Group #14312420 — subgroups

Subgroups — C₃×#10601319 #14312420

Order 1920.

316 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 122 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	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	C₃×#65414	C₂	1	1	yes	maximal, characteristic
3	C₃×#64779	C₂	1	2	yes	maximal
4	C₃×#64777	C₂	1	1	yes	maximal, characteristic
5	C₃×#64743	C₂	1	2	yes	maximal
6	#10601319	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×#4072	C₂²	1	2	yes	—
11	C₂×C₃×#681	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊C₈	—	2	2	no	—
13	C₃×#3951	—	2	2	no	—
14	C₃×#2102	—	5	1	no	maximal, Hall
15	C₅×#423	C₂×C₃	1	1	yes	characteristic
16	#65414	C₂×C₃	1	1	yes	characteristic
17	#64779	C₂×C₃	1	2	yes	—
18	#64777	C₂×C₃	1	1	yes	characteristic
19	#64743	C₂×C₃	1	2	yes	—
20	C₂²×C₃×C₄×C₅	—	2	1	no	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₃×C₅×M₁₆	—	2	2	no	—
23	C₃×C₅×M₁₆	—	2	2	no	—
24	C₂×C₃×C₅×C₈	—	2	1	no	—
25	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
26	C₃×C₄²×C₅	—	2	1	no	—
27	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
28	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
29	C₃×C₂²⋊Dic₅	—	2	2	no	—
30	C₃×C₄⋊Dic₅	—	2	2	no	—
31	C₃×C₄×Dic₅	—	2	2	no	—
32	C₃×#681	—	2	2	no	—
33	C₃×#681	—	2	2	no	—
34	C₂×C₃×#168	—	2	2	no	—
35	C₃×#430	—	5	1	no	—
36	C₃×#423	D₅	1	1	yes	characteristic
37	C₃×#344	—	5	2	no	—
38	C₃×#339	—	5	2	no	—
39	C₃×H(3,ℤ/4)	—	5	1	no	—
40	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
43	#4072	C₂²×C₃	1	2	yes	—
44	C₂×#681	C₂²×C₃	1	2	yes	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	#3951	—	2	2	no	—
47	#2102	—	5	1	no	Sylow
48	C₂³×C₃×C₅	#41	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	—	2	1	no	—
50	C₂×C₃×C₄×C₅	—	2	1	no	—
51	C₂×C₃×C₄×C₅	—	4	2	no	—
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	#41	1	2	yes	—
55	C₂×C₃×C₄×C₅	#41	1	2	yes	—
56	C₂×C₃×Dic₅	—	4	2	no	—
57	C₃×C₅×C₈	—	2	2	no	—
58	C₃×C₅×C₈	—	4	1	no	—
59	C₃×#168	—	4	2	no	—
60	C₂×C₃×M₁₆	—	5	2	no	—
61	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
62	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
63	C₃×#117	—	5	2	no	—
64	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
65	C₃×#108	—	10	2	no	—
66	C₃×C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
69	C₅×M₁₆	—	2	2	no	—
70	C₅×M₁₆	—	2	2	no	—
71	C₂×C₅×C₈	—	2	1	no	—
72	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
73	C₄²×C₅	—	2	1	no	—
74	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
75	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
76	C₂²⋊Dic₅	—	2	2	no	—
77	C₄⋊Dic₅	—	2	2	no	—
78	C₄×Dic₅	—	2	2	no	—
79	#681	—	2	2	no	—
80	#681	—	2	2	no	—
81	C₂×#168	—	2	2	no	—
82	#430	—	5	1	no	—
83	#423	C₃×D₅	1	1	yes	characteristic
84	#344	—	5	2	no	—
85	#339	—	5	2	no	—
86	H(3,ℤ/4)	—	5	1	no	—
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	—	2	2	no	—
91	C₃×C₄×C₅	—	4	1	no	—
92	C₃×C₄×C₅	#123	1	1	yes	characteristic
93	C₃×C₄×C₅	#124	1	1	yes	characteristic
94	C₃×C₄×C₅	#125	1	2	yes	—
95	C₃×Dic₅	—	8	2	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
98	C₃×M₁₆	—	2	2	no	—
99	C₃×M₁₆	—	2	2	no	—
100	C₃×M₁₆	—	10	2	no	—
101	C₃×M₁₆	—	10	2	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	—	10	2	no	—
104	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
105	C₃×C₄⋊C₄	—	10	2	no	—
106	C₃×C₂²⋊C₄	—	10	2	no	—
107	C₃×C₄²	—	2	1	no	—
108	C₃×C₄²	—	10	2	no	—
109	C₃×C₄²	Dic₁₀	1	1	yes	characteristic
110	C₃×C₄²	Dic₁₀	1	1	yes	characteristic
111	C₂³×C₅	C₃×#41	1	1	yes	characteristic
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₃×#41	1	2	yes	—
118	C₂×C₄×C₅	C₃×#41	1	2	yes	—
119	C₂×Dic₅	—	4	2	no	—
120	C₅×C₈	—	2	2	no	—
121	C₅×C₈	—	4	1	no	—
122	#168	—	4	2	no	—
123	C₂×M₁₆	—	5	2	no	—
124	C₂×M₁₆	C₂×C₃×D₅	1	1	yes	characteristic
125	C₂×M₁₆	C₂×C₃×D₅	1	1	yes	characteristic
126	#117	—	5	2	no	—
127	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
128	#108	—	10	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₂×C₃×C₅	—	2	2	no	—
131	C₂×C₃×C₅	—	2	1	no	—
132	C₂×C₃×C₅	#400	1	1	yes	characteristic
133	C₂³×C₃	#709	1	1	yes	characteristic
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	—	20	2	no	—
138	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
139	C₂×C₃×C₄	#711	1	2	yes	—
140	C₂×C₃×C₄	#713	1	2	yes	—
141	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	—	4	1	no	—
144	C₃×C₈	—	20	2	no	—
145	C₂²×C₅	—	4	1	no	—
146	C₂²×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
147	C₂²×C₅	C₃×#126	1	2	yes	—
148	C₄×C₅	—	2	2	no	—
149	C₄×C₅	—	4	1	no	—
150	C₄×C₅	C₃×#123	1	1	yes	characteristic
151	C₄×C₅	C₃×#124	1	1	yes	characteristic
152	C₄×C₅	C₃×#125	1	2	yes	—
153	Dic₅	—	8	2	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
156	M₁₆	—	2	2	no	—
157	M₁₆	—	2	2	no	—
158	M₁₆	—	10	2	no	—
159	M₁₆	—	10	2	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	10	2	no	—
162	C₂×C₈	C₃×C₂²⋊D₅	1	2	yes	—
163	C₄⋊C₄	—	10	2	no	—
164	C₂²⋊C₄	—	10	2	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	10	2	no	—
167	C₄²	C₃×Dic₁₀	1	1	yes	characteristic
168	C₄²	C₃×Dic₁₀	1	1	yes	characteristic
169	C₃×C₅	#2102	1	1	yes	characteristic, complemented, Hall
170	C₂²×C₃	—	4	1	no	—
171	C₂²×C₃	#4044	1	2	yes	—
172	C₂²×C₃	#4070	1	1	yes	characteristic
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	4	1	no	—
175	C₃×C₄	—	40	2	no	—
176	C₃×C₄	#4036	1	2	yes	—
177	C₃×C₄	#4081	1	1	yes	characteristic
178	C₃×C₄	#4093	1	1	yes	characteristic
179	C₂×C₅	—	2	2	no	—
180	C₂×C₅	—	2	1	no	—
181	C₂×C₅	C₃×#400	1	1	yes	characteristic
182	C₂³	C₃×#709	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	20	2	no	—
187	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
188	C₂×C₄	C₃×#711	1	2	yes	—
189	C₂×C₄	C₃×#713	1	2	yes	—
190	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	4	1	no	—
193	C₈	—	20	2	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	2	2	no	—
196	C₂×C₃	#65275	1	1	yes	characteristic
197	C₅	C₃×#2102	1	1	yes	characteristic, complemented, Sylow
198	C₂²	—	4	1	no	—
199	C₂²	C₃×#4044	1	2	yes	—
200	C₂²	C₃×#4070	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	4	1	no	—
203	C₄	—	40	2	no	—
204	C₄	C₃×#4036	1	2	yes	—
205	C₄	C₃×#4081	1	1	yes	characteristic
206	C₄	C₃×#4093	1	1	yes	characteristic
207	C₃	#10601319	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	—	2	1	no	—
209	C₂	—	2	2	no	—
210	C₂	C₃×#65275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×#423

C₂

yes

maximal, characteristic

C₃×#65414

C₂

yes

maximal, characteristic

C₃×#64779

C₂

yes

maximal

C₃×#64777

C₂

yes

maximal, characteristic

C₃×#64743

C₂

yes

maximal

#10601319

C₃

yes

maximal, characteristic, direct product factor, Hall

C₂×C₃×C₅×M₁₆

C₂²

yes

characteristic

C₂×C₃×C₅×M₁₆

C₂²

yes

characteristic

C₂×C₃×C₄²×C₅

C₂²

yes

characteristic

C₃×#4072

C₂²

yes

—

C₂×C₃×#681

C₂²

yes

—

C₃×C₅×C₄⋊C₈

—

—

—

C₂×C₃

yes

characteristic

#65414

C₂×C₃

yes

characteristic

#64779

C₂×C₃

yes

—

#64777

C₂×C₃

yes

characteristic

#64743

C₂×C₃

yes

—

C₂²×C₃×C₄×C₅

—

C₂²×C₃×C₄×C₅

C₂³

yes

characteristic

C₃×C₅×M₁₆

—

C₃×C₅×M₁₆

—

C₂×C₃×C₅×C₈

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₃×C₄²×C₅

—

C₃×C₄²×C₅

Q₈

yes

characteristic

C₃×C₄²×C₅

Q₈

yes

characteristic

C₃×C₂²⋊Dic₅

—

C₃×C₄⋊Dic₅

—

C₃×C₄×Dic₅

—

C₃×#681

—

C₃×#681

—

C₂×C₃×#168

—

C₃×#430

—

C₃×#423

D₅

yes

—

—

—

C₂×C₅×M₁₆

C₂²×C₃

yes

characteristic

C₂×C₅×M₁₆

C₂²×C₃

yes

characteristic

C₂×C₄²×C₅

C₂²×C₃

yes

characteristic

#4072

C₂²×C₃

yes

—

C₂×#681

C₂²×C₃

yes

—

C₅×C₄⋊C₈

—

—

—

C₂³×C₃×C₅

yes

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

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×C₃×Dic₅

—

C₃×C₅×C₈

—

C₃×C₅×C₈

—

C₃×#168

—

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

C₂×D₅

yes

characteristic

C₂×C₃×M₁₆

C₂×D₅

yes

characteristic

C₃×#117

—

C₂×C₃×C₄²

C₂×D₅

yes

characteristic

C₃×#108

—

C₃×C₄⋊C₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂³×C₃

yes

characteristic

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₃×D₄

yes

—

C₄²×C₅

—

C₄²×C₅

C₃×Q₈

yes

characteristic

C₄²×C₅

C₃×Q₈

yes

characteristic

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

—

—

—

—

C₃×D₅

yes

—

—

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂²⋊Q₈

yes

characteristic

C₂²×C₃×C₅

#126

yes

—

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

—

C₃×Dic₅

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂²×D₅

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

100

C₃×M₁₆

—

101

C₃×M₁₆

—

102

C₂×C₃×C₈

—

103

C₂×C₃×C₈

—

104

C₂×C₃×C₈

C₂²⋊D₅

yes

—

105

C₃×C₄⋊C₄

—

106

C₃×C₂²⋊C₄

—

107

C₃×C₄²

—

108

C₃×C₄²

—

109

C₃×C₄²

Dic₁₀

yes

characteristic

110

C₃×C₄²

Dic₁₀

yes

characteristic

111

C₂³×C₅

C₃×#41

yes

characteristic

112

C₂×C₄×C₅

—

113

C₂×C₄×C₅

—

114

C₂×C₄×C₅

—

115

C₂×C₄×C₅

C₂×C₃×D₄

yes

characteristic

116

C₂×C₄×C₅

C₂×C₃×Q₈

yes

characteristic

117

C₂×C₄×C₅

C₃×#41

yes

—

118

C₂×C₄×C₅

C₃×#41

yes

—

119

C₂×Dic₅

—

120

C₅×C₈

—

121

C₅×C₈

—

122

#168

—

123

C₂×M₁₆

—

124

C₂×M₁₆

C₂×C₃×D₅

yes

characteristic

125

C₂×M₁₆

C₂×C₃×D₅

yes

characteristic

126

#117

—

127

C₂×C₄²

C₂×C₃×D₅

yes

characteristic

128

#108

—

129

C₄⋊C₈

—

130

C₂×C₃×C₅

—

131

C₂×C₃×C₅

—

132

C₂×C₃×C₅

#400

yes

characteristic

133

C₂³×C₃

#709

yes

characteristic

134

C₂×C₃×C₄

—

135

C₂×C₃×C₄

—

136

C₂×C₃×C₄

—

137

C₂×C₃×C₄

—

138

C₂×C₃×C₄

C₂×Dic₁₀

yes

characteristic

139

C₂×C₃×C₄

#711

yes

—

140

C₂×C₃×C₄

#713

yes

—

141

C₂×C₃×C₄

C₂×C₂²⋊D₅

yes

characteristic

142

C₃×C₈

—

143

C₃×C₈

—

144

C₃×C₈

—

145

C₂²×C₅

—

146

C₂²×C₅

C₃×C₂²⋊Q₈

yes

characteristic

147

C₂²×C₅

C₃×#126

yes

—

148

C₄×C₅

—

149

C₄×C₅

—

150

C₄×C₅

C₃×#123

yes

characteristic

151

C₄×C₅

C₃×#124

yes

characteristic

152

C₄×C₅

C₃×#125

yes

—

153

Dic₅

—

154

C₂²×C₄

—

155

C₂²×C₄

C₂²×C₃×D₅

yes

characteristic

156

M₁₆

—

157

M₁₆

—

158

M₁₆

—

159

M₁₆

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

C₃×C₂²⋊D₅

yes

—

163

C₄⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

C₃×Dic₁₀

yes

characteristic

168

C₄²

C₃×Dic₁₀

yes

characteristic

169

C₃×C₅

#2102

yes

characteristic, complemented, Hall

170

C₂²×C₃

—

171

C₂²×C₃

#4044

yes

—

172

C₂²×C₃

#4070

yes

characteristic

173

C₃×C₄

—

174

C₃×C₄

—

175

C₃×C₄

—

176

C₃×C₄

#4036

yes

—

177

C₃×C₄

#4081

yes

characteristic

178

C₃×C₄

#4093

yes

characteristic

179

C₂×C₅

—

180

C₂×C₅

—

181

C₂×C₅

C₃×#400

yes

characteristic

182

C₂³

C₃×#709

yes

characteristic

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

C₂×C₃×Dic₁₀

yes

characteristic

188

C₂×C₄

C₃×#711

yes

—

189

C₂×C₄

C₃×#713

yes

—

190

C₂×C₄

C₂×C₃×C₂²⋊D₅

yes

characteristic

191

C₈

—

192

C₈

—

193

C₈

—

194

C₂×C₃

—

195

C₂×C₃

—

196

C₂×C₃

#65275

yes

characteristic

197

C₅

C₃×#2102

yes

characteristic, complemented, Sylow

198

C₂²

—

199

C₂²

C₃×#4044

yes

—

200

C₂²

C₃×#4070

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

C₃×#4036

yes

—

205

C₄

C₃×#4081

yes

characteristic

206

C₄

C₃×#4093

yes

characteristic

207

C₃

#10601319

yes

characteristic, direct product factor, Sylow

208

C₂

—

209

C₂

—

210

C₂

C₃×#65275

yes

characteristic

	subgroup	G/H
center	C₃×C₄	#4093
commutator, cosocle	C₂²×C₄×C₅	C₂³×C₃
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#423	C₂
socle	C₂×C₃×C₅	#400
prime radical	C₂³×C₃×C₅	#41
hypercenter	C₃×#423	D₅
hypocenter	C₅	C₃×#2102
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#423

subgroup

G/H

center

C₃×C₄

#4093

commutator, cosocle

C₂²×C₄×C₅

C₂³×C₃

Frattini

C₂²×C₄

C₂²×C₃×D₅

Fitting

C₃×C₅×#423

C₂

C₂×C₃×C₅

C₂³×C₃×C₅

D₅

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#423

N	H ≅ G/N	type
6. #10601319	C₃	direct (N × H)
169. C₃×C₅	#2102	semidirect
197. C₅	C₃×#2102	semidirect
207. C₃	#10601319	direct (N × H)

H ≅ G/N

type

6. #10601319

C₃

direct (N × H)

169. C₃×C₅

#2102

semidirect

197. C₅

C₃×#2102

semidirect

207. C₃

#10601319

direct (N × H)

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

subgroup	index
C₃×C₅×#423	2
C₃×#65414	2
C₃×#64779	2
C₃×#64777	2
C₃×#64743	2
#10601319	3
C₃×#2102	5

subgroup

index

C₃×C₅×#423

Subgroups — C₃×#10601319 #14312420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#10601319 #14312420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601319 #14312420