Group #12015638 — subgroups

Subgroups — #12015638

Order 1280.

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

normal subgroups only

This group has 366 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	C₅×#2916	C₂	1	1	yes	maximal, characteristic
2	#10602805	C₂	1	1	yes	maximal, characteristic
3	#10602797	C₂	1	1	yes	maximal, characteristic
4	#10598226	C₂	1	2	yes	maximal, complemented
5	#10598220	C₂	1	2	yes	maximal, complemented
6	C₅×#568	—	2	1	no	—
7	C₂×C₅×#131	C₂²	1	1	yes	characteristic
8	C₅×#430	—	2	1	no	—
9	C₅×#430	C₄	1	4	yes	complemented
10	C₂×C₅×#108	—	2	2	no	—
11	C₂×C₅×#108	C₂²	1	2	yes	—
12	#65529	—	2	1	no	—
13	#65395	C₂²	1	2	yes	—
14	C₂×#3944	C₂²	1	1	yes	characteristic
15	C₂²×#672	C₂²	1	1	yes	characteristic
16	#64777	—	2	1	no	—
17	C₂²⋊#672	—	2	2	no	—
18	#12877	—	5	1	no	maximal, Sylow
19	C₂×C₅×#41	D₄	1	1	yes	characteristic
20	C₅×#131	—	2	2	no	—
21	C₅×#131	—	2	2	no	—
22	C₅×#131	—	4	4	no	—
23	C₂×C₅×M₁₆	—	2	4	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	D₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	#4072	—	2	2	no	—
30	C₅×#108	—	2	4	no	—
31	C₅×#108	—	2	2	no	—
32	C₅×#108	C₂×C₄	1	4	yes	—
33	#3966	—	4	2	no	—
34	#3950	—	2	2	no	—
35	#3949	—	2	2	no	—
36	#3944	—	2	2	no	—
37	C₂×#672	—	2	1	no	—
38	C₂×#672	—	2	1	no	—
39	C₂×#672	—	2	2	no	—
40	#2916	D₅	1	1	yes	characteristic
41	#2113	—	5	1	no	—
42	#1812	—	5	1	no	—
43	#1389	—	5	2	no	—
44	#1382	—	5	2	no	—
45	C₅×#41	—	2	2	no	—
46	C₅×#41	—	2	2	no	—
47	C₂×C₅×Q₈	Q₁₆	1	1	yes	characteristic
48	C₂×C₅×D₄	SD₁₆	1	2	yes	—
49	C₂×C₅×D₄	Q₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₅×M₁₆	—	2	2	no	—
53	C₅×M₁₆	—	2	4	no	—
54	C₅×M₁₆	—	2	4	no	—
55	C₅×M₁₆	—	4	4	no	—
56	C₅×M₁₆	—	4	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	4	4	no	—
60	C₂×C₅×C₈	—	4	2	no	—
61	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
62	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	#41	1	2	yes	—
65	C₂×C₅×C₈	#41	1	2	yes	—
66	C₂²⋊Dic₅	—	4	2	no	—
67	C₄⋊Dic₅	—	4	2	no	—
68	C₄×Dic₅	—	4	2	no	—
69	#672	—	4	1	no	—
70	#672	—	4	1	no	—
71	#568	—	2	1	no	—
72	C₂×#131	C₂×D₅	1	1	yes	characteristic
73	C₂×M₃₂	—	5	1	no	—
74	C₂²×C₁₆	—	5	1	no	—
75	#430	—	2	1	no	—
76	#430	Dic₅	1	4	yes	complemented
77	C₂×#108	—	2	2	no	—
78	C₂×#108	C₂×D₅	1	2	yes	—
79	#427	—	5	2	no	—
80	#410	—	10	1	no	—
81	C₂²⋊C₁₆	—	10	2	no	—
82	H(3,ℤ/4)	—	10	1	no	—
83	C₂³×C₅	#103	1	2	yes	—
84	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
85	C₅×Q₈	—	2	2	no	—
86	C₅×D₄	—	2	4	no	—
87	C₅×D₄	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	#103	1	2	yes	—
91	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅	#123	1	2	yes	—
93	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×Dic₅	—	8	2	no	—
96	C₅×C₈	—	2	2	no	—
97	C₅×C₈	—	4	4	no	—
98	C₅×C₈	#117	1	2	yes	—
99	C₅×C₈	#123	1	1	yes	characteristic
100	C₅×C₈	#123	1	1	yes	characteristic
101	C₂×#41	C₂²⋊D₅	1	1	yes	characteristic
102	#131	—	2	2	no	—
103	#131	—	2	2	no	—
104	#131	—	4	4	no	—
105	C₂×M₁₆	—	2	4	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
108	C₂×M₁₆	C₂²⋊D₅	1	1	yes	characteristic
109	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
110	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
111	#117	—	10	2	no	—
112	M₃₂	—	10	2	no	—
113	C₂×C₁₆	—	10	2	no	—
114	C₂×C₁₆	—	10	1	no	—
115	C₂×C₁₆	—	10	1	no	—
116	#108	—	2	4	no	—
117	#108	—	2	2	no	—
118	#108	C₂×Dic₅	1	4	yes	—
119	#107	—	10	2	no	—
120	#106	—	10	2	no	—
121	#99	—	20	2	no	—
122	C₂²×C₅	—	2	2	no	—
123	C₂²×C₅	—	4	1	no	—
124	C₂²×C₅	—	4	2	no	—
125	C₂²×C₅	C₂×#103	1	1	yes	characteristic
126	C₂²×C₅	#481	1	2	yes	—
127	C₄×C₅	—	2	2	no	—
128	C₄×C₅	#381	1	1	yes	characteristic
129	C₄×C₅	#418	1	1	yes	characteristic
130	C₄×C₅	#484	1	2	yes	—
131	Dic₅	—	16	2	no	—
132	#41	—	2	2	no	—
133	#41	—	2	2	no	—
134	C₂×Q₈	#689	1	1	yes	characteristic
135	C₂×D₄	#688	1	2	yes	—
136	C₂×D₄	#689	1	1	yes	characteristic
137	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
139	M₁₆	—	2	2	no	—
140	M₁₆	—	2	4	no	—
141	M₁₆	—	2	4	no	—
142	M₁₆	—	4	4	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
149	C₂×C₈	#709	1	2	yes	—
150	C₂×C₈	#709	1	2	yes	—
151	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
153	C₄⋊C₄	—	20	2	no	—
154	C₂²⋊C₄	—	20	2	no	—
155	C₄²	—	20	2	no	—
156	C₁₆	—	20	1	no	—
157	C₁₆	—	20	1	no	—
158	C₂×C₅	—	2	2	no	—
159	C₂×C₅	—	2	1	no	—
160	C₂×C₅	—	4	2	no	—
161	C₂×C₅	#1824	1	1	yes	characteristic
162	C₂³	#3967	1	2	yes	—
163	C₂³	C₂×#688	1	1	yes	characteristic
164	Q₈	—	2	2	no	—
165	D₄	—	2	4	no	—
166	D₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	40	2	no	—
170	C₂×C₄	#3967	1	2	yes	—
171	C₂×C₄	#4075	1	2	yes	—
172	C₂×C₄	#4078	1	2	yes	—
173	C₂×C₄	C₂×#689	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	4	4	no	—
177	C₈	#4072	1	2	yes	—
178	C₈	#4075	1	1	yes	characteristic
179	C₈	#4075	1	1	yes	characteristic
180	C₅	#12877	1	1	yes	characteristic, complemented, Sylow
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	#65257	1	2	yes	—
185	C₂²	C₂×#3967	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	80	2	no	—
188	C₄	#65259	1	2	yes	—
189	C₄	#65500	1	1	yes	characteristic
190	C₄	#65504	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#10602347	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2916

C₂

yes

maximal, characteristic

#10602805

C₂

yes

maximal, characteristic

#10602797

C₂

yes

maximal, characteristic

#10598226

C₂

yes

maximal, complemented

#10598220

C₂

yes

maximal, complemented

C₅×#568

—

C₂×C₅×#131

C₂²

yes

characteristic

C₅×#430

—

C₅×#430

C₄

yes

complemented

C₂×C₅×#108

—

C₂×C₅×#108

C₂²

yes

—

#65529

—

#65395

C₂²

yes

—

C₂×#3944

C₂²

yes

characteristic

C₂²×#672

C₂²

yes

characteristic

#64777

—

C₂²⋊#672

—

#12877

—

maximal, Sylow

C₂×C₅×#41

D₄

yes

—

—

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

C₂×C₄

yes

—

C₂×C₅×M₁₆

D₄

yes

characteristic

C₂²×C₅×C₈

D₄

yes

—

C₂²×C₅×C₈

C₂³

yes

—

—

—

C₂×C₄

yes

—

—

—

—

—

—

—

—

D₅

yes

—

—

—

—

C₅×#41

—

C₅×#41

—

C₂×C₅×Q₈

Q₁₆

yes

characteristic

C₂×C₅×D₄

SD₁₆

yes

—

C₂×C₅×D₄

Q₁₆

yes

characteristic

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

#41

yes

—

C₂×C₅×C₈

#41

yes

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

—

—

—

C₂×D₅

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#430

—

#430

Dic₅

yes

complemented

C₂×#108

—

C₂×#108

C₂×D₅

yes

—

#427

—

#410

—

C₂²⋊C₁₆

—

H(3,ℤ/4)

—

C₂³×C₅

#103

yes

—

C₂³×C₅

C₂×SD₁₆

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#103

yes

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×Dic₅

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

#117

yes

—

C₅×C₈

#123

yes

characteristic

100

C₅×C₈

#123

yes

characteristic

101

C₂×#41

C₂²⋊D₅

yes

characteristic

102

#131

—

103

#131

—

104

#131

—

105

C₂×M₁₆

—

106

C₂×M₁₆

—

107

C₂×M₁₆

C₂×Dic₅

yes

—

108

C₂×M₁₆

C₂²⋊D₅

yes

characteristic

109

C₂²×C₈

C₂²⋊D₅

yes

—

110

C₂²×C₈

C₂²×D₅

yes

characteristic

111

#117

—

112

M₃₂

—

113

C₂×C₁₆

—

114

C₂×C₁₆

—

115

C₂×C₁₆

—

116

#108

—

117

#108

—

118

#108

C₂×Dic₅

yes

—

119

#107

—

120

#106

—

121

#99

—

122

C₂²×C₅

—

123

C₂²×C₅

—

124

C₂²×C₅

—

125

C₂²×C₅

C₂×#103

yes

characteristic

126

C₂²×C₅

#481

yes

—

127

C₄×C₅

—

128

C₄×C₅

#381

yes

characteristic

129

C₄×C₅

#418

yes

characteristic

130

C₄×C₅

#484

yes

—

131

Dic₅

—

132

#41

—

133

#41

—

134

C₂×Q₈

#689

yes

characteristic

135

C₂×D₄

#688

yes

—

136

C₂×D₄

#689

yes

characteristic

137

C₂²×C₄

C₂²⋊Dic₅

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

139

M₁₆

—

140

M₁₆

—

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

C₂²⋊Dic₅

yes

—

149

C₂×C₈

#709

yes

—

150

C₂×C₈

#709

yes

—

151

C₂×C₈

C₂²×Dic₅

yes

characteristic

152

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

153

C₄⋊C₄

—

154

C₂²⋊C₄

—

155

C₄²

—

156

C₁₆

—

157

C₁₆

—

158

C₂×C₅

—

159

C₂×C₅

—

160

C₂×C₅

—

161

C₂×C₅

#1824

yes

characteristic

162

C₂³

#3967

yes

—

163

C₂³

C₂×#688

yes

characteristic

164

Q₈

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#3967

yes

—

171

C₂×C₄

#4075

yes

—

172

C₂×C₄

#4078

yes

—

173

C₂×C₄

C₂×#689

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊Dic₅

yes

characteristic

175

C₈

—

176

C₈

—

177

C₈

#4072

yes

—

178

C₈

#4075

yes

characteristic

179

C₈

#4075

yes

characteristic

180

C₅

#12877

yes

characteristic, complemented, Sylow

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

#65257

yes

—

185

C₂²

C₂×#3967

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

#65259

yes

—

189

C₄

#65500

yes

characteristic

190

C₄

#65504

yes

characteristic

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#10602347

yes

characteristic

	subgroup	G/H
center	C₄	#65504
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2916	C₂
socle	C₂×C₅	#1824
prime radical	C₂×C₅×#41	D₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2916	D₅
hypocenter	C₅	#12877
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

subgroup

G/H

center

C₄

#65504

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅

C₂

C₂×C₅

C₂×C₅×#41

D₄

cosocle

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

N	H ≅ G/N	type
4. #10598226	C₂	semidirect
5. #10598220	C₂	semidirect
9. C₅×#430	C₄	semidirect
76. #430	Dic₅	semidirect
180. C₅	#12877	semidirect

H ≅ G/N

type

4. #10598226

C₂

semidirect

5. #10598220

C₂

semidirect

9. C₅×#430

C₄

semidirect

76. #430

Dic₅

semidirect

180. C₅

#12877

semidirect

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

subgroup	index
C₅×#2916	2
#10602805	2
#10602797	2
#10598226	2
#10598220	2
#12877	5

subgroup

index

Subgroups — #12015638

Conjugacy classes of subgroups

Complements

Maximal subgroups