Group #14298915 — subgroups

Subgroups — #14298915

Order 1920.

360 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 127 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	#11751219	C₂	1	2	yes	maximal
2	C₂×#91099	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#341	C₂	1	1	yes	maximal, characteristic
4	#11746379	C₂	1	2	yes	maximal
5	#11746373	C₂	1	1	yes	maximal, characteristic
6	#10599722	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
10	C₂×#6710	C₂²	1	2	yes	—
11	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
12	#91099	C₄	1	4	yes	—
13	#91098	—	2	4	no	—
14	#68929	—	5	1	no	maximal, Hall
15	#65407	—	3	2	no	—
16	C₂×#3947	—	3	1	no	—
17	C₅×#341	D₃	1	1	yes	characteristic
18	#64778	—	3	2	no	—
19	#64772	—	3	1	no	—
20	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×Dic₁₅	Q₈	1	2	yes	—
22	C₃×C₅×M₁₆	—	2	2	no	—
23	C₃×C₅×M₁₆	—	2	2	no	—
24	C₂×C₃×C₅×C₈	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	2	no	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₅	C₂×C₄	1	2	yes	—
28	C₄×Dic₁₅	D₄	1	2	yes	—
29	#6710	—	2	2	no	—
30	#6710	—	2	2	no	—
31	C₂×C₃⋊#168	—	2	2	no	—
32	C₂×C₃⋊#168	—	2	2	no	—
33	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
34	#5118	—	5	2	no	—
35	C₂×#815	—	5	1	no	—
36	C₃×#341	D₅	1	1	yes	characteristic
37	#4558	—	5	2	no	—
38	#4552	—	5	1	no	—
39	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
40	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂×C₄×Dic₅	—	3	1	no	—
42	C₂×#681	—	3	2	no	—
43	C₂²×#168	—	3	1	no	—
44	#3947	—	3	4	no	—
45	#3946	—	6	4	no	—
46	#1838	—	15	1	no	Sylow
47	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	#41	1	2	yes	—
51	C₂×C₃×C₄×C₅	#41	1	2	yes	—
52	C₂×Dic₁₅	—	4	2	no	—
53	C₂×Dic₁₅	C₄⋊C₄	1	4	yes	—
54	C₃×C₅×C₈	—	2	2	no	—
55	C₃×C₅×C₈	—	4	2	no	—
56	C₃⋊#168	—	2	2	no	—
57	C₃⋊#168	—	4	2	no	—
58	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
59	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
60	C₂×C₄×Dic₃	—	5	1	no	—
61	C₂×#208	—	5	2	no	—
62	C₂²×C₃⋊C₈	—	5	1	no	—
63	#815	—	5	4	no	—
64	Dic₃⋊C₈	—	10	4	no	—
65	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
66	C₂²×Dic₅	—	3	2	no	—
67	C₅×M₁₆	—	2	2	no	—
68	C₅×M₁₆	—	2	2	no	—
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
72	C₄×Dic₅	—	3	2	no	—
73	C₄×Dic₅	—	3	2	no	—
74	#681	—	6	2	no	—
75	#681	—	6	2	no	—
76	C₂×#168	—	3	2	no	—
77	C₂×#168	—	6	2	no	—
78	C₂×#168	—	6	2	no	—
79	#423	—	15	2	no	—
80	C₂×#97	—	15	1	no	—
81	#341	—	15	2	no	—
82	#341	—	15	1	no	—
83	#341	D₁₅	1	1	yes	characteristic
84	C₂²×C₃×C₅	—	2	2	no	—
85	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₃×C₅	#125	1	2	yes	—
87	C₃×C₄×C₅	#117	1	1	yes	characteristic
88	C₃×C₄×C₅	#117	1	1	yes	characteristic
89	C₃×C₄×C₅	#124	1	2	yes	—
90	Dic₁₅	—	2	4	no	—
91	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
92	C₂²×Dic₃	—	5	2	no	—
93	C₃×M₁₆	—	2	2	no	—
94	C₃×M₁₆	—	2	2	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
98	C₄×Dic₃	—	5	2	no	—
99	C₄×Dic₃	—	5	2	no	—
100	#208	—	10	2	no	—
101	#208	—	10	2	no	—
102	C₂×C₃⋊C₈	—	5	2	no	—
103	C₂×C₃⋊C₈	—	10	2	no	—
104	C₂×C₃⋊C₈	—	10	2	no	—
105	C₂³×C₅	D₃×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
107	C₂×C₄×C₅	#235	1	2	yes	—
108	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅	#237	1	2	yes	—
110	C₂×Dic₅	—	3	4	no	—
111	C₂×Dic₅	—	12	2	no	—
112	C₅×C₈	—	2	2	no	—
113	C₅×C₈	—	4	2	no	—
114	#168	—	6	2	no	—
115	#168	—	12	2	no	—
116	C₂×M₁₆	—	15	2	no	—
117	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
118	C₂²×C₈	—	15	1	no	—
119	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
120	C₂×C₄²	—	15	1	no	—
121	C₄⋊C₈	—	30	4	no	—
122	#97	—	15	4	no	—
123	C₂×C₃×C₅	—	2	2	no	—
124	C₂×C₃×C₅	#382	1	1	yes	characteristic
125	C₂×C₃×C₅	#430	1	2	yes	—
126	C₂³×C₃	Q₈×D₅	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
128	C₂×C₃×C₄	#709	1	2	yes	—
129	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
130	C₂×C₃×C₄	#711	1	2	yes	—
131	C₂×Dic₃	—	5	4	no	—
132	C₂×Dic₃	—	20	2	no	—
133	C₃×C₈	—	2	2	no	—
134	C₃×C₈	—	4	2	no	—
135	C₃⋊C₈	—	10	2	no	—
136	C₃⋊C₈	—	20	2	no	—
137	C₂²×C₅	—	2	2	no	—
138	C₂²×C₅	#889	1	2	yes	—
139	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₅	#872	1	1	yes	characteristic
141	C₄×C₅	#877	1	1	yes	characteristic
142	C₄×C₅	#885	1	2	yes	—
143	Dic₅	—	6	4	no	—
144	C₂²×C₄	—	15	2	no	—
145	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
146	M₁₆	—	2	2	no	—
147	M₁₆	—	2	2	no	—
148	M₁₆	—	30	2	no	—
149	M₁₆	—	30	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	15	2	no	—
153	C₂×C₈	—	30	2	no	—
154	C₂×C₈	—	30	2	no	—
155	C₂×C₈	C₄×D₁₅	1	2	yes	—
156	C₄²	—	15	2	no	—
157	C₄²	—	15	2	no	—
158	C₃×C₅	#1838	1	1	yes	characteristic, complemented, Hall
159	C₂²×C₃	—	2	2	no	—
160	C₂²×C₃	#4035	1	2	yes	—
161	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
162	C₃×C₄	#4018	1	1	yes	characteristic
163	C₃×C₄	#4023	1	1	yes	characteristic
164	C₃×C₄	#4031	1	2	yes	—
165	Dic₃	—	10	4	no	—
166	C₂×C₅	—	2	2	no	—
167	C₂×C₅	#4650	1	1	yes	characteristic
168	C₂×C₅	#4893	1	2	yes	—
169	C₂³	Q₈×D₁₅	1	1	yes	characteristic
170	C₂×C₄	—	15	4	no	—
171	C₂×C₄	—	60	2	no	—
172	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
173	C₂×C₄	#6817	1	2	yes	—
174	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
175	C₂×C₄	#6819	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	4	2	no	—
178	C₈	—	30	2	no	—
179	C₈	—	60	2	no	—
180	C₂×C₃	—	2	2	no	—
181	C₂×C₃	#64939	1	1	yes	characteristic
182	C₂×C₃	#65182	1	2	yes	—
183	C₅	#68929	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	#91778	1	2	yes	—
186	C₂²	C₄⋊C₄×D₁₅	1	1	yes	characteristic
187	C₄	—	30	4	no	—
188	C₄	#91761	1	1	yes	characteristic
189	C₄	#91766	1	1	yes	characteristic
190	C₄	#91774	1	2	yes	—
191	C₃	#10599722	1	1	yes	characteristic, complemented, Sylow
192	C₂	—	2	2	no	—
193	C₂	#11750751	1	1	yes	characteristic
194	C₂	#11750994	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11751219

C₂

yes

maximal

C₂×#91099

C₂

yes

maximal, characteristic

C₃×C₅×#341

C₂

yes

maximal, characteristic

#11746379

C₂

yes

maximal

#11746373

C₂

yes

maximal, characteristic

#10599722

—

maximal, Hall

C₂×C₃×C₅×M₁₆

C₂²

yes

—

C₂²×C₃×C₅×C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₅

C₂²

yes

characteristic

C₂×#6710

C₂²

yes

—

C₂²×C₃⋊#168

C₂²

yes

characteristic

#91099

C₄

yes

—

—

—

—

—

D₃

yes

characteristic

#64778

—

#64772

—

C₂²×C₃×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₁₅

Q₈

yes

—

C₃×C₅×M₁₆

—

C₃×C₅×M₁₆

—

C₂×C₃×C₅×C₈

—

C₂×C₃×C₅×C₈

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

—

C₄×Dic₁₅

C₂×C₄

yes

—

C₄×Dic₁₅

D₄

yes

—

#6710

—

#6710

—

C₂×C₃⋊#168

—

C₂×C₃⋊#168

—

C₂×C₃⋊#168

C₂×C₄

yes

—

#5118

—

C₂×#815

—

C₃×#341

D₅

yes

characteristic

#4558

—

#4552

—

C₂×C₅×M₁₆

C₂×D₃

yes

—

C₂²×C₅×C₈

C₂×D₃

yes

characteristic

C₂×C₄×Dic₅

—

C₂×#681

—

C₂²×#168

—

—

—

—

C₂³×C₃×C₅

C₂×Q₈

yes

characteristic

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

#41

yes

—

C₂×C₃×C₄×C₅

#41

yes

—

C₂×Dic₁₅

—

C₂×Dic₁₅

C₄⋊C₄

yes

—

C₃×C₅×C₈

—

C₃×C₅×C₈

—

C₃⋊#168

—

C₃⋊#168

—

C₂×C₃×M₁₆

C₂×D₅

yes

—

C₂²×C₃×C₈

C₂×D₅

yes

characteristic

C₂×C₄×Dic₃

—

C₂×#208

—

C₂²×C₃⋊C₈

—

#815

—

Dic₃⋊C₈

—

C₂²×C₄×C₅

C₂²×D₃

yes

characteristic

C₂²×Dic₅

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄×D₃

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

—

—

—

—

—

—

C₂×#97

—

#341

—

#341

—

#341

D₁₅

yes

characteristic

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃×C₅

#125

yes

—

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

—

Dic₁₅

—

C₂²×C₃×C₄

C₂²×D₅

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₅

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

100

#208

—

101

#208

—

102

C₂×C₃⋊C₈

—

103

C₂×C₃⋊C₈

—

104

C₂×C₃⋊C₈

—

105

C₂³×C₅

D₃×Q₈

yes

characteristic

106

C₂×C₄×C₅

C₂×C₄×D₃

yes

characteristic

107

C₂×C₄×C₅

#235

yes

—

108

C₂×C₄×C₅

D₃×D₄

yes

characteristic

109

C₂×C₄×C₅

#237

yes

—

110

C₂×Dic₅

—

111

C₂×Dic₅

—

112

C₅×C₈

—

113

C₅×C₈

—

114

#168

—

115

#168

—

116

C₂×M₁₆

—

117

C₂×M₁₆

C₂×D₁₅

yes

—

118

C₂²×C₈

—

119

C₂²×C₈

C₂×D₁₅

yes

characteristic

120

C₂×C₄²

—

121

C₄⋊C₈

—

122

#97

—

123

C₂×C₃×C₅

—

124

C₂×C₃×C₅

#382

yes

characteristic

125

C₂×C₃×C₅

#430

yes

—

126

C₂³×C₃

Q₈×D₅

yes

characteristic

127

C₂×C₃×C₄

C₂×C₄×D₅

yes

characteristic

128

C₂×C₃×C₄

#709

yes

—

129

C₂×C₃×C₄

D₄×D₅

yes

characteristic

130

C₂×C₃×C₄

#711

yes

—

131

C₂×Dic₃

—

132

C₂×Dic₃

—

133

C₃×C₈

—

134

C₃×C₈

—

135

C₃⋊C₈

—

136

C₃⋊C₈

—

137

C₂²×C₅

—

138

C₂²×C₅

#889

yes

—

139

C₂²×C₅

D₃×C₄⋊C₄

yes

characteristic

140

C₄×C₅

#872

yes

characteristic

141

C₄×C₅

#877

yes

characteristic

142

C₄×C₅

#885

yes

—

143

Dic₅

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂²×D₁₅

yes

characteristic

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

C₄×D₁₅

yes

—

156

C₄²

—

157

C₄²

—

158

C₃×C₅

#1838

yes

characteristic, complemented, Hall

159

C₂²×C₃

—

160

C₂²×C₃

#4035

yes

—

161

C₂²×C₃

D₅×C₄⋊C₄

yes

characteristic

162

C₃×C₄

#4018

yes

characteristic

163

C₃×C₄

#4023

yes

characteristic

164

C₃×C₄

#4031

yes

—

165

Dic₃

—

166

C₂×C₅

—

167

C₂×C₅

#4650

yes

characteristic

168

C₂×C₅

#4893

yes

—

169

C₂³

Q₈×D₁₅

yes

characteristic

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

C₂×C₄×D₁₅

yes

characteristic

173

C₂×C₄

#6817

yes

—

174

C₂×C₄

D₄×D₁₅

yes

characteristic

175

C₂×C₄

#6819

yes

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₂×C₃

—

181

C₂×C₃

#64939

yes

characteristic

182

C₂×C₃

#65182

yes

—

183

C₅

#68929

yes

characteristic, complemented, Sylow

184

C₂²

—

185

C₂²

#91778

yes

—

186

C₂²

C₄⋊C₄×D₁₅

yes

characteristic

187

C₄

—

188

C₄

#91761

yes

characteristic

189

C₄

#91766

yes

characteristic

190

C₄

#91774

yes

—

191

C₃

#10599722

yes

characteristic, complemented, Sylow

192

C₂

—

193

C₂

#11750751

yes

characteristic

194

C₂

#11750994

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#341	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₂×Q₈
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#341	D₁₅
hypocenter	C₃×C₅	#1838
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₅

commutator

C₂×C₃×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅

Fitting

C₃×C₅×#341

C₂

socle

C₂²×C₃×C₅

C₂×C₄⋊C₄

prime radical

C₂³×C₃×C₅

C₂×Q₈

cosocle

C₂²×C₃×C₄×C₅

C₂³

D₁₅

C₃×C₅

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
158. C₃×C₅	#1838	semidirect
183. C₅	#68929	semidirect
191. C₃	#10599722	semidirect

H ≅ G/N

type

158. C₃×C₅

#1838

semidirect

183. C₅

#68929

semidirect

191. C₃

#10599722

semidirect

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

subgroup	index
#11751219	2
C₂×#91099	2
C₃×C₅×#341	2
#11746379	2
#11746373	2
#10599722	3
#68929	5

subgroup

index

C₃×C₅×#341

Subgroups — #14298915

Conjugacy classes of subgroups

Complements

Maximal subgroups