Group #12019191 — subgroups

Subgroups — #12019191

Order 1280.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₅×#2161	C₂	1	1	yes	maximal, characteristic, complemented
2	#10602224	C₂	1	1	yes	maximal, characteristic
3	#10601504	C₂	1	2	yes	maximal
4	#10598311	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598224	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	#65415	—	2	2	no	—
10	C₂×#3951	C₂²	1	2	yes	—
11	C₂×#3944	C₂²	1	1	yes	characteristic
12	C₅×#368	—	2	2	no	—
13	#64778	—	2	2	no	—
14	#64727	—	2	2	no	—
15	#13498	—	5	1	no	maximal, Sylow
16	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
17	C₂×C₅×M₁₆	—	2	2	no	—
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₂×#681	—	2	2	no	—
20	C₂²×#168	D₄	1	1	yes	characteristic
21	C₂×C₈×D₅	—	2	1	no	—
22	C₂×C₈×D₅	—	2	2	no	—
23	C₂×C₈×D₅	—	2	1	no	—
24	C₂×C₈×D₅	—	2	2	no	—
25	C₅×M₃₂	—	2	2	no	—
26	C₂×C₅×C₁₆	D₄	1	2	yes	—
27	C₅×#108	—	2	2	no	—
28	C₅×#108	—	2	2	no	—
29	#3952	—	4	4	no	—
30	#3951	—	2	2	no	—
31	#3951	—	2	2	no	—
32	#3944	—	2	2	no	—
33	C₂×#672	—	2	1	no	—
34	#2161	D₅	1	1	yes	characteristic, complemented
35	#2112	—	5	1	no	—
36	#1810	—	5	2	no	—
37	#1388	—	5	2	no	—
38	#1385	—	5	1	no	—
39	C₂³×D₅	Q₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₅	Q₁₆	1	1	yes	characteristic
42	C₂×C₄×D₅	—	2	1	no	—
43	C₂×C₄×D₅	—	2	2	no	—
44	C₂×C₄×D₅	—	2	1	no	—
45	C₂×C₄×D₅	—	2	2	no	—
46	C₅×M₁₆	—	4	2	no	—
47	C₅×M₁₆	—	4	2	no	—
48	C₂×C₅×C₈	—	4	2	no	—
49	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₅×C₈	#41	1	2	yes	—
51	C₂×C₅×C₈	#41	1	1	yes	characteristic
52	C₂×C₅×C₈	#41	1	2	yes	—
53	#681	—	4	2	no	—
54	#681	—	4	2	no	—
55	C₂×#168	—	2	1	no	—
56	C₂×#168	—	2	2	no	—
57	C₂×#168	—	4	2	no	—
58	C₈×D₅	—	4	1	no	—
59	C₈×D₅	—	4	1	no	—
60	C₈×D₅	—	4	2	no	—
61	C₅×C₁₆	—	2	2	no	—
62	#672	—	4	1	no	—
63	C₂³×C₈	—	5	1	no	—
64	C₂×M₃₂	—	5	1	no	—
65	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
66	C₂×#108	—	5	2	no	—
67	C₂×#108	C₂×D₅	1	2	yes	—
68	#407	—	10	2	no	—
69	#368	—	2	2	no	—
70	C₂²⋊C₁₆	—	10	2	no	—
71	#341	—	10	2	no	—
72	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
73	C₂²×D₅	—	2	1	no	—
74	C₂²×D₅	—	2	2	no	—
75	C₂²×D₅	—	4	2	no	—
76	C₂×C₄×C₅	#121	1	1	yes	characteristic
77	C₂×C₄×C₅	#123	1	2	yes	—
78	C₂×C₄×C₅	#135	1	1	yes	characteristic
79	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×Dic₅	—	2	1	no	—
81	C₂×Dic₅	—	2	2	no	—
82	C₄×D₅	—	4	1	no	—
83	C₄×D₅	—	4	1	no	—
84	C₄×D₅	—	4	2	no	—
85	C₅×C₈	—	8	2	no	—
86	C₅×C₈	#123	1	1	yes	characteristic
87	C₅×C₈	#124	1	1	yes	characteristic
88	C₅×C₈	#126	1	2	yes	—
89	#168	—	4	1	no	—
90	#168	—	8	2	no	—
91	C₂³×C₄	—	5	1	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	10	2	no	—
94	C₂²×C₈	—	5	1	no	—
95	C₂²×C₈	—	10	2	no	—
96	C₂²×C₈	—	10	1	no	—
97	C₂²×C₈	—	10	2	no	—
98	C₂²×C₈	—	10	1	no	—
99	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	10	2	no	—
102	C₂×C₁₆	—	10	1	no	—
103	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
104	#108	—	2	2	no	—
105	#108	—	2	2	no	—
106	#108	—	10	2	no	—
107	#108	—	10	2	no	—
108	C₂²⋊C₈	—	20	4	no	—
109	C₂²×C₅	—	4	1	no	—
110	C₂²×C₅	#467	1	1	yes	characteristic
111	C₂²×C₅	#484	1	2	yes	—
112	C₂×D₅	—	2	2	no	—
113	C₂×D₅	—	2	4	no	—
114	C₂×D₅	—	4	1	no	—
115	C₂×D₅	—	4	2	no	—
116	C₂×D₅	—	4	1	no	—
117	C₄×C₅	#397	1	1	yes	characteristic
118	C₄×C₅	#469	1	1	yes	characteristic
119	C₄×C₅	#482	1	2	yes	—
120	Dic₅	—	4	1	no	—
121	C₂⁴	—	5	1	no	—
122	C₂²×C₄	—	5	1	no	—
123	C₂²×C₄	—	10	2	no	—
124	C₂²×C₄	—	10	2	no	—
125	C₂²×C₄	—	10	1	no	—
126	C₂²×C₄	—	10	1	no	—
127	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
128	M₁₆	—	4	2	no	—
129	M₁₆	—	4	2	no	—
130	M₁₆	—	20	2	no	—
131	M₁₆	—	20	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	10	2	no	—
134	C₂×C₈	—	10	1	no	—
135	C₂×C₈	—	20	2	no	—
136	C₂×C₈	—	20	1	no	—
137	C₂×C₈	—	20	1	no	—
138	C₂×C₈	—	20	2	no	—
139	C₂×C₈	#709	1	2	yes	—
140	C₂×C₈	#711	1	1	yes	characteristic
141	C₂×C₈	#713	1	2	yes	—
142	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
143	C₁₆	—	2	2	no	—
144	C₁₆	—	20	1	no	—
145	C₂×C₅	—	2	1	no	—
146	C₂×C₅	—	2	2	no	—
147	C₂×C₅	#2076	1	1	yes	characteristic
148	D₅	—	4	2	no	—
149	C₂³	—	10	2	no	—
150	C₂³	—	10	1	no	—
151	C₂³	—	20	2	no	—
152	C₂³	D₅×Q₁₆	1	1	yes	characteristic
153	C₂×C₄	—	10	2	no	—
154	C₂×C₄	—	10	1	no	—
155	C₂×C₄	—	20	1	no	—
156	C₂×C₄	—	20	2	no	—
157	C₂×C₄	—	20	1	no	—
158	C₂×C₄	#4040	1	2	yes	—
159	C₂×C₄	#4057	1	2	yes	—
160	C₂×C₄	#4065	1	1	yes	characteristic
161	C₂×C₄	#4084	1	1	yes	characteristic
162	C₈	—	8	2	no	—
163	C₈	—	20	1	no	—
164	C₈	—	40	2	no	—
165	C₈	#4044	1	2	yes	—
166	C₈	#4075	1	1	yes	characteristic
167	C₈	#4093	1	1	yes	characteristic
168	C₅	#13498	1	1	yes	characteristic, complemented, Sylow
169	C₂²	—	4	1	no	—
170	C₂²	—	10	4	no	—
171	C₂²	—	10	2	no	—
172	C₂²	—	20	2	no	—
173	C₂²	—	20	1	no	—
174	C₂²	—	20	1	no	—
175	C₂²	#65173	1	2	yes	—
176	C₂²	#65477	1	1	yes	characteristic
177	C₄	—	20	1	no	—
178	C₄	#65174	1	2	yes	—
179	C₄	#65282	1	1	yes	characteristic
180	C₄	#65448	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	—	2	1	no	—
183	C₂	—	20	2	no	—
184	C₂	#10601430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2161

C₂

yes

maximal, characteristic, complemented

#10602224

C₂

yes

maximal, characteristic

#10601504

C₂

yes

maximal

#10598311

C₂

yes

maximal, characteristic, complemented

#10598224

C₂

yes

maximal, complemented

C₂²×C₈×D₅

C₂²

yes

characteristic

C₂×C₅×M₃₂

C₂²

yes

characteristic

C₂×C₅×#108

C₂²

yes

—

#65415

—

C₂×#3951

C₂²

yes

—

C₂×#3944

C₂²

yes

—

—

—

—

C₂²×C₄×D₅

D₄

yes

characteristic

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×#681

—

C₂²×#168

D₄

yes

characteristic

C₂×C₈×D₅

—

C₂×C₈×D₅

—

C₂×C₈×D₅

—

C₂×C₈×D₅

—

C₅×M₃₂

—

C₂×C₅×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

D₅

yes

characteristic, complemented

—

—

—

—

C₂³×D₅

Q₁₆

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

Q₁₆

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

#41

yes

—

C₂×C₅×C₈

#41

yes

characteristic

C₂×C₅×C₈

#41

yes

—

—

—

—

—

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₅×C₁₆

—

#672

—

C₂³×C₈

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₅

yes

—

#407

—

#368

—

C₂²⋊C₁₆

—

#341

—

C₂³×C₅

C₂×Q₁₆

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

#135

yes

characteristic

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

yes

C₅×C₈

yes

C₅×C₈

yes

—

#168

—

#168

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₅

yes

characteristic

100

M₃₂

—

101

M₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₅

yes

—

104

#108

—

105

#108

—

106

#108

—

107

#108

—

108

C₂²⋊C₈

—

109

C₂²×C₅

—

110

C₂²×C₅

#467

yes

characteristic

111

C₂²×C₅

#484

yes

—

112

C₂×D₅

—

113

C₂×D₅

—

114

C₂×D₅

—

115

C₂×D₅

—

116

C₂×D₅

—

117

C₄×C₅

#397

yes

characteristic

118

C₄×C₅

#469

yes

characteristic

119

C₄×C₅

#482

yes

—

120

Dic₅

—

121

C₂⁴

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

D₄×D₅

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

131

M₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

#709

yes

—

140

C₂×C₈

#711

yes

characteristic

141

C₂×C₈

#713

yes

—

142

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

143

C₁₆

—

144

C₁₆

—

145

C₂×C₅

—

146

C₂×C₅

—

147

C₂×C₅

#2076

yes

characteristic

148

D₅

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

D₅×Q₁₆

yes

characteristic

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#4040

yes

—

159

C₂×C₄

#4057

yes

—

160

C₂×C₄

#4065

yes

characteristic

161

C₂×C₄

#4084

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

#4044

yes

—

166

C₈

#4075

yes

characteristic

167

C₈

#4093

yes

characteristic

168

C₅

#13498

yes

characteristic, complemented, Sylow

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#65173

yes

—

176

C₂²

#65477

yes

characteristic

177

C₄

—

178

C₄

#65174

yes

—

179

C₄

#65282

yes

characteristic

180

C₄

#65448

yes

characteristic

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

#10601430

yes

characteristic

	subgroup	G/H
center	C₄	#65448
commutator, cosocle	C₂²×C₅×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2161	C₂
socle	C₂×C₅	#2076
prime radical	C₂³×D₅	Q₁₆
hypercenter	#2161	D₅
hypocenter	C₅	#13498
LCS	G ▷ C₂²×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

subgroup

G/H

center

C₄

#65448

commutator, cosocle

C₂²×C₅×C₈

C₂³

Frattini

C₂²×C₈

C₂²×D₅

C₂

C₂×C₅

C₂³×D₅

Q₁₆

D₅

C₅

G ▷ C₂²×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

N	H ≅ G/N	type
1. C₅×#2161	C₂	semidirect
4. #10598311	C₂	semidirect
5. #10598224	C₂	semidirect
34. #2161	D₅	semidirect
168. C₅	#13498	semidirect

H ≅ G/N

type

1. C₅×#2161

C₂

semidirect

4. #10598311

C₂

semidirect

5. #10598224

C₂

semidirect

34. #2161

D₅

semidirect

168. C₅

#13498

semidirect

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

subgroup	index
C₅×#2161	2
#10602224	2
#10601504	2
#10598311	2
#10598224	2
#13498	5

subgroup

index

Subgroups — #12019191

Conjugacy classes of subgroups

Complements

Maximal subgroups