Group #14510736 — subgroups

Subgroups — C₅×#73012 #14510736

Order 1920.

182 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 44 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	C₅×#5391	C₂	1	1	yes	maximal, characteristic
2	C₅×C₄⋊#226	C₂	1	1	yes	maximal, characteristic
3	C₅×C₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
4	C₅×#1672	—	3	1	no	maximal, Hall
5	C₂×C₅×#226	—	2	1	no	—
6	C₄×C₅×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₅×#859	—	2	1	no	—
8	C₅×#817	—	4	1	no	maximal
9	#73012	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₅×#498	—	3	1	no	—
11	C₅×#477	—	3	1	no	—
12	C₅×#462	—	3	1	no	—
13	C₅×C₈×Q₈	D₃	1	1	yes	characteristic
14	C₅×#334	—	3	1	no	—
15	C₅×#333	—	3	1	no	—
16	C₅×#332	—	3	1	no	—
17	C₂×C₅×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
18	C₅×#226	—	4	1	no	—
19	C₂×C₃×C₅×C₈	—	4	1	no	—
20	C₅×C₄⋊Dic₃	—	4	1	no	—
21	C₅×C₄⋊Dic₃	—	4	1	no	—
22	#5391	C₂×C₅	1	1	yes	characteristic
23	C₄⋊#226	C₂×C₅	1	1	yes	characteristic
24	C₈×SL(2,𝔽₃)	C₂×C₅	1	1	yes	characteristic
25	C₂×C₅×Q₁₆	—	6	1	no	—
26	C₅×C₄⋊Q₈	—	3	1	no	—
27	C₅×C₄⋊Q₈	—	3	1	no	—
28	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
29	C₅×#107	—	6	1	no	—
30	C₅×#106	—	6	1	no	—
31	C₅×#106	—	6	1	no	—
32	C₅×C₄⋊C₈	—	3	1	no	—
33	C₅×C₄⋊C₈	—	3	1	no	—
34	C₅×C₄⋊C₈	—	3	1	no	—
35	C₅×#103	—	6	1	no	—
36	C₅×#103	—	6	1	no	—
37	C₄×C₅×C₈	—	3	1	no	—
38	#1672	—	3	1	no	Sylow
39	C₂×C₃×C₄×C₅	—	4	1	no	—
40	C₂×C₅×Dic₃	—	8	1	no	—
41	C₂×C₅×Dic₃	—	8	1	no	—
42	C₅×SL(2,𝔽₃)	SD₁₆	1	1	yes	characteristic
43	C₃×C₅×C₈	—	4	2	no	—
44	C₂×#226	—	2	1	no	—
45	C₄×SL(2,𝔽₃)	C₂²×C₅	1	1	yes	characteristic
46	#859	—	2	1	no	—
47	#817	—	4	1	no	—
48	C₂×C₅×Q₈	—	6	1	no	—
49	C₂×C₅×Q₈	—	6	1	no	—
50	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
51	C₅×Q₁₆	—	12	1	no	—
52	C₅×Q₁₆	—	12	1	no	—
53	C₂×C₅×C₈	—	3	1	no	—
54	C₂×C₅×C₈	—	6	1	no	—
55	C₂×C₅×C₈	—	6	1	no	—
56	C₂×C₅×C₈	S₄	1	1	yes	characteristic
57	C₅×C₄⋊C₄	—	3	1	no	—
58	C₅×C₄⋊C₄	—	6	1	no	—
59	C₅×C₄⋊C₄	—	6	1	no	—
60	C₅×C₄⋊C₄	—	6	1	no	—
61	C₅×C₄⋊C₄	—	6	1	no	—
62	C₄²×C₅	—	3	1	no	—
63	#498	—	3	1	no	—
64	#477	—	3	1	no	—
65	#462	—	3	1	no	—
66	C₈×Q₈	C₅×D₃	1	1	yes	characteristic
67	#334	—	3	1	no	—
68	#333	—	3	1	no	—
69	#332	—	3	1	no	—
70	C₂²×C₃×C₅	—	4	1	no	—
71	C₃×C₄×C₅	—	4	1	no	—
72	C₃×C₄×C₅	—	4	1	no	—
73	C₅×Dic₃	—	16	1	no	—
74	C₅×Dic₃	—	16	1	no	—
75	C₂×SL(2,𝔽₃)	C₅×D₄	1	1	yes	characteristic
76	#226	—	4	1	no	—
77	C₂×C₃×C₈	—	4	1	no	—
78	C₄⋊Dic₃	—	4	1	no	—
79	C₄⋊Dic₃	—	4	1	no	—
80	C₅×Q₈	—	3	1	no	—
81	C₅×Q₈	—	12	1	no	—
82	C₅×Q₈	—	12	1	no	—
83	C₅×Q₈	—	24	1	no	—
84	C₅×Q₈	#204	1	1	yes	characteristic
85	C₂×C₄×C₅	—	3	1	no	—
86	C₂×C₄×C₅	—	3	1	no	—
87	C₂×C₄×C₅	—	12	1	no	—
88	C₂×C₄×C₅	—	12	1	no	—
89	C₂×C₄×C₅	C₂×S₄	1	1	yes	characteristic
90	C₅×C₈	—	6	1	no	—
91	C₅×C₈	—	12	1	no	—
92	C₅×C₈	—	12	1	no	—
93	C₅×C₈	#226	1	2	yes	complemented
94	C₂×Q₁₆	—	6	1	no	—
95	C₄⋊Q₈	—	3	1	no	—
96	C₄⋊Q₈	—	3	1	no	—
97	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
98	#107	—	6	1	no	—
99	#106	—	6	1	no	—
100	#106	—	6	1	no	—
101	C₄⋊C₈	—	3	1	no	—
102	C₄⋊C₈	—	3	1	no	—
103	C₄⋊C₈	—	3	1	no	—
104	#103	—	6	1	no	—
105	#103	—	6	1	no	—
106	C₄×C₈	—	3	1	no	—
107	C₂×C₃×C₅	—	4	1	no	—
108	C₂×C₃×C₅	—	4	1	no	—
109	C₂×C₃×C₅	—	4	1	no	—
110	C₂×C₃×C₄	—	4	1	no	—
111	C₂×Dic₃	—	8	1	no	—
112	C₂×Dic₃	—	8	1	no	—
113	SL(2,𝔽₃)	C₅×SD₁₆	1	1	yes	characteristic
114	C₃×C₈	—	4	2	no	—
115	C₂²×C₅	C₄⋊S₄	1	1	yes	characteristic
116	C₄×C₅	—	3	1	no	—
117	C₄×C₅	—	3	1	no	—
118	C₄×C₅	—	6	1	no	—
119	C₄×C₅	—	24	1	no	—
120	C₄×C₅	—	24	1	no	—
121	C₄×C₅	C₂×#226	1	1	yes	characteristic
122	C₄×C₅	#984	1	1	yes	characteristic
123	C₂×Q₈	—	6	1	no	—
124	C₂×Q₈	—	6	1	no	—
125	C₂×Q₈	C₅×D₁₂	1	1	yes	characteristic
126	Q₁₆	—	12	1	no	—
127	Q₁₆	—	12	1	no	—
128	C₂×C₈	—	3	1	no	—
129	C₂×C₈	—	6	1	no	—
130	C₂×C₈	—	6	1	no	—
131	C₂×C₈	C₅×S₄	1	1	yes	characteristic
132	C₄⋊C₄	—	3	1	no	—
133	C₄⋊C₄	—	6	1	no	—
134	C₄⋊C₄	—	6	1	no	—
135	C₄⋊C₄	—	6	1	no	—
136	C₄⋊C₄	—	6	1	no	—
137	C₄²	—	3	1	no	—
138	C₃×C₅	—	4	1	no	Hall
139	C₂²×C₃	—	4	1	no	—
140	C₃×C₄	—	4	1	no	—
141	C₃×C₄	—	4	1	no	—
142	Dic₃	—	16	1	no	—
143	Dic₃	—	16	1	no	—
144	C₂×C₅	C₄⋊#226	1	1	yes	characteristic
145	C₂×C₅	#5401	1	1	yes	characteristic
146	C₂×C₅	#5406	1	1	yes	characteristic
147	Q₈	—	3	1	no	—
148	Q₈	—	12	1	no	—
149	Q₈	—	12	1	no	—
150	Q₈	—	24	1	no	—
151	Q₈	C₅×#204	1	1	yes	characteristic
152	C₂×C₄	—	3	1	no	—
153	C₂×C₄	—	3	1	no	—
154	C₂×C₄	—	12	1	no	—
155	C₂×C₄	—	12	1	no	—
156	C₂×C₄	C₂×C₅×S₄	1	1	yes	characteristic
157	C₈	—	6	1	no	—
158	C₈	—	12	1	no	—
159	C₈	—	12	1	no	—
160	C₈	C₅×#226	1	2	yes	complemented
161	C₂×C₃	—	4	1	no	—
162	C₂×C₃	—	4	1	no	—
163	C₂×C₃	—	4	1	no	—
164	C₅	#73012	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²	C₅×C₄⋊S₄	1	1	yes	characteristic
166	C₄	—	3	1	no	—
167	C₄	—	3	1	no	—
168	C₄	—	6	1	no	—
169	C₄	—	24	1	no	—
170	C₄	—	24	1	no	—
171	C₄	C₂×C₅×#226	1	1	yes	characteristic
172	C₄	C₅×#984	1	1	yes	characteristic
173	C₃	—	4	1	no	Sylow
174	C₂	C₅×C₄⋊#226	1	1	yes	characteristic
175	C₂	C₅×#5401	1	1	yes	characteristic
176	C₂	C₅×#5406	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#5391

C₂

yes

maximal, characteristic

C₅×C₄⋊#226

C₂

yes

maximal, characteristic

C₅×C₈×SL(2,𝔽₃)

C₂

yes

maximal, characteristic

C₅×#1672

—

maximal, Hall

C₂×C₅×#226

—

C₄×C₅×SL(2,𝔽₃)

C₂²

yes

—

—

C₅

yes

maximal, characteristic, direct product factor, Hall

C₅×#498

—

C₅×#477

—

C₅×#462

—

C₅×C₈×Q₈

D₃

yes

—

—

—

C₂×C₅×SL(2,𝔽₃)

D₄

yes

characteristic

C₅×#226

—

C₂×C₃×C₅×C₈

—

C₅×C₄⋊Dic₃

—

C₅×C₄⋊Dic₃

—

#5391

C₂×C₅

yes

characteristic

C₄⋊#226

C₂×C₅

yes

characteristic

C₈×SL(2,𝔽₃)

C₂×C₅

yes

characteristic

C₂×C₅×Q₁₆

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₄×C₅×Q₈

C₂×D₃

yes

—

—

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×#103

—

C₅×#103

—

C₄×C₅×C₈

—

#1672

—

Sylow

C₂×C₃×C₄×C₅

—

C₂×C₅×Dic₃

—

C₂×C₅×Dic₃

—

C₅×SL(2,𝔽₃)

SD₁₆

yes

characteristic

C₃×C₅×C₈

—

C₂×#226

—

C₄×SL(2,𝔽₃)

C₂²×C₅

yes

characteristic

#859

—

#817

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

D₁₂

yes

characteristic

C₅×Q₁₆

—

C₅×Q₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

S₄

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

#498

—

#477

—

#462

—

C₈×Q₈

C₅×D₃

yes

—

—

—

C₂²×C₃×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₅×Dic₃

—

C₅×Dic₃

—

C₂×SL(2,𝔽₃)

C₅×D₄

yes

characteristic

#226

—

C₂×C₃×C₈

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

#204

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×S₄

yes

characteristic

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

#226

yes

complemented

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×C₅×D₃

yes

characteristic

#107

—

#106

—

100

#106

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

#103

—

105

#103

—

106

C₄×C₈

—

107

C₂×C₃×C₅

—

108

C₂×C₃×C₅

—

109

C₂×C₃×C₅

—

110

C₂×C₃×C₄

—

111

C₂×Dic₃

—

112

C₂×Dic₃

—

113

SL(2,𝔽₃)

C₅×SD₁₆

yes

characteristic

114

C₃×C₈

—

115

C₂²×C₅

C₄⋊S₄

yes

characteristic

116

C₄×C₅

—

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

—

120

C₄×C₅

—

121

C₄×C₅

C₂×#226

yes

characteristic

122

C₄×C₅

#984

yes

characteristic

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

C₅×D₁₂

yes

characteristic

126

Q₁₆

—

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

C₅×S₄

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

—

138

C₃×C₅

—

Hall

139

C₂²×C₃

—

140

C₃×C₄

—

141

C₃×C₄

—

142

Dic₃

—

143

Dic₃

—

144

C₂×C₅

C₄⋊#226

yes

characteristic

145

C₂×C₅

#5401

yes

characteristic

146

C₂×C₅

#5406

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

Q₈

—

151

Q₈

C₅×#204

yes

characteristic

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

C₂×C₅×S₄

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

C₅×#226

yes

complemented

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

—

164

C₅

#73012

yes

characteristic, direct product factor, Sylow

165

C₂²

C₅×C₄⋊S₄

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

C₂×C₅×#226

yes

characteristic

172

C₄

C₅×#984

yes

characteristic

173

C₃

—

Sylow

174

C₂

C₅×C₄⋊#226

yes

characteristic

175

C₂

C₅×#5401

yes

characteristic

176

C₂

C₅×#5406

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₄⋊S₄
commutator, cosocle	C₄×SL(2,𝔽₃)	C₂²×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₅×C₈×Q₈	D₃
prime radical	C₂×C₅×SL(2,𝔽₃)	D₄
hypercenter	C₂×C₅×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₅×SD₁₆
LCS	G ▷ C₄×SL(2,𝔽₃) ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

subgroup

G/H

center, socle

C₂²×C₅

C₄⋊S₄

commutator, cosocle

C₄×SL(2,𝔽₃)

C₂²×C₅

Frattini

C₂×C₄

C₂×C₅×S₄

Fitting

C₅×C₈×Q₈

D₃

prime radical

C₂×C₅×SL(2,𝔽₃)

D₄

hypercenter

C₂×C₅×C₈

S₄

hypocenter

SL(2,𝔽₃)

C₅×SD₁₆

LCS

G ▷ C₄×SL(2,𝔽₃) ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

N	H ≅ G/N	type
9. #73012	C₅	direct (N × H)
93. C₅×C₈	#226	semidirect
160. C₈	C₅×#226	semidirect
164. C₅	#73012	direct (N × H)

H ≅ G/N

type

9. #73012

C₅

direct (N × H)

93. C₅×C₈

#226

semidirect

160. C₈

C₅×#226

semidirect

164. C₅

#73012

direct (N × H)

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

subgroup	index
C₅×#5391	2
C₅×C₄⋊#226	2
C₅×C₈×SL(2,𝔽₃)	2
C₅×#1672	3
C₅×#817	4
#73012	5

subgroup

index

C₅×#5391

C₅×C₄⋊#226

C₅×C₈×SL(2,𝔽₃)

C₅×#1672

C₅×#817

#73012

Subgroups — C₅×#73012 #14510736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#73012 #14510736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#73012 #14510736