Group #14294101 — subgroups

Subgroups — #14294101

Order 1920.

396 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 116 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	#11750794	C₂	1	1	yes	maximal, characteristic
2	C₃×#64988	C₂	1	1	yes	maximal, characteristic, complemented
3	#11746960	C₂	1	2	yes	maximal, complemented
4	#11746910	C₂	1	2	yes	maximal
5	C₅×#4455	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598891	—	3	1	no	maximal, Hall
7	C₂×#6706	—	2	2	no	—
8	#91763	C₂²	1	1	yes	characteristic
9	#91758	C₂²	1	1	yes	characteristic
10	C₂×C₃×Dic₂₀	—	2	2	no	—
11	C₃×#4015	C₂²	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	#90990	—	2	4	no	—
15	#90968	—	2	4	no	—
16	#71301	—	5	1	no	maximal, Hall
17	#64988	D₃	1	1	yes	characteristic, complemented
18	#64983	—	3	2	no	—
19	#64982	—	3	1	no	—
20	#64971	—	3	2	no	—
21	C₅×#322	—	3	1	no	—
22	C₂×D₆₀	—	2	2	no	—
23	C₂×D₆₀	—	2	1	no	—
24	C₂×Dic₃₀	—	2	1	no	—
25	C₂×C₃×Dic₁₀	—	2	2	no	—
26	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
27	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₁₅	—	2	2	no	—
29	#6706	—	4	4	no	—
30	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
31	C₃×C₄⋊Dic₅	—	2	2	no	—
32	C₃×C₄⋊Dic₅	—	2	2	no	—
33	C₃×Dic₂₀	—	4	4	no	—
34	C₃×#494	—	5	1	no	—
35	#5088	—	5	2	no	—
36	#5078	—	5	2	no	—
37	#4693	—	5	1	no	—
38	#4455	D₅	1	1	yes	characteristic, complemented
39	C₂×Dic₂₀	—	2	2	no	—
40	C₂×#677	—	6	2	no	—
41	#4020	—	3	1	no	—
42	#4015	—	3	1	no	—
43	#4015	C₂×D₃	1	2	yes	—
44	C₄×C₅×C₈	—	3	2	no	—
45	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
46	#3953	—	6	4	no	—
47	#3950	—	6	4	no	—
48	#1722	—	15	1	no	Sylow
49	C₂²×D₁₅	—	4	1	no	—
50	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
52	C₂×Dic₁₅	—	4	1	no	—
53	D₆₀	—	4	2	no	—
54	D₆₀	—	4	4	no	—
55	Dic₃₀	—	4	2	no	—
56	C₂×C₃×Dic₅	—	4	2	no	—
57	C₃×Dic₁₀	—	4	4	no	—
58	C₃×C₅×C₈	D₈	1	4	yes	—
59	C₅×C₃⋊C₈	—	2	4	no	—
60	C₂×C₃×Q₁₆	—	10	2	no	—
61	C₃×C₄⋊Q₈	—	5	2	no	—
62	C₂×#204	—	10	2	no	—
63	#874	—	5	1	no	—
64	#869	—	5	1	no	—
65	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
66	#809	—	10	4	no	—
67	#807	—	10	4	no	—
68	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
69	C₂×D₂₀	—	6	2	no	—
70	C₂×D₂₀	—	6	1	no	—
71	C₂×Dic₁₀	—	2	2	no	—
72	C₂×Dic₁₀	—	6	1	no	—
73	C₂×C₅×C₈	—	3	4	no	—
74	C₂×C₅×C₈	#67	1	2	yes	—
75	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
76	C₄⋊Dic₅	—	2	2	no	—
77	C₄⋊Dic₅	—	2	2	no	—
78	C₄⋊Dic₅	—	6	2	no	—
79	Dic₂₀	—	4	4	no	—
80	#677	—	12	4	no	—
81	#500	—	15	2	no	—
82	#494	—	5	1	no	—
83	#492	—	15	1	no	—
84	#486	—	15	2	no	—
85	#322	—	3	1	no	—
86	C₂²×C₃×C₅	#127	1	1	yes	characteristic
87	C₂×D₁₅	—	8	1	no	—
88	C₂×D₁₅	—	8	2	no	—
89	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
90	C₃×C₄×C₅	#137	1	4	yes	—
91	Dic₁₅	—	8	1	no	—
92	C₃×Dic₅	—	8	2	no	—
93	C₂×C₃×Q₈	—	10	2	no	—
94	C₂×D₁₂	—	10	2	no	—
95	C₂×D₁₂	—	10	1	no	—
96	C₂×Dic₆	—	10	1	no	—
97	C₃×Q₁₆	—	20	4	no	—
98	C₂×C₃×C₈	D₂₀	1	2	yes	—
99	C₃×C₄⋊C₄	—	10	2	no	—
100	C₃×C₄⋊C₄	—	10	2	no	—
101	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
102	C₄⋊Dic₃	—	10	2	no	—
103	C₂×C₃⋊C₈	D₂₀	1	4	yes	—
104	#204	—	20	4	no	—
105	C₂²×D₅	—	12	1	no	—
106	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
107	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
108	C₂×Dic₅	—	4	2	no	—
109	C₂×Dic₅	—	12	1	no	—
110	D₂₀	—	12	4	no	—
111	D₂₀	—	12	2	no	—
112	Dic₁₀	—	4	4	no	—
113	Dic₁₀	—	12	2	no	—
114	C₅×C₈	—	6	4	no	—
115	C₅×C₈	#213	1	4	yes	—
116	C₂×Q₁₆	—	10	2	no	—
117	C₂×SD₁₆	—	30	2	no	—
118	C₄⋊Q₈	—	5	2	no	—
119	C₄⋊Q₈	—	15	1	no	—
120	#127	—	15	1	no	—
121	#107	—	30	4	no	—
122	#102	—	30	4	no	—
123	C₄×C₈	—	3	2	no	—
124	C₄×C₈	D₃×D₅	1	1	yes	characteristic
125	C₂×C₃×C₅	#493	1	1	yes	characteristic
126	C₂×C₃×C₅	#497	1	2	yes	—
127	D₁₅	—	16	1	no	—
128	C₂²×D₃	—	20	1	no	—
129	C₃×Q₈	—	20	4	no	—
130	C₂×C₃×C₄	—	20	2	no	—
131	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
132	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
133	C₂×Dic₃	—	20	1	no	—
134	D₁₂	—	20	2	no	—
135	D₁₂	—	20	4	no	—
136	Dic₆	—	20	2	no	—
137	C₃×C₈	D₄₀	1	4	yes	—
138	C₃⋊C₈	—	2	4	no	—
139	C₂²×C₅	#940	1	1	yes	characteristic
140	C₂×D₅	—	24	2	no	—
141	C₂×D₅	—	24	1	no	—
142	C₄×C₅	#918	1	4	yes	—
143	C₄×C₅	C₂×#213	1	2	yes	—
144	Dic₅	—	8	2	no	—
145	Dic₅	—	24	1	no	—
146	C₂×Q₈	—	10	2	no	—
147	C₂×Q₈	—	30	1	no	—
148	C₂×D₄	—	30	2	no	—
149	C₂×D₄	—	30	1	no	—
150	Q₁₆	—	20	4	no	—
151	SD₁₆	—	60	4	no	—
152	C₂×C₈	—	3	4	no	—
153	C₂×C₈	#1202	1	2	yes	—
154	C₄⋊C₄	—	10	2	no	—
155	C₄⋊C₄	—	10	2	no	—
156	C₄⋊C₄	—	30	2	no	—
157	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
158	C₃×C₅	#1722	1	1	yes	characteristic, complemented, Hall
159	C₂²×C₃	#4020	1	1	yes	characteristic
160	C₂×D₃	—	40	1	no	—
161	C₂×D₃	—	40	2	no	—
162	C₃×C₄	—	40	2	no	—
163	C₃×C₄	C₂×D₄₀	1	2	yes	—
164	C₃×C₄	#4055	1	4	yes	—
165	Dic₃	—	40	1	no	—
166	C₂×C₅	#5073	1	1	yes	characteristic
167	C₂×C₅	#5190	1	2	yes	—
168	D₅	—	48	1	no	—
169	C₂³	—	60	1	no	—
170	Q₈	—	20	4	no	—
171	Q₈	—	60	2	no	—
172	D₄	—	60	4	no	—
173	D₄	—	60	2	no	—
174	C₂×C₄	—	20	2	no	—
175	C₂×C₄	—	60	1	no	—
176	C₂×C₄	D₃×D₂₀	1	2	yes	—
177	C₂×C₄	C₂×#1202	1	1	yes	characteristic
178	C₈	—	6	4	no	—
179	C₈	#6653	1	4	yes	—
180	C₂×C₃	#64984	1	1	yes	characteristic
181	C₂×C₃	#65004	1	2	yes	—
182	D₃	—	80	1	no	—
183	C₅	#71301	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	120	2	no	—
185	C₂²	—	120	1	no	—
186	C₂²	#91458	1	1	yes	characteristic
187	C₄	—	40	2	no	—
188	C₄	—	120	1	no	—
189	C₄	#91263	1	4	yes	—
190	C₄	C₂×#6653	1	2	yes	—
191	C₃	#10598891	1	1	yes	characteristic, complemented, Sylow
192	C₂	—	240	1	no	—
193	C₂	#11747035	1	1	yes	characteristic
194	C₂	#11748034	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11750794

C₂

yes

maximal, characteristic

C₃×#64988

C₂

yes

maximal, characteristic, complemented

#11746960

C₂

yes

maximal, complemented

#11746910

C₂

yes

maximal

C₅×#4455

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

characteristic

#91758

C₂²

yes

characteristic

C₂×C₃×Dic₂₀

—

C₃×#4015

C₂²

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

C₄×C₅×C₃⋊C₈

C₂²

yes

—

—

—

—

D₃

yes

characteristic, complemented

—

—

—

—

C₂×D₆₀

—

C₂×D₆₀

—

C₂×Dic₃₀

—

C₂×C₃×Dic₁₀

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₁₅

—

#6706

—

C₂×C₅×C₃⋊C₈

D₄

yes

—

C₃×C₄⋊Dic₅

—

C₃×C₄⋊Dic₅

—

C₃×Dic₂₀

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂×Dic₂₀

—

—

—

—

C₂×D₃

yes

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

—

C₂²×D₁₅

—

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₂×Dic₁₅

—

D₆₀

—

D₆₀

—

Dic₃₀

—

C₂×C₃×Dic₅

—

C₃×Dic₁₀

—

C₃×C₅×C₈

D₈

yes

—

C₅×C₃⋊C₈

—

C₂×C₃×Q₁₆

—

C₃×C₄⋊Q₈

—

C₂×#204

—

#874

—

#869

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#809

—

#807

—

C₄×C₃⋊C₈

C₂×D₅

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

#67

yes

—

C₄²×C₅

C₂²×D₃

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

Dic₂₀

—

—

—

—

—

—

—

C₂²×C₃×C₅

#127

yes

characteristic

C₂×D₁₅

—

C₂×D₁₅

—

C₃×C₄×C₅

C₂×D₈

yes

—

C₃×C₄×C₅

#137

yes

—

Dic₁₅

—

C₃×Dic₅

—

C₂×C₃×Q₈

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

D₂₀

yes

—

C₃×C₄⋊C₄

—

100

C₃×C₄⋊C₄

—

101

C₃×C₄²

C₂²×D₅

yes

characteristic

102

C₄⋊Dic₃

—

103

C₂×C₃⋊C₈

D₂₀

yes

—

104

#204

—

105

C₂²×D₅

—

106

C₂×C₄×C₅

D₃×D₄

yes

—

107

C₂×C₄×C₅

C₂×#67

yes

characteristic

108

C₂×Dic₅

—

109

C₂×Dic₅

—

110

D₂₀

—

111

D₂₀

—

112

Dic₁₀

—

113

Dic₁₀

—

114

C₅×C₈

—

115

C₅×C₈

#213

yes

—

116

C₂×Q₁₆

—

117

C₂×SD₁₆

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

#127

—

121

#107

—

122

#102

—

123

C₄×C₈

—

124

C₄×C₈

D₃×D₅

yes

characteristic

125

C₂×C₃×C₅

#493

yes

characteristic

126

C₂×C₃×C₅

#497

yes

—

127

D₁₅

—

128

C₂²×D₃

—

129

C₃×Q₈

—

130

C₂×C₃×C₄

—

131

C₂×C₃×C₄

C₂×D₂₀

yes

—

132

C₂×C₃×C₄

C₂×D₂₀

yes

characteristic

133

C₂×Dic₃

—

134

D₁₂

—

135

D₁₂

—

136

Dic₆

—

137

C₃×C₈

D₄₀

yes

—

138

C₃⋊C₈

—

139

C₂²×C₅

#940

yes

characteristic

140

C₂×D₅

—

141

C₂×D₅

—

142

C₄×C₅

#918

yes

—

143

C₄×C₅

C₂×#213

yes

—

144

Dic₅

—

145

Dic₅

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×D₄

—

149

C₂×D₄

—

150

Q₁₆

—

151

SD₁₆

—

152

C₂×C₈

—

153

C₂×C₈

#1202

yes

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄²

C₂×D₃×D₅

yes

characteristic

158

C₃×C₅

#1722

yes

characteristic, complemented, Hall

159

C₂²×C₃

#4020

yes

characteristic

160

C₂×D₃

—

161

C₂×D₃

—

162

C₃×C₄

—

163

C₃×C₄

C₂×D₄₀

yes

—

164

C₃×C₄

#4055

yes

—

165

Dic₃

—

166

C₂×C₅

#5073

yes

characteristic

167

C₂×C₅

#5190

yes

—

168

D₅

—

169

C₂³

—

170

Q₈

—

171

Q₈

—

172

D₄

—

173

D₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

D₃×D₂₀

yes

—

177

C₂×C₄

C₂×#1202

yes

characteristic

178

C₈

—

179

C₈

#6653

yes

—

180

C₂×C₃

#64984

yes

characteristic

181

C₂×C₃

#65004

yes

—

182

D₃

—

183

C₅

#71301

yes

characteristic, complemented, Sylow

184

C₂²

—

120

—

185

C₂²

—

120

—

186

C₂²

#91458

yes

characteristic

187

C₄

—

188

C₄

—

120

—

189

C₄

#91263

yes

—

190

C₄

C₂×#6653

yes

—

191

C₃

#10598891

yes

characteristic, complemented, Sylow

192

C₂

—

240

—

193

C₂

#11747035

yes

characteristic

194

C₂

#11748034

yes

—

	subgroup	G/H
center	C₂²	#91458
commutator, cosocle	C₃×C₄²×C₅	C₂³
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	#127
prime radical	#91763	C₂²
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1722
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

#91458

commutator, cosocle

C₃×C₄²×C₅

C₂³

Frattini

C₄²

C₂×D₃×D₅

Fitting

C₃×C₄×C₅×C₈

C₂²

C₂²×C₃×C₅

C₂²

C₄×C₈

D₃×D₅

hypocenter

C₃×C₅

#1722

LCS

G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. C₃×#64988	C₂	semidirect
3. #11746960	C₂	semidirect
5. C₅×#4455	C₂	semidirect
17. #64988	D₃	semidirect
38. #4455	D₅	semidirect
158. C₃×C₅	#1722	semidirect
183. C₅	#71301	semidirect
191. C₃	#10598891	semidirect

H ≅ G/N

type

2. C₃×#64988

C₂

semidirect

3. #11746960

C₂

semidirect

5. C₅×#4455

C₂

semidirect

17. #64988

D₃

semidirect

38. #4455

D₅

semidirect

158. C₃×C₅

#1722

semidirect

183. C₅

#71301

semidirect

191. C₃

#10598891

semidirect

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

subgroup	index
#11750794	2
C₃×#64988	2
#11746960	2
#11746910	2
C₅×#4455	2
#10598891	3
#71301	5

subgroup

index

Subgroups — #14294101

Conjugacy classes of subgroups

Complements

Maximal subgroups