Group #14293773 — subgroups

Subgroups — #14293773

Order 1920.

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

normal subgroups only

This group has 352 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	#11750800	C₂	1	1	yes	maximal, characteristic
2	C₅×#4680	C₂	1	1	yes	maximal, characteristic
3	#11746941	C₂	1	2	yes	maximal
4	#11746929	C₂	1	2	yes	maximal
5	C₃×#64675	C₂	1	1	yes	maximal, characteristic
6	#10602131	—	3	1	no	maximal, Hall
7	C₂×Dic₆₀	—	2	2	no	—
8	#91758	C₂²	1	1	yes	characteristic
9	#91758	C₂²	1	1	yes	characteristic
10	C₅×#869	C₂²	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#817	—	2	2	no	—
13	C₃×C₄×#168	C₂²	1	2	yes	—
14	#90985	—	2	4	no	—
15	#90973	—	2	4	no	—
16	#68095	—	5	1	no	maximal, Hall
17	C₅×#498	—	3	1	no	—
18	#65370	—	3	2	no	—
19	#65368	—	3	2	no	—
20	#64988	—	3	1	no	—
21	#64675	D₃	1	1	yes	characteristic
22	C₂×Dic₃₀	—	2	1	no	—
23	C₂×Dic₃₀	—	2	1	no	—
24	C₂×C₅×Dic₆	—	2	2	no	—
25	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
26	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₁₅	—	2	2	no	—
28	C₄⋊Dic₁₅	—	2	2	no	—
29	Dic₆₀	—	4	4	no	—
30	C₅×C₄⋊Dic₃	—	2	2	no	—
31	C₅×C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₃×#168	D₄	1	4	yes	—
33	#4699	—	5	1	no	—
34	#4681	—	5	2	no	—
35	#4680	—	5	2	no	—
36	#4680	D₅	1	1	yes	characteristic
37	C₃×#322	—	5	1	no	—
38	C₅×C₄⋊Q₈	—	3	2	no	—
39	C₂×Dic₂₀	—	6	2	no	—
40	#4015	—	3	1	no	—
41	#4015	—	3	1	no	—
42	C₅×#106	—	6	2	no	—
43	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
44	#3967	—	6	4	no	—
45	#3940	—	6	4	no	—
46	C₄×#168	C₂×D₃	1	2	yes	—
47	#1716	—	15	1	no	Sylow
48	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
50	C₂×Dic₁₅	—	4	1	no	—
51	C₂×Dic₁₅	—	4	1	no	—
52	Dic₃₀	—	4	2	no	—
53	Dic₃₀	—	4	2	no	—
54	C₂×C₅×Dic₃	—	4	2	no	—
55	C₅×Dic₆	—	4	4	no	—
56	C₃×C₅×C₈	SD₁₆	1	4	yes	—
57	C₃×#168	—	2	4	no	—
58	C₂×Dic₁₂	—	10	2	no	—
59	#869	—	5	1	no	—
60	#869	—	5	1	no	—
61	#869	C₂×D₅	1	2	yes	—
62	C₃×C₄×C₈	—	5	2	no	—
63	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
64	#817	—	2	2	no	—
65	#817	—	10	4	no	—
66	#816	—	10	4	no	—
67	C₂×C₅×Q₈	—	6	2	no	—
68	C₂×Dic₁₀	—	6	1	no	—
69	C₂×Dic₁₀	—	6	1	no	—
70	C₂×C₅×C₈	D₁₂	1	2	yes	—
71	C₅×C₄⋊C₄	—	6	2	no	—
72	C₅×C₄⋊C₄	—	6	2	no	—
73	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
74	C₄⋊Dic₅	—	6	2	no	—
75	C₄⋊Dic₅	—	6	2	no	—
76	C₂×#168	D₁₂	1	4	yes	—
77	Dic₂₀	—	12	4	no	—
78	#498	—	3	1	no	—
79	#498	—	15	2	no	—
80	#494	—	15	1	no	—
81	#487	—	15	2	no	—
82	#322	—	5	1	no	—
83	C₂²×C₃×C₅	#127	1	1	yes	characteristic
84	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
85	C₃×C₄×C₅	#137	1	4	yes	—
86	Dic₁₅	—	8	1	no	—
87	Dic₁₅	—	8	1	no	—
88	C₅×Dic₃	—	8	2	no	—
89	C₂×Dic₆	—	2	2	no	—
90	C₂×Dic₆	—	10	1	no	—
91	C₂×Dic₆	—	10	1	no	—
92	C₂×C₃×C₈	—	5	4	no	—
93	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
94	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
95	C₄⋊Dic₃	—	2	2	no	—
96	C₄⋊Dic₃	—	2	2	no	—
97	C₄⋊Dic₃	—	10	2	no	—
98	C₄⋊Dic₃	—	10	2	no	—
99	Dic₁₂	—	20	4	no	—
100	C₅×Q₈	—	12	4	no	—
101	C₂×C₄×C₅	—	12	2	no	—
102	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
103	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
104	C₂×Dic₅	—	12	1	no	—
105	C₂×Dic₅	—	12	1	no	—
106	Dic₁₀	—	12	2	no	—
107	Dic₁₀	—	12	2	no	—
108	C₅×C₈	#204	1	4	yes	—
109	#168	—	2	4	no	—
110	C₂×Q₁₆	—	30	2	no	—
111	C₄⋊Q₈	—	3	2	no	—
112	C₄⋊Q₈	—	15	1	no	—
113	C₄⋊Q₈	—	15	1	no	—
114	#106	—	6	2	no	—
115	#106	—	30	4	no	—
116	#103	—	30	4	no	—
117	C₄×C₈	—	5	2	no	—
118	C₄×C₈	D₃×D₅	1	1	yes	characteristic
119	C₂×C₃×C₅	#492	1	1	yes	characteristic
120	C₂×C₃×C₅	#497	1	2	yes	—
121	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
122	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
123	C₂×Dic₃	—	4	2	no	—
124	C₂×Dic₃	—	20	1	no	—
125	C₂×Dic₃	—	20	1	no	—
126	Dic₆	—	4	4	no	—
127	Dic₆	—	20	2	no	—
128	Dic₆	—	20	2	no	—
129	C₃×C₈	—	10	4	no	—
130	C₃×C₈	#688	1	4	yes	—
131	C₂²×C₅	#874	1	1	yes	characteristic
132	C₄×C₅	—	24	2	no	—
133	C₄×C₅	C₂×#204	1	2	yes	—
134	C₄×C₅	#909	1	4	yes	—
135	Dic₅	—	24	1	no	—
136	Dic₅	—	24	1	no	—
137	C₂×Q₈	—	6	2	no	—
138	C₂×Q₈	—	30	1	no	—
139	C₂×Q₈	—	30	1	no	—
140	Q₁₆	—	60	4	no	—
141	C₂×C₈	—	5	4	no	—
142	C₂×C₈	#1203	1	2	yes	—
143	C₄⋊C₄	—	6	2	no	—
144	C₄⋊C₄	—	6	2	no	—
145	C₄⋊C₄	—	30	2	no	—
146	C₄⋊C₄	—	30	2	no	—
147	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
148	C₃×C₅	#1716	1	1	yes	characteristic, complemented, Hall
149	C₂²×C₃	#4086	1	1	yes	characteristic
150	C₃×C₄	#4061	1	4	yes	—
151	C₃×C₄	C₂×#688	1	2	yes	—
152	Dic₃	—	8	2	no	—
153	Dic₃	—	40	1	no	—
154	Dic₃	—	40	1	no	—
155	C₂×C₅	#4693	1	1	yes	characteristic
156	C₂×C₅	#4715	1	2	yes	—
157	Q₈	—	12	4	no	—
158	Q₈	—	60	2	no	—
159	Q₈	—	60	2	no	—
160	C₂×C₄	—	12	2	no	—
161	C₂×C₄	—	60	1	no	—
162	C₂×C₄	—	60	1	no	—
163	C₂×C₄	D₅×D₁₂	1	2	yes	—
164	C₂×C₄	C₂×#1203	1	1	yes	characteristic
165	C₈	—	10	4	no	—
166	C₈	#6660	1	4	yes	—
167	C₂×C₃	#65374	1	1	yes	characteristic
168	C₂×C₃	#65456	1	2	yes	—
169	C₅	#68095	1	1	yes	characteristic, complemented, Sylow
170	C₂²	#91451	1	1	yes	characteristic
171	C₄	—	24	2	no	—
172	C₄	—	120	1	no	—
173	C₄	—	120	1	no	—
174	C₄	#91260	1	4	yes	—
175	C₄	C₂×#6660	1	2	yes	—
176	C₃	#10602131	1	1	yes	characteristic, complemented, Sylow
177	C₂	#11747067	1	1	yes	characteristic
178	C₂	#11748024	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11750800

C₂

yes

maximal, characteristic

C₅×#4680

C₂

yes

maximal, characteristic

#11746941

C₂

yes

maximal

#11746929

C₂

yes

maximal

C₃×#64675

C₂

yes

maximal, characteristic

#10602131

—

maximal, Hall

C₂×Dic₆₀

—

#91758

C₂²

yes

characteristic

#91758

C₂²

yes

characteristic

C₅×#869

C₂²

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#817

—

C₃×C₄×#168

C₂²

yes

—

—

—

—

—

—

—

—

D₃

yes

characteristic

C₂×Dic₃₀

—

C₂×Dic₃₀

—

C₂×C₅×Dic₆

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₁₅

—

C₄⋊Dic₁₅

—

Dic₆₀

—

C₅×C₄⋊Dic₃

—

C₅×C₄⋊Dic₃

—

C₂×C₃×#168

D₄

yes

—

—

—

—

D₅

yes

characteristic

C₃×#322

—

C₅×C₄⋊Q₈

—

C₂×Dic₂₀

—

#4015

—

#4015

—

C₅×#106

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

C₂×D₃

yes

—

#1716

—

Sylow

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₂×Dic₁₅

—

C₂×Dic₁₅

—

Dic₃₀

—

Dic₃₀

—

C₂×C₅×Dic₃

—

C₅×Dic₆

—

C₃×C₅×C₈

SD₁₆

yes

—

C₃×#168

—

C₂×Dic₁₂

—

#869

—

#869

—

#869

C₂×D₅

yes

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₅

yes

—

—

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₁₂

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²×D₃

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

D₁₂

yes

—

Dic₂₀

—

—

—

—

—

—

C₂²×C₃×C₅

#127

yes

characteristic

C₃×C₄×C₅

C₂×SD₁₆

yes

—

C₃×C₄×C₅

#137

yes

—

Dic₁₅

—

Dic₁₅

—

C₅×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊D₅

yes

—

C₃×C₄²

C₂²×D₅

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

100

C₅×Q₈

—

101

C₂×C₄×C₅

—

102

C₂×C₄×C₅

C₂×D₁₂

yes

—

103

C₂×C₄×C₅

C₂×D₁₂

yes

characteristic

104

C₂×Dic₅

—

105

C₂×Dic₅

—

106

Dic₁₀

—

107

Dic₁₀

—

108

C₅×C₈

#204

yes

—

109

#168

—

110

C₂×Q₁₆

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

#106

—

115

#106

—

116

#103

—

117

C₄×C₈

—

118

C₄×C₈

D₃×D₅

yes

characteristic

119

C₂×C₃×C₅

#492

yes

characteristic

120

C₂×C₃×C₅

#497

yes

—

121

C₂×C₃×C₄

D₄×D₅

yes

—

122

C₂×C₃×C₄

C₂×C₂²⋊D₅

yes

characteristic

123

C₂×Dic₃

—

124

C₂×Dic₃

—

125

C₂×Dic₃

—

126

Dic₆

—

127

Dic₆

—

128

Dic₆

—

129

C₃×C₈

—

130

C₃×C₈

#688

yes

—

131

C₂²×C₅

#874

yes

characteristic

132

C₄×C₅

—

133

C₄×C₅

C₂×#204

yes

—

134

C₄×C₅

#909

yes

—

135

Dic₅

—

136

Dic₅

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

Q₁₆

—

141

C₂×C₈

—

142

C₂×C₈

#1203

yes

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄²

C₂×D₃×D₅

yes

characteristic

148

C₃×C₅

#1716

yes

characteristic, complemented, Hall

149

C₂²×C₃

#4086

yes

characteristic

150

C₃×C₄

#4061

yes

—

151

C₃×C₄

C₂×#688

yes

—

152

Dic₃

—

153

Dic₃

—

154

Dic₃

—

155

C₂×C₅

#4693

yes

characteristic

156

C₂×C₅

#4715

yes

—

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

D₅×D₁₂

yes

—

164

C₂×C₄

C₂×#1203

yes

characteristic

165

C₈

—

166

C₈

#6660

yes

—

167

C₂×C₃

#65374

yes

characteristic

168

C₂×C₃

#65456

yes

—

169

C₅

#68095

yes

characteristic, complemented, Sylow

170

C₂²

#91451

yes

characteristic

171

C₄

—

172

C₄

—

120

—

173

C₄

—

120

—

174

C₄

#91260

yes

—

175

C₄

C₂×#6660

yes

—

176

C₃

#10602131

yes

characteristic, complemented, Sylow

177

C₂

#11747067

yes

characteristic

178

C₂

#11748024

yes

—

	subgroup	G/H
center	C₂²	#91451
commutator, cosocle	C₃×C₄²×C₅	C₂³
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle, prime radical	C₂²×C₃×C₅	#127
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1716
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

#91451

commutator, cosocle

C₃×C₄²×C₅

C₂³

Frattini

C₄²

C₂×D₃×D₅

Fitting

C₃×C₄×C₅×C₈

C₂²

socle, prime radical

C₂²×C₃×C₅

#127

hypercenter

C₄×C₈

D₃×D₅

hypocenter

C₃×C₅

#1716

LCS

G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
148. C₃×C₅	#1716	semidirect
169. C₅	#68095	semidirect
176. C₃	#10602131	semidirect

H ≅ G/N

type

148. C₃×C₅

#1716

semidirect

169. C₅

#68095

semidirect

176. C₃

#10602131

semidirect

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

subgroup	index
#11750800	2
C₅×#4680	2
#11746941	2
#11746929	2
C₃×#64675	2
#10602131	3
#68095	5

subgroup

index

Subgroups — #14293773

Conjugacy classes of subgroups

Complements

Maximal subgroups