Group #14510735 — subgroups

Subgroups — C₅×#73011 #14510735

Order 1920.

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

normal subgroups only

This group has 188 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	C₅×C₄⋊#226	C₂	1	1	yes	maximal, characteristic
2	C₅×#5386	C₂	1	1	yes	maximal, characteristic
3	C₅×C₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
4	C₅×#1680	—	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×#816	—	4	1	no	maximal
9	#73011	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₅×#487	—	3	1	no	—
11	C₅×#475	—	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×#332	—	3	1	no	—
16	C₅×#332	—	3	1	no	—
17	C₂×C₅×Dic₆	—	4	1	no	—
18	C₂×C₅×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
19	C₅×#226	—	4	1	no	—
20	C₂×C₃×C₅×C₈	—	4	1	no	—
21	C₅×C₄⋊Dic₃	—	4	1	no	—
22	C₄⋊#226	C₂×C₅	1	1	yes	characteristic
23	#5386	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×#106	—	6	1	no	—
30	C₅×C₄⋊C₈	—	3	1	no	—
31	C₅×C₄⋊C₈	—	3	1	no	—
32	C₅×C₄⋊C₈	—	3	1	no	—
33	C₅×#103	—	6	1	no	—
34	C₅×#103	—	6	1	no	—
35	C₅×#103	—	6	1	no	—
36	C₅×#103	—	6	1	no	—
37	C₄×C₅×C₈	—	3	1	no	—
38	#1680	—	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×Dic₆	—	4	2	no	—
43	C₅×Dic₆	—	8	1	no	—
44	C₅×SL(2,𝔽₃)	SD₁₆	1	1	yes	characteristic
45	C₃×C₅×C₈	—	8	1	no	—
46	C₂×#226	—	2	1	no	—
47	C₄×SL(2,𝔽₃)	C₂²×C₅	1	1	yes	characteristic
48	#859	—	2	1	no	—
49	#816	—	4	1	no	—
50	C₂×C₅×Q₈	—	6	1	no	—
51	C₂×C₅×Q₈	—	6	1	no	—
52	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
53	C₅×Q₁₆	—	12	1	no	—
54	C₅×Q₁₆	—	12	1	no	—
55	C₂×C₅×C₈	—	3	1	no	—
56	C₂×C₅×C₈	—	6	1	no	—
57	C₂×C₅×C₈	—	6	1	no	—
58	C₂×C₅×C₈	S₄	1	1	yes	characteristic
59	C₅×C₄⋊C₄	—	3	1	no	—
60	C₅×C₄⋊C₄	—	6	1	no	—
61	C₅×C₄⋊C₄	—	6	1	no	—
62	C₅×C₄⋊C₄	—	6	1	no	—
63	C₅×C₄⋊C₄	—	6	1	no	—
64	C₄²×C₅	—	3	1	no	—
65	#487	—	3	1	no	—
66	#475	—	3	1	no	—
67	#462	—	3	1	no	—
68	C₈×Q₈	C₅×D₃	1	1	yes	characteristic
69	#334	—	3	1	no	—
70	#332	—	3	1	no	—
71	#332	—	3	1	no	—
72	C₂²×C₃×C₅	—	4	1	no	—
73	C₃×C₄×C₅	—	4	1	no	—
74	C₃×C₄×C₅	—	4	1	no	—
75	C₅×Dic₃	—	8	2	no	—
76	C₅×Dic₃	—	16	1	no	—
77	C₂×Dic₆	—	4	1	no	—
78	C₂×SL(2,𝔽₃)	C₅×D₄	1	1	yes	characteristic
79	#226	—	4	1	no	—
80	C₂×C₃×C₈	—	4	1	no	—
81	C₄⋊Dic₃	—	4	1	no	—
82	C₅×Q₈	—	3	1	no	—
83	C₅×Q₈	—	12	1	no	—
84	C₅×Q₈	—	12	1	no	—
85	C₅×Q₈	—	12	1	no	—
86	C₅×Q₈	—	12	1	no	—
87	C₅×Q₈	#204	1	1	yes	characteristic
88	C₂×C₄×C₅	—	3	1	no	—
89	C₂×C₄×C₅	—	3	1	no	—
90	C₂×C₄×C₅	—	12	1	no	—
91	C₂×C₄×C₅	—	12	1	no	—
92	C₂×C₄×C₅	C₂×S₄	1	1	yes	characteristic
93	C₅×C₈	—	2	1	no	—
94	C₅×C₈	—	6	1	no	—
95	C₅×C₈	—	12	1	no	—
96	C₅×C₈	—	12	1	no	—
97	C₂×Q₁₆	—	6	1	no	—
98	C₄⋊Q₈	—	3	1	no	—
99	C₄⋊Q₈	—	3	1	no	—
100	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
101	#106	—	6	1	no	—
102	C₄⋊C₈	—	3	1	no	—
103	C₄⋊C₈	—	3	1	no	—
104	C₄⋊C₈	—	3	1	no	—
105	#103	—	6	1	no	—
106	#103	—	6	1	no	—
107	#103	—	6	1	no	—
108	#103	—	6	1	no	—
109	C₄×C₈	—	3	1	no	—
110	C₂×C₃×C₅	—	4	1	no	—
111	C₂×C₃×C₅	—	4	1	no	—
112	C₂×C₃×C₅	—	4	1	no	—
113	C₂×C₃×C₄	—	4	1	no	—
114	C₂×Dic₃	—	8	1	no	—
115	C₂×Dic₃	—	8	1	no	—
116	Dic₆	—	4	2	no	—
117	Dic₆	—	8	1	no	—
118	SL(2,𝔽₃)	C₅×SD₁₆	1	1	yes	characteristic
119	C₃×C₈	—	8	1	no	—
120	C₂²×C₅	C₄⋊S₄	1	1	yes	characteristic
121	C₄×C₅	—	3	1	no	—
122	C₄×C₅	—	3	1	no	—
123	C₄×C₅	—	6	1	no	—
124	C₄×C₅	—	24	1	no	—
125	C₄×C₅	—	24	1	no	—
126	C₄×C₅	#983	1	1	yes	characteristic
127	C₄×C₅	#985	1	1	yes	characteristic
128	C₂×Q₈	—	6	1	no	—
129	C₂×Q₈	—	6	1	no	—
130	C₂×Q₈	C₅×D₁₂	1	1	yes	characteristic
131	Q₁₆	—	12	1	no	—
132	Q₁₆	—	12	1	no	—
133	C₂×C₈	—	3	1	no	—
134	C₂×C₈	—	6	1	no	—
135	C₂×C₈	—	6	1	no	—
136	C₂×C₈	C₅×S₄	1	1	yes	characteristic
137	C₄⋊C₄	—	3	1	no	—
138	C₄⋊C₄	—	6	1	no	—
139	C₄⋊C₄	—	6	1	no	—
140	C₄⋊C₄	—	6	1	no	—
141	C₄⋊C₄	—	6	1	no	—
142	C₄²	—	3	1	no	—
143	C₃×C₅	—	4	1	no	Hall
144	C₂²×C₃	—	4	1	no	—
145	C₃×C₄	—	4	1	no	—
146	C₃×C₄	—	4	1	no	—
147	Dic₃	—	8	2	no	—
148	Dic₃	—	16	1	no	—
149	C₂×C₅	#5395	1	1	yes	characteristic
150	C₂×C₅	#5401	1	1	yes	characteristic
151	C₂×C₅	#5403	1	1	yes	characteristic
152	Q₈	—	3	1	no	—
153	Q₈	—	12	1	no	—
154	Q₈	—	12	1	no	—
155	Q₈	—	12	1	no	—
156	Q₈	—	12	1	no	—
157	Q₈	C₅×#204	1	1	yes	characteristic
158	C₂×C₄	—	3	1	no	—
159	C₂×C₄	—	3	1	no	—
160	C₂×C₄	—	12	1	no	—
161	C₂×C₄	—	12	1	no	—
162	C₂×C₄	C₂×C₅×S₄	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	6	1	no	—
165	C₈	—	12	1	no	—
166	C₈	—	12	1	no	—
167	C₂×C₃	—	4	1	no	—
168	C₂×C₃	—	4	1	no	—
169	C₂×C₃	—	4	1	no	—
170	C₅	#73011	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²	C₅×C₄⋊S₄	1	1	yes	characteristic
172	C₄	—	3	1	no	—
173	C₄	—	3	1	no	—
174	C₄	—	6	1	no	—
175	C₄	—	24	1	no	—
176	C₄	—	24	1	no	—
177	C₄	C₅×#983	1	1	yes	characteristic
178	C₄	C₅×#985	1	1	yes	characteristic
179	C₃	—	4	1	no	Sylow
180	C₂	C₅×#5395	1	1	yes	characteristic
181	C₂	C₅×#5401	1	1	yes	characteristic
182	C₂	C₅×#5403	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×C₄⋊#226

C₂

yes

maximal, characteristic

C₅×#5386

C₂

yes

maximal, characteristic

C₅×C₈×SL(2,𝔽₃)

C₂

yes

maximal, characteristic

C₅×#1680

—

maximal, Hall

C₂×C₅×#226

—

C₄×C₅×SL(2,𝔽₃)

C₂²

yes

—

—

C₅

yes

maximal, characteristic, direct product factor, Hall

C₅×#487

—

C₅×#475

—

C₅×#462

—

C₅×C₈×Q₈

D₃

yes

—

—

—

C₂×C₅×Dic₆

—

C₂×C₅×SL(2,𝔽₃)

D₄

yes

characteristic

C₅×#226

—

C₂×C₃×C₅×C₈

—

C₅×C₄⋊Dic₃

—

C₄⋊#226

C₂×C₅

yes

characteristic

#5386

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

characteristic

C₅×#106

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

—

—

—

—

C₄×C₅×C₈

—

#1680

—

Sylow

C₂×C₃×C₄×C₅

—

C₂×C₅×Dic₃

—

C₂×C₅×Dic₃

—

C₅×Dic₆

—

C₅×Dic₆

—

C₅×SL(2,𝔽₃)

SD₁₆

yes

characteristic

C₃×C₅×C₈

—

C₂×#226

—

C₄×SL(2,𝔽₃)

C₂²×C₅

yes

characteristic

#859

—

#816

—

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

—

#487

—

#475

—

#462

—

C₈×Q₈

C₅×D₃

yes

—

—

—

C₂²×C₃×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₅×Dic₃

—

C₅×Dic₃

—

C₂×Dic₆

—

C₂×SL(2,𝔽₃)

C₅×D₄

yes

characteristic

#226

—

C₂×C₃×C₈

—

C₄⋊Dic₃

—

C₅×Q₈

—

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

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₄×Q₈

C₂×C₅×D₃

yes

characteristic

101

#106

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

106

#103

—

107

#103

—

108

#103

—

109

C₄×C₈

—

110

C₂×C₃×C₅

—

111

C₂×C₃×C₅

—

112

C₂×C₃×C₅

—

113

C₂×C₃×C₄

—

114

C₂×Dic₃

—

115

C₂×Dic₃

—

116

Dic₆

—

117

Dic₆

—

118

SL(2,𝔽₃)

C₅×SD₁₆

yes

characteristic

119

C₃×C₈

—

120

C₂²×C₅

C₄⋊S₄

yes

characteristic

121

C₄×C₅

—

122

C₄×C₅

—

123

C₄×C₅

—

124

C₄×C₅

—

125

C₄×C₅

—

126

C₄×C₅

#983

yes

characteristic

127

C₄×C₅

#985

yes

characteristic

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

C₅×D₁₂

yes

characteristic

131

Q₁₆

—

132

Q₁₆

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

C₅×S₄

yes

characteristic

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

—

143

C₃×C₅

—

Hall

144

C₂²×C₃

—

145

C₃×C₄

—

146

C₃×C₄

—

147

Dic₃

—

148

Dic₃

—

149

C₂×C₅

#5395

yes

characteristic

150

C₂×C₅

#5401

yes

characteristic

151

C₂×C₅

#5403

yes

characteristic

152

Q₈

—

153

Q₈

—

154

Q₈

—

155

Q₈

—

156

Q₈

—

157

Q₈

C₅×#204

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂×C₅×S₄

yes

characteristic

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₂×C₃

—

168

C₂×C₃

—

169

C₂×C₃

—

170

C₅

#73011

yes

characteristic, direct product factor, Sylow

171

C₂²

C₅×C₄⋊S₄

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

C₅×#983

yes

characteristic

178

C₄

C₅×#985

yes

characteristic

179

C₃

—

Sylow

180

C₂

C₅×#5395

yes

characteristic

181

C₂

C₅×#5401

yes

characteristic

182

C₂

C₅×#5403

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. #73011	C₅	direct (N × H)
170. C₅	#73011	direct (N × H)

H ≅ G/N

type

9. #73011

C₅

direct (N × H)

170. C₅

#73011

direct (N × H)

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

subgroup	index
C₅×C₄⋊#226	2
C₅×#5386	2
C₅×C₈×SL(2,𝔽₃)	2
C₅×#1680	3
C₅×#816	4
#73011	5

subgroup

index

C₅×C₄⋊#226

C₅×C₈×SL(2,𝔽₃)

Subgroups — C₅×#73011 #14510735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#73011 #14510735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#73011 #14510735