Group #14312422 — subgroups

Subgroups — C₅×#68701 #14312422

Order 1920.

316 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 118 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×#5125	C₂	1	1	yes	maximal, characteristic
2	C₅×#5118	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
4	C₅×#4533	C₂	1	2	yes	maximal
5	C₅×#4532	C₂	1	2	yes	maximal
6	C₅×#2102	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	C₂²	1	2	yes	—
9	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₅×#208	C₂²	1	2	yes	—
11	C₂×C₅×#208	C₂²	1	1	yes	characteristic
12	C₅×#819	—	2	2	no	—
13	C₅×Dic₃⋊C₈	—	2	2	no	—
14	#68701	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅×#430	—	3	1	no	—
16	C₅×#423	—	3	1	no	—
17	C₅×#344	—	3	2	no	—
18	C₅×#339	—	3	2	no	—
19	C₅×H(3,ℤ/4)	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×C₅×Dic₃	—	2	1	no	—
22	C₃×C₅×M₁₆	—	2	2	no	—
23	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
24	C₃×C₅×C₄⋊C₄	—	2	2	no	—
25	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
26	C₃×C₄²×C₅	—	2	2	no	—
27	C₄×C₅×Dic₃	—	2	1	no	—
28	C₄×C₅×Dic₃	Q₈	1	1	yes	characteristic
29	C₄×C₅×Dic₃	Q₈	1	1	yes	characteristic
30	C₅×#208	—	2	2	no	—
31	C₅×#208	—	2	2	no	—
32	C₅×#208	—	2	2	no	—
33	C₂×C₅×C₃⋊C₈	—	2	1	no	—
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	#5125	C₂×C₅	1	1	yes	characteristic
36	#5118	C₂×C₅	1	1	yes	characteristic
37	C₃×H(3,ℤ/4)	C₂×C₅	1	1	yes	characteristic
38	#4533	C₂×C₅	1	2	yes	—
39	#4532	C₂×C₅	1	2	yes	—
40	C₂×C₅×M₁₆	—	3	2	no	—
41	C₂×C₅×M₁₆	—	3	1	no	—
42	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
43	C₅×#117	C₂×D₃	1	2	yes	—
44	C₂×C₄²×C₅	—	3	1	no	—
45	C₅×#108	—	6	2	no	—
46	C₅×C₄⋊C₈	—	6	2	no	—
47	#2102	—	3	1	no	Sylow
48	C₂³×C₃×C₅	#41	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	—	4	2	no	—
50	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	#41	1	2	yes	—
53	C₂×C₃×C₄×C₅	#41	1	2	yes	—
54	C₂×C₅×Dic₃	—	2	1	no	—
55	C₂×C₅×Dic₃	—	2	1	no	—
56	C₂×C₅×Dic₃	—	4	2	no	—
57	C₃×C₅×C₈	—	2	2	no	—
58	C₅×C₃⋊C₈	—	4	1	no	—
59	C₅×C₃⋊C₈	—	4	2	no	—
60	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
61	C₃×#117	C₂²×C₅	1	2	yes	—
62	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
63	C₂×#208	C₂²×C₅	1	2	yes	—
64	C₂×#208	C₂²×C₅	1	1	yes	characteristic
65	#819	—	2	2	no	—
66	Dic₃⋊C₈	—	2	2	no	—
67	C₂²×C₄×C₅	—	6	1	no	—
68	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
69	C₅×M₁₆	—	2	2	no	—
70	C₅×M₁₆	—	6	2	no	—
71	C₅×M₁₆	—	6	2	no	—
72	C₅×M₁₆	—	6	2	no	—
73	C₂×C₅×C₈	—	6	1	no	—
74	C₂×C₅×C₈	—	6	2	no	—
75	C₂×C₅×C₈	D₁₂	1	2	yes	—
76	C₅×C₄⋊C₄	—	2	2	no	—
77	C₅×C₂²⋊C₄	—	2	2	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	3	1	no	—
80	C₄²×C₅	—	3	1	no	—
81	C₄²×C₅	—	6	1	no	—
82	#430	—	3	1	no	—
83	#423	—	3	1	no	—
84	#344	—	3	2	no	—
85	#339	—	3	2	no	—
86	H(3,ℤ/4)	C₅×D₃	1	1	yes	characteristic
87	C₂²×C₃×C₅	—	4	1	no	—
88	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
89	C₂²×C₃×C₅	#126	1	2	yes	—
90	C₃×C₄×C₅	—	8	2	no	—
91	C₃×C₄×C₅	#123	1	1	yes	characteristic
92	C₃×C₄×C₅	#124	1	1	yes	characteristic
93	C₃×C₄×C₅	#125	1	2	yes	—
94	C₅×Dic₃	—	2	2	no	—
95	C₅×Dic₃	—	4	1	no	—
96	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
97	C₂²×Dic₃	—	2	1	no	—
98	C₃×M₁₆	—	2	2	no	—
99	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
100	C₃×C₄⋊C₄	—	2	2	no	—
101	C₃×C₂²⋊C₄	—	2	2	no	—
102	C₃×C₄²	—	2	2	no	—
103	C₄×Dic₃	—	2	1	no	—
104	C₄×Dic₃	C₅×Q₈	1	1	yes	characteristic
105	C₄×Dic₃	C₅×Q₈	1	1	yes	characteristic
106	#208	—	2	2	no	—
107	#208	—	2	2	no	—
108	#208	—	2	2	no	—
109	C₂×C₃⋊C₈	—	2	1	no	—
110	C₂×C₃⋊C₈	—	2	2	no	—
111	C₂³×C₅	#237	1	1	yes	characteristic
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	—	6	1	no	—
114	C₂×C₄×C₅	—	6	1	no	—
115	C₂×C₄×C₅	—	12	2	no	—
116	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
117	C₂×C₄×C₅	#235	1	2	yes	—
118	C₂×C₄×C₅	#237	1	2	yes	—
119	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
120	C₅×C₈	—	2	2	no	—
121	C₅×C₈	—	12	2	no	—
122	C₅×C₈	—	12	1	no	—
123	C₂×M₁₆	—	3	2	no	—
124	C₂×M₁₆	—	3	1	no	—
125	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
126	#117	C₂×C₅×D₃	1	2	yes	—
127	C₂×C₄²	—	3	1	no	—
128	#108	—	6	2	no	—
129	C₄⋊C₈	—	6	2	no	—
130	C₂×C₃×C₅	—	2	1	no	—
131	C₂×C₃×C₅	—	2	2	no	—
132	C₂×C₃×C₅	#400	1	1	yes	characteristic
133	C₂³×C₃	C₅×#41	1	1	yes	characteristic
134	C₂×C₃×C₄	—	4	2	no	—
135	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
137	C₂×C₃×C₄	C₅×#41	1	2	yes	—
138	C₂×C₃×C₄	C₅×#41	1	2	yes	—
139	C₂×Dic₃	—	2	1	no	—
140	C₂×Dic₃	—	2	1	no	—
141	C₂×Dic₃	—	4	2	no	—
142	C₃×C₈	—	2	2	no	—
143	C₃⋊C₈	—	4	1	no	—
144	C₃⋊C₈	—	4	2	no	—
145	C₂²×C₅	—	4	1	no	—
146	C₂²×C₅	#879	1	2	yes	—
147	C₂²×C₅	#897	1	1	yes	characteristic
148	C₄×C₅	—	6	2	no	—
149	C₄×C₅	—	8	2	no	—
150	C₄×C₅	—	12	1	no	—
151	C₄×C₅	#875	1	1	yes	characteristic
152	C₄×C₅	#886	1	1	yes	characteristic
153	C₄×C₅	#889	1	2	yes	—
154	C₂²×C₄	—	6	1	no	—
155	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
156	M₁₆	—	2	2	no	—
157	M₁₆	—	6	2	no	—
158	M₁₆	—	6	2	no	—
159	M₁₆	—	6	2	no	—
160	C₂×C₈	—	6	1	no	—
161	C₂×C₈	—	6	2	no	—
162	C₂×C₈	C₅×D₁₂	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	3	1	no	—
167	C₄²	—	3	1	no	—
168	C₄²	—	6	1	no	—
169	C₃×C₅	#2102	1	1	yes	characteristic, complemented, Hall
170	C₂²×C₃	—	4	1	no	—
171	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
172	C₂²×C₃	C₅×#126	1	2	yes	—
173	C₃×C₄	—	8	2	no	—
174	C₃×C₄	C₅×#123	1	1	yes	characteristic
175	C₃×C₄	C₅×#124	1	1	yes	characteristic
176	C₃×C₄	C₅×#125	1	2	yes	—
177	Dic₃	—	2	2	no	—
178	Dic₃	—	4	1	no	—
179	C₂×C₅	—	2	1	no	—
180	C₂×C₅	—	2	2	no	—
181	C₂×C₅	#4659	1	1	yes	characteristic
182	C₂³	C₅×#237	1	1	yes	characteristic
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	6	1	no	—
185	C₂×C₄	—	6	1	no	—
186	C₂×C₄	—	12	2	no	—
187	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
188	C₂×C₄	C₅×#235	1	2	yes	—
189	C₂×C₄	C₅×#237	1	2	yes	—
190	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	12	2	no	—
193	C₈	—	12	1	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	2	2	no	—
196	C₂×C₃	C₅×#400	1	1	yes	characteristic
197	C₅	#68701	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²	—	4	1	no	—
199	C₂²	C₅×#879	1	2	yes	—
200	C₂²	C₅×#897	1	1	yes	characteristic
201	C₄	—	6	2	no	—
202	C₄	—	8	2	no	—
203	C₄	—	12	1	no	—
204	C₄	C₅×#875	1	1	yes	characteristic
205	C₄	C₅×#886	1	1	yes	characteristic
206	C₄	C₅×#889	1	2	yes	—
207	C₃	C₅×#2102	1	1	yes	characteristic, complemented, Sylow
208	C₂	—	2	1	no	—
209	C₂	—	2	2	no	—
210	C₂	C₅×#4659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#5125

C₂

yes

maximal, characteristic

C₅×#5118

C₂

yes

maximal, characteristic

C₃×C₅×H(3,ℤ/4)

C₂

yes

maximal, characteristic

C₅×#4533

C₂

yes

maximal

C₅×#4532

C₂

yes

maximal

C₅×#2102

—

maximal, Hall

C₂×C₃×C₅×M₁₆

C₂²

yes

characteristic

C₃×C₅×#117

C₂²

yes

—

C₂×C₄×C₅×Dic₃

C₂²

yes

characteristic

C₂×C₅×#208

C₂²

yes

—

C₂×C₅×#208

C₂²

yes

characteristic

C₅×#819

—

C₅×Dic₃⋊C₈

—

#68701

C₅

yes

maximal, characteristic, direct product factor, Hall

—

—

—

—

D₃

yes

characteristic

C₂²×C₃×C₄×C₅

C₂³

yes

characteristic

C₂²×C₅×Dic₃

—

C₃×C₅×M₁₆

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₃×C₅×C₄⋊C₄

—

C₃×C₅×C₂²⋊C₄

—

C₃×C₄²×C₅

—

C₄×C₅×Dic₃

—

C₄×C₅×Dic₃

Q₈

yes

characteristic

C₄×C₅×Dic₃

Q₈

yes

—

—

—

C₂×C₅×C₃⋊C₈

—

C₂×C₅×C₃⋊C₈

—

#5125

C₂×C₅

yes

characteristic

#5118

C₂×C₅

yes

characteristic

C₃×H(3,ℤ/4)

C₂×C₅

yes

characteristic

#4533

C₂×C₅

yes

—

#4532

C₂×C₅

yes

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

C₂×D₃

yes

characteristic

C₅×#117

C₂×D₃

yes

—

C₂×C₄²×C₅

—

C₅×#108

—

C₅×C₄⋊C₈

—

—

C₂³×C₃×C₅

yes

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×Q₈

yes

characteristic

C₂×C₃×C₄×C₅

#41

yes

—

C₂×C₃×C₄×C₅

#41

yes

—

C₂×C₅×Dic₃

—

C₂×C₅×Dic₃

—

C₂×C₅×Dic₃

—

C₃×C₅×C₈

—

C₅×C₃⋊C₈

—

C₅×C₃⋊C₈

—

C₂×C₃×M₁₆

C₂²×C₅

yes

characteristic

C₃×#117

C₂²×C₅

yes

—

C₂×C₄×Dic₃

C₂²×C₅

yes

characteristic

C₂×#208

C₂²×C₅

yes

—

C₂×#208

C₂²×C₅

yes

characteristic

#819

—

Dic₃⋊C₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²×D₃

yes

characteristic

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

D₁₂

yes

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

—

—

—

—

C₅×D₃

yes

characteristic

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂²⋊Q₈

yes

characteristic

C₂²×C₃×C₅

#126

yes

—

C₃×C₄×C₅

—

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

—

C₅×Dic₃

—

C₅×Dic₃

—

C₂²×C₃×C₄

C₂³×C₅

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₂×C₃×C₈

C₅×D₄

yes

—

100

C₃×C₄⋊C₄

—

101

C₃×C₂²⋊C₄

—

102

C₃×C₄²

—

103

C₄×Dic₃

—

104

C₄×Dic₃

C₅×Q₈

yes

characteristic

105

C₄×Dic₃

C₅×Q₈

yes

characteristic

106

#208

—

107

#208

—

108

#208

—

109

C₂×C₃⋊C₈

—

110

C₂×C₃⋊C₈

—

111

C₂³×C₅

#237

yes

characteristic

112

C₂×C₄×C₅

—

113

C₂×C₄×C₅

—

114

C₂×C₄×C₅

—

115

C₂×C₄×C₅

—

116

C₂×C₄×C₅

C₂×D₁₂

yes

characteristic

117

C₂×C₄×C₅

#235

yes

—

118

C₂×C₄×C₅

#237

yes

—

119

C₂×C₄×C₅

D₃×Q₈

yes

characteristic

120

C₅×C₈

—

121

C₅×C₈

—

122

C₅×C₈

—

123

C₂×M₁₆

—

124

C₂×M₁₆

—

125

C₂×M₁₆

C₂×C₅×D₃

yes

characteristic

126

#117

C₂×C₅×D₃

yes

—

127

C₂×C₄²

—

128

#108

—

129

C₄⋊C₈

—

130

C₂×C₃×C₅

—

131

C₂×C₃×C₅

—

132

C₂×C₃×C₅

#400

yes

characteristic

133

C₂³×C₃

C₅×#41

yes

characteristic

134

C₂×C₃×C₄

—

135

C₂×C₃×C₄

C₂×C₅×D₄

yes

characteristic

136

C₂×C₃×C₄

C₂×C₅×Q₈

yes

characteristic

137

C₂×C₃×C₄

C₅×#41

yes

—

138

C₂×C₃×C₄

C₅×#41

yes

—

139

C₂×Dic₃

—

140

C₂×Dic₃

—

141

C₂×Dic₃

—

142

C₃×C₈

—

143

C₃⋊C₈

—

144

C₃⋊C₈

—

145

C₂²×C₅

—

146

C₂²×C₅

#879

yes

—

147

C₂²×C₅

#897

yes

characteristic

148

C₄×C₅

—

149

C₄×C₅

—

150

C₄×C₅

—

151

C₄×C₅

#875

yes

characteristic

152

C₄×C₅

#886

yes

characteristic

153

C₄×C₅

#889

yes

—

154

C₂²×C₄

—

155

C₂²×C₄

C₂²×C₅×D₃

yes

characteristic

156

M₁₆

—

157

M₁₆

—

158

M₁₆

—

159

M₁₆

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

C₅×D₁₂

yes

—

163

C₄⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

169

C₃×C₅

#2102

yes

characteristic, complemented, Hall

170

C₂²×C₃

—

171

C₂²×C₃

C₅×C₂²⋊Q₈

yes

characteristic

172

C₂²×C₃

C₅×#126

yes

—

173

C₃×C₄

—

174

C₃×C₄

C₅×#123

yes

characteristic

175

C₃×C₄

C₅×#124

yes

characteristic

176

C₃×C₄

C₅×#125

yes

—

177

Dic₃

—

178

Dic₃

—

179

C₂×C₅

—

180

C₂×C₅

—

181

C₂×C₅

#4659

yes

characteristic

182

C₂³

C₅×#237

yes

characteristic

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

C₂×C₅×D₁₂

yes

characteristic

188

C₂×C₄

C₅×#235

yes

—

189

C₂×C₄

C₅×#237

yes

—

190

C₂×C₄

C₅×D₃×Q₈

yes

characteristic

191

C₈

—

192

C₈

—

193

C₈

—

194

C₂×C₃

—

195

C₂×C₃

—

196

C₂×C₃

C₅×#400

yes

characteristic

197

C₅

#68701

yes

characteristic, direct product factor, Sylow

198

C₂²

—

199

C₂²

C₅×#879

yes

—

200

C₂²

C₅×#897

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

C₅×#875

yes

characteristic

205

C₄

C₅×#886

yes

characteristic

206

C₄

C₅×#889

yes

—

207

C₃

C₅×#2102

yes

characteristic, complemented, Sylow

208

C₂

—

209

C₂

—

210

C₂

C₅×#4659

yes

characteristic

	subgroup	G/H
center	C₄×C₅	#875
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×H(3,ℤ/4)	C₂
socle	C₂×C₃×C₅	#400
prime radical	C₂³×C₃×C₅	#41
hypercenter	C₅×H(3,ℤ/4)	D₃
hypocenter	C₃	C₅×#2102
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×H(3,ℤ/4)

subgroup

G/H

center

C₄×C₅

#875

commutator, cosocle

C₂²×C₃×C₄

C₂³×C₅

Frattini

C₂²×C₄

C₂²×C₅×D₃

C₂

C₂×C₃×C₅

C₂³×C₃×C₅

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×H(3,ℤ/4)

N	H ≅ G/N	type
14. #68701	C₅	direct (N × H)
169. C₃×C₅	#2102	semidirect
197. C₅	#68701	direct (N × H)
207. C₃	C₅×#2102	semidirect

H ≅ G/N

type

14. #68701

C₅

direct (N × H)

169. C₃×C₅

#2102

semidirect

197. C₅

#68701

direct (N × H)

207. C₃

C₅×#2102

semidirect

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

subgroup	index
C₅×#5125	2
C₅×#5118	2
C₃×C₅×H(3,ℤ/4)	2
C₅×#4533	2
C₅×#4532	2
C₅×#2102	3
#68701	5

subgroup

index

Subgroups — C₅×#68701 #14312422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#68701 #14312422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68701 #14312422