Group #14298932 — subgroups

Subgroups — #14298932

Order 1920.

360 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 156 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	#11751025	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic
3	C₃×#65213	C₂	1	1	yes	maximal, characteristic
4	#11745983	C₂	1	4	yes	maximal
5	#10600634	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×#6710	C₂²	1	2	yes	—
8	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃×#681	C₂²	1	2	yes	—
10	C₄⋊(C₃⋊#168)	—	2	2	no	—
11	C₅×#803	C₄	1	4	yes	—
12	C₃×C₄⋊#168	—	2	2	no	—
13	#70517	—	5	1	no	maximal, Hall
14	C₂×C₅×#97	—	3	1	no	—
15	#65213	—	3	1	no	—
16	#65213	D₃	1	1	yes	characteristic
17	#64772	—	3	4	no	—
18	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
19	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
21	C₃×C₄²×C₅	D₄	1	2	yes	—
22	#6710	—	2	4	no	—
23	C₂×C₃⋊#168	—	2	2	no	—
24	C₂×C₅×C₃⋊C₈	—	2	2	no	—
25	C₂×C₅×C₃⋊C₈	—	2	2	no	—
26	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
27	C₃×#681	—	2	4	no	—
28	C₂×C₃×#168	—	2	2	no	—
29	C₃×#423	—	5	1	no	—
30	#4924	—	5	1	no	—
31	C₂×#803	D₅	1	1	yes	characteristic
32	#4558	—	5	4	no	—
33	C₂²×C₅×C₈	—	3	2	no	—
34	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
35	C₂×#681	—	3	2	no	—
36	C₂×#681	C₂×D₃	1	2	yes	—
37	C₅×#97	—	3	4	no	—
38	C₄⋊#168	—	2	2	no	—
39	C₄⋊#168	—	6	2	no	—
40	#1838	—	15	1	no	Sylow
41	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	4	2	no	—
43	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
44	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	#41	1	2	yes	—
47	C₂×C₃×C₄×C₅	#41	1	2	yes	—
48	C₃⋊#168	—	4	2	no	—
49	C₃×#168	—	4	2	no	—
50	C₅×C₃⋊C₈	—	2	4	no	—
51	C₂×C₃×M₁₆	—	5	2	no	—
52	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
53	C₂×#208	—	5	2	no	—
54	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
55	C₃×C₄⋊C₈	—	10	2	no	—
56	C₄⋊(C₃⋊C₈)	—	10	2	no	—
57	#803	Dic₅	1	4	yes	—
58	C₂²×C₄×C₅	Dic₆	1	2	yes	—
59	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
60	C₂×C₅×C₈	—	3	4	no	—
61	C₂×C₅×C₈	—	6	2	no	—
62	C₂×C₅×C₈	—	6	2	no	—
63	C₄²×C₅	C₄×D₃	1	2	yes	—
64	C₄²×C₅	#67	1	2	yes	—
65	#681	—	2	4	no	—
66	#681	—	6	4	no	—
67	C₂×#168	—	2	2	no	—
68	C₂×#168	—	6	2	no	—
69	#423	—	5	1	no	—
70	#423	—	15	1	no	—
71	C₂×#97	—	3	1	no	—
72	#341	—	15	4	no	—
73	C₂²×C₃×C₅	—	2	2	no	—
74	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₃×C₅	#125	1	2	yes	—
76	C₃×C₄×C₅	—	2	4	no	—
77	C₃×C₄×C₅	#117	1	2	yes	—
78	C₃×C₄×C₅	#124	1	1	yes	characteristic
79	C₃×C₄×C₅	#124	1	1	yes	characteristic
80	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
81	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
82	C₃×M₁₆	—	10	4	no	—
83	C₂×C₃×C₈	—	10	2	no	—
84	C₃×C₄²	D₂₀	1	2	yes	—
85	C₃×C₄²	C₂×Dic₅	1	2	yes	—
86	#208	—	10	4	no	—
87	C₂×C₃⋊C₈	—	2	2	no	—
88	C₂×C₃⋊C₈	—	2	2	no	—
89	C₂×C₃⋊C₈	—	10	2	no	—
90	C₂×C₃⋊C₈	C₂×Dic₅	1	4	yes	—
91	C₂³×C₅	C₂×Dic₆	1	1	yes	characteristic
92	C₂×C₄×C₅	—	4	2	no	—
93	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
94	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
95	C₂×C₄×C₅	#237	1	2	yes	—
96	C₂×C₄×C₅	#239	1	2	yes	—
97	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
98	C₅×C₈	—	6	4	no	—
99	#168	—	4	2	no	—
100	#168	—	12	2	no	—
101	C₂×M₁₆	—	5	2	no	—
102	C₂×M₁₆	—	15	2	no	—
103	C₂²×C₈	—	3	2	no	—
104	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
105	C₄⋊C₈	—	10	2	no	—
106	C₄⋊C₈	—	30	2	no	—
107	#97	—	3	4	no	—
108	C₂×C₃×C₅	—	2	2	no	—
109	C₂×C₃×C₅	#382	1	1	yes	characteristic
110	C₂×C₃×C₅	#430	1	2	yes	—
111	C₂³×C₃	C₂×Dic₁₀	1	1	yes	characteristic
112	C₂×C₃×C₄	—	4	2	no	—
113	C₂×C₃×C₄	C₄⋊Dic₅	1	4	yes	—
114	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
115	C₂×C₃×C₄	#709	1	2	yes	—
116	C₂×C₃×C₄	#709	1	2	yes	—
117	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
118	C₃×C₈	—	20	2	no	—
119	C₃⋊C₈	—	2	4	no	—
120	C₃⋊C₈	—	20	2	no	—
121	C₂²×C₅	—	2	2	no	—
122	C₂²×C₅	#890	1	2	yes	—
123	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
124	C₄×C₅	—	2	4	no	—
125	C₄×C₅	#892	1	2	yes	—
126	C₄×C₅	#936	1	1	yes	characteristic
127	C₄×C₅	#947	1	1	yes	characteristic
128	C₂²×C₄	#1204	1	2	yes	—
129	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
130	M₁₆	—	10	4	no	—
131	M₁₆	—	30	4	no	—
132	C₂×C₈	—	3	4	no	—
133	C₂×C₈	—	6	2	no	—
134	C₂×C₈	—	6	2	no	—
135	C₂×C₈	—	10	2	no	—
136	C₂×C₈	—	30	2	no	—
137	C₄²	D₃×Dic₅	1	2	yes	—
138	C₄²	#1202	1	2	yes	—
139	C₃×C₅	#1838	1	1	yes	characteristic, complemented, Hall
140	C₂²×C₃	—	2	2	no	—
141	C₂²×C₃	#4016	1	2	yes	—
142	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
143	C₃×C₄	—	2	4	no	—
144	C₃×C₄	#4021	1	1	yes	characteristic
145	C₃×C₄	#4021	1	1	yes	characteristic
146	C₃×C₄	#4072	1	2	yes	—
147	C₂×C₅	—	2	2	no	—
148	C₂×C₅	#4977	1	1	yes	characteristic
149	C₂×C₅	#5124	1	2	yes	—
150	C₂³	C₂×#1204	1	1	yes	characteristic
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	Dic₃⋊Dic₅	1	4	yes	—
153	C₂×C₄	#6765	1	2	yes	—
154	C₂×C₄	#6769	1	2	yes	—
155	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
156	C₂×C₄	C₂×#1202	1	1	yes	characteristic
157	C₈	—	6	4	no	—
158	C₈	—	20	2	no	—
159	C₈	—	60	2	no	—
160	C₂×C₃	—	2	2	no	—
161	C₂×C₃	#65225	1	1	yes	characteristic
162	C₂×C₃	#65414	1	2	yes	—
163	C₅	#70517	1	1	yes	characteristic, complemented, Sylow
164	C₂²	—	2	2	no	—
165	C₂²	#91381	1	2	yes	—
166	C₂²	C₂×Dic₃⋊Dic₅	1	1	yes	characteristic
167	C₄	—	2	4	no	—
168	C₄	#91338	1	2	yes	—
169	C₄	#91364	1	1	yes	characteristic
170	C₄	#91374	1	1	yes	characteristic
171	C₃	#10600634	1	1	yes	characteristic, complemented, Sylow
172	C₂	—	2	2	no	—
173	C₂	#11748480	1	2	yes	—
174	C₂	#11748619	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11751025

C₂

yes

maximal, characteristic

C₂×C₅×#803

C₂

yes

maximal, characteristic

C₃×#65213

C₂

yes

maximal, characteristic

#11745983

C₂

yes

maximal

#10600634

—

maximal, Hall

C₂×C₃×C₄²×C₅

C₂²

yes

characteristic

C₂×#6710

C₂²

yes

—

C₂²×C₅×C₃⋊C₈

C₂²

yes

—

C₂×C₃×#681

C₂²

yes

—

C₄⋊(C₃⋊#168)

—

C₅×#803

C₄

yes

—

C₃×C₄⋊#168

—

#70517

—

maximal, Hall

C₂×C₅×#97

—

#65213

—

#65213

D₃

yes

characteristic

#64772

—

C₂²×C₃×C₄×C₅

Q₈

yes

—

C₂²×C₃×C₄×C₅

C₂³

yes

characteristic

C₃×C₄²×C₅

C₂×C₄

yes

—

C₃×C₄²×C₅

D₄

yes

—

#6710

—

C₂×C₃⋊#168

—

C₂×C₅×C₃⋊C₈

—

C₂×C₅×C₃⋊C₈

—

C₂×C₅×C₃⋊C₈

C₂×C₄

yes

—

C₃×#681

—

C₂×C₃×#168

—

C₃×#423

—

#4924

—

C₂×#803

D₅

yes

characteristic

#4558

—

C₂²×C₅×C₈

—

C₂×C₄²×C₅

C₂×D₃

yes

characteristic

C₂×#681

—

C₂×#681

C₂×D₃

yes

—

C₅×#97

—

—

—

—

C₂³×C₃×C₅

C₂×Q₈

yes

characteristic

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₄⋊C₄

yes

—

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

—

C₃⋊#168

—

C₃×#168

—

C₅×C₃⋊C₈

—

C₂×C₃×M₁₆

—

C₂×C₃×C₄²

C₂×D₅

yes

characteristic

C₂×#208

—

C₂²×C₃⋊C₈

C₂×D₅

yes

—

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

#803

Dic₅

yes

—

C₂²×C₄×C₅

Dic₆

yes

—

C₂²×C₄×C₅

C₂²×D₃

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

C₄×D₃

yes

—

C₄²×C₅

#67

yes

—

—

—

—

—

—

—

C₂×#97

—

#341

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃×C₅

#125

yes

—

C₃×C₄×C₅

—

C₃×C₄×C₅

#117

yes

—

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

C₂²×C₃×C₄

Dic₁₀

yes

—

C₂²×C₃×C₄

C₂²×D₅

yes

characteristic

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄²

D₂₀

yes

—

C₃×C₄²

C₂×Dic₅

yes

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

C₂×Dic₅

yes

—

C₂³×C₅

C₂×Dic₆

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

Dic₃⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄×D₃

yes

characteristic

C₂×C₄×C₅

#237

yes

—

C₂×C₄×C₅

#239

yes

—

C₂×C₄×C₅

C₂×#67

yes

characteristic

C₅×C₈

—

#168

—

100

#168

—

101

C₂×M₁₆

—

102

C₂×M₁₆

—

103

C₂²×C₈

—

104

C₂×C₄²

D₃×D₅

yes

characteristic

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

#97

—

108

C₂×C₃×C₅

—

109

C₂×C₃×C₅

#382

yes

characteristic

110

C₂×C₃×C₅

#430

yes

—

111

C₂³×C₃

C₂×Dic₁₀

yes

characteristic

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

C₄⋊Dic₅

yes

—

114

C₂×C₃×C₄

C₂×D₂₀

yes

characteristic

115

C₂×C₃×C₄

#709

yes

—

116

C₂×C₃×C₄

#709

yes

—

117

C₂×C₃×C₄

C₂²×Dic₅

yes

characteristic

118

C₃×C₈

—

119

C₃⋊C₈

—

120

C₃⋊C₈

—

121

C₂²×C₅

—

122

C₂²×C₅

#890

yes

—

123

C₂²×C₅

C₂×Dic₃⋊C₄

yes

characteristic

124

C₄×C₅

—

125

C₄×C₅

#892

yes

—

126

C₄×C₅

#936

yes

characteristic

127

C₄×C₅

#947

yes

characteristic

128

C₂²×C₄

#1204

yes

—

129

C₂²×C₄

C₂×D₃×D₅

yes

characteristic

130

M₁₆

—

131

M₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₄²

D₃×Dic₅

yes

—

138

C₄²

#1202

yes

—

139

C₃×C₅

#1838

yes

characteristic, complemented, Hall

140

C₂²×C₃

—

141

C₂²×C₃

#4016

yes

—

142

C₂²×C₃

C₂×C₄⋊Dic₅

yes

characteristic

143

C₃×C₄

—

144

C₃×C₄

#4021

yes

characteristic

145

C₃×C₄

#4021

yes

characteristic

146

C₃×C₄

#4072

yes

—

147

C₂×C₅

—

148

C₂×C₅

#4977

yes

characteristic

149

C₂×C₅

#5124

yes

—

150

C₂³

C₂×#1204

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

Dic₃⋊Dic₅

yes

—

153

C₂×C₄

#6765

yes

—

154

C₂×C₄

#6769

yes

—

155

C₂×C₄

C₂×D₃×Dic₅

yes

characteristic

156

C₂×C₄

C₂×#1202

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

160

C₂×C₃

—

161

C₂×C₃

#65225

yes

characteristic

162

C₂×C₃

#65414

yes

—

163

C₅

#70517

yes

characteristic, complemented, Sylow

164

C₂²

—

165

C₂²

#91381

yes

—

166

C₂²

C₂×Dic₃⋊Dic₅

yes

characteristic

167

C₄

—

168

C₄

#91338

yes

—

169

C₄

#91364

yes

characteristic

170

C₄

#91374

yes

characteristic

171

C₃

#10600634

yes

characteristic, complemented, Sylow

172

C₂

—

173

C₂

#11748480

yes

—

174

C₂

#11748619

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1202
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₂×Q₈
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1838
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂×C₄

C₂×#1202

commutator

C₂×C₃×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂×D₃×D₅

Fitting

C₂×C₃×C₄²×C₅

C₂²

socle

C₂²×C₃×C₅

C₂×C₄⋊C₄

prime radical

C₂³×C₃×C₅

C₂×Q₈

cosocle

C₂²×C₃×C₄×C₅

C₂³

hypercenter

C₂×C₄²

D₃×D₅

hypocenter

C₃×C₅

#1838

LCS

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
139. C₃×C₅	#1838	semidirect
163. C₅	#70517	semidirect
171. C₃	#10600634	semidirect

H ≅ G/N

type

139. C₃×C₅

#1838

semidirect

163. C₅

#70517

semidirect

171. C₃

#10600634

semidirect

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

subgroup	index
#11751025	2
C₂×C₅×#803	2
C₃×#65213	2
#11745983	2
#10600634	3
#70517	5

subgroup

index

C₂×C₅×#803

Subgroups — #14298932

Conjugacy classes of subgroups

Complements

Maximal subgroups