂×M₃₂

—

C₂²×C₁₆

—

#350

—

#350

—

C₂×C₃×D₅

—

C₃×C₄×C₅

—

C₃×Dic₅

—

—

yes

C₂×SL(2,𝔽₃)

—

C₂×C₃×C₈

—

C₃⋊C₁₆

—

C₂²×D₅

—

C₅×Q₈

C₂×C₃⋊C₈

yes

characteristic

C₅×D₄

—

C₂×C₄×C₅

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

SL(2,ℤ/4)

yes

characteristic

Dic₁₀

—

C₅×C₈

—

C₅×C₈

yes

—

C₂×S₄

yes

characteristic

C₂×#41

—

—

—

—

ΓL(1,𝔽₁₆)

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₃×C₅

—

104

C₃×D₅

—

105

C₂×C₃×C₄

—

106

SL(2,𝔽₃)

D₅⋊C₈

yes

characteristic

107

C₃×C₈

—

108

C₃×C₈

—

109

C₂²×C₅

—

110

C₂×D₅

—

111

C₂×D₅

—

112

C₂×D₅

A₄⋊C₈

yes

characteristic

113

C₄×C₅

—

114

C₄×C₅

C₂×SL(2,ℤ/4)

yes

characteristic

115

Dic₅

—

116

Dic₅

A₄⋊C₈

yes

characteristic

117

#41

—

118

#41

—

119

#41

—

120

#41

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

121

C₂×Q₈

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₁₆

—

134

C₁₆

—

135

C₃×C₅

—

Hall

136

C₂²×C₃

—

137

C₃×C₄

—

138

C₃×C₄

—

139

C₂×C₅

—

140

C₂×C₅

C₂×A₄⋊C₈

yes

characteristic

141

D₅

—

142

D₅

#4628

yes

complemented

143

C₂³

—

144

Q₈

—

145

Q₈

D₅⋊(C₃⋊C₈)

yes

characteristic

146

D₄

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

A₄⋊AGL(1,𝔽₅)

yes

characteristic

159

C₂×C₃

—

160

C₂×C₃

—

161

C₅

C₂×#4628

yes

characteristic, complemented, Sylow

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

C₂×A₄⋊AGL(1,𝔽₅)

yes

characteristic

170

C₃

—

Sylow

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

D₅⋊(A₄⋊C₈)

yes

characteristic

	subgroup	G/H
center, hypercenter	C₈	A₄⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×SL(2,𝔽₃)	C₂×C₈
Frattini	C₄	C₂×A₄⋊AGL(1,𝔽₅)
Fitting	C₅×#131	Dic₃
socle	C₂×C₅	C₂×A₄⋊C₈
prime radical	D₅×#41⋊C₃	C₄
cosocle	#4094⋊C₃	C₂²
LCS	G ▷ C₅×SL(2,𝔽₃)
UCS	C₁ ◁ C₈

subgroup

G/H

center, hypercenter

C₈

A₄⋊AGL(1,𝔽₅)

commutator, hypocenter

C₅×SL(2,𝔽₃)

C₂×C₈

Frattini

C₄

C₂×A₄⋊AGL(1,𝔽₅)

Fitting

C₅×#131

Dic₃

socle

C₂×C₅

C₂×A₄⋊C₈

prime radical

D₅×#41⋊C₃

C₄

C₂²

G ▷ C₅×SL(2,𝔽₃)

C₁ ◁ C₈

N	H ≅ G/N	type
2. #11751558	C₂	semidirect
142. D₅	#4628	semidirect
161. C₅	C₂×#4628	semidirect

H ≅ G/N

type

2. #11751558

C₂

semidirect

142. D₅

#4628

semidirect

161. C₅

C₂×#4628

semidirect

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

subgroup	index
D₅×#131⋊C₃	2
#11751558	2
D₅⋊#350	3
D₅⋊(C₃⋊C₁₆)	4
C₂×#4628	5

subgroup

index

D₅×#131⋊C₃

#11751558

D₅⋊#350

D₅⋊(C₃⋊C₁₆)

C₂×#4628

Subgroups — D₅⋊#4628 #14512465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D5⋊#4628 #14512465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅⋊#4628 #14512465