Group #12019196 — subgroups

Subgroups — #12019196

Order 1280.

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

normal subgroups only

This group has 276 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	#10602225	C₂	1	1	yes	maximal, characteristic
3	#10601512	C₂	1	2	yes	maximal
4	#10598309	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598220	C₂	1	2	yes	maximal, complemented
6	C₂×#4047	C₂²	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	#65428	—	2	2	no	—
10	#65395	C₂²	1	2	yes	—
11	C₂²×#672	C₂²	1	1	yes	characteristic
12	C₅×#368	—	2	2	no	—
13	#64776	—	2	2	no	—
14	#64727	—	2	2	no	—
15	#13499	—	5	1	no	maximal, Sylow
16	C₂×#709	D₄	1	1	yes	characteristic
17	C₂×C₅×M₁₆	—	2	2	no	—
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	#4072	—	2	2	no	—
20	C₂×#681	D₄	1	1	yes	characteristic
21	#4047	—	2	2	no	—
22	#4047	—	2	2	no	—
23	C₂×#676	—	2	1	no	—
24	C₂×C₈×D₅	—	2	1	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	#3957	—	4	4	no	—
30	#3950	—	2	2	no	—
31	#3949	—	2	2	no	—
32	C₂×#672	—	2	1	no	—
33	C₂×#672	—	2	2	no	—
34	#2161	D₅	1	1	yes	characteristic, complemented
35	#2113	—	5	1	no	—
36	#1814	—	5	2	no	—
37	#1384	—	5	1	no	—
38	#1382	—	5	2	no	—
39	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂×C₂²⋊D₅	—	2	1	no	—
41	#709	—	2	2	no	—
42	#709	—	2	2	no	—
43	C₂×D₂₀	Q₁₆	1	1	yes	characteristic
44	C₂×C₄×D₅	—	2	1	no	—
45	C₂×Dic₁₀	Q₁₆	1	1	yes	characteristic
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	C₂²⋊Dic₅	—	4	2	no	—
54	C₄⋊Dic₅	—	4	2	no	—
55	C₄×Dic₅	—	4	2	no	—
56	#681	—	2	2	no	—
57	C₂×#168	—	2	1	no	—
58	#676	—	4	2	no	—
59	C₈×D₅	—	4	1	no	—
60	C₈×D₅	—	4	1	no	—
61	C₅×C₁₆	—	2	2	no	—
62	#672	—	4	1	no	—
63	C₂×#131	—	5	1	no	—
64	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
65	C₂²×C₁₆	—	5	1	no	—
66	C₂×#108	C₂×D₅	1	2	yes	—
67	#427	—	5	2	no	—
68	#421	—	10	2	no	—
69	#368	—	2	2	no	—
70	C₂²⋊C₁₆	—	10	2	no	—
71	#344	—	10	2	no	—
72	C₂³×C₅	#135	1	1	yes	characteristic
73	C₂²×D₅	—	2	1	no	—
74	C₂×C₄×C₅	#121	1	1	yes	characteristic
75	C₂×C₄×C₅	#123	1	2	yes	—
76	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
78	C₂²⋊D₅	—	4	2	no	—
79	C₂×Dic₅	—	2	1	no	—
80	C₂×Dic₅	—	8	2	no	—
81	D₂₀	—	2	2	no	—
82	C₄×D₅	—	4	1	no	—
83	C₄×D₅	—	4	1	no	—
84	Dic₁₀	—	2	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	C₂×#41	—	5	1	no	—
91	#131	—	10	2	no	—
92	#131	—	10	2	no	—
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	5	1	no	—
95	C₂×M₁₆	—	10	1	no	—
96	C₂²×C₈	—	10	1	no	—
97	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
98	#117	—	10	2	no	—
99	M₃₂	—	2	2	no	—
100	C₂×C₁₆	—	10	2	no	—
101	C₂×C₁₆	—	10	1	no	—
102	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
103	#108	—	2	2	no	—
104	#108	—	2	2	no	—
105	#107	—	10	2	no	—
106	#106	—	10	2	no	—
107	#104	—	20	4	no	—
108	C₂²×C₅	—	4	1	no	—
109	C₂²×C₅	#467	1	1	yes	characteristic
110	C₂²×C₅	#482	1	2	yes	—
111	C₂×D₅	—	4	2	no	—
112	C₂×D₅	—	4	1	no	—
113	C₄×C₅	#397	1	1	yes	characteristic
114	C₄×C₅	#469	1	1	yes	characteristic
115	C₄×C₅	#484	1	2	yes	—
116	Dic₅	—	4	1	no	—
117	Dic₅	—	16	2	no	—
118	#41	—	10	2	no	—
119	#41	—	10	2	no	—
120	C₂×Q₈	—	5	1	no	—
121	C₂×D₄	—	5	1	no	—
122	C₂×D₄	—	10	1	no	—
123	C₂²×C₄	—	10	1	no	—
124	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	M₁₆	—	4	2	no	—
127	M₁₆	—	10	2	no	—
128	M₁₆	—	20	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	10	1	no	—
131	C₂×C₈	—	20	1	no	—
132	C₂×C₈	—	20	1	no	—
133	C₂×C₈	#709	1	2	yes	—
134	C₂×C₈	#711	1	1	yes	characteristic
135	C₂×C₈	#713	1	2	yes	—
136	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
137	C₄⋊C₄	—	20	2	no	—
138	C₂²⋊C₄	—	20	2	no	—
139	C₄²	—	20	2	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	20	1	no	—
142	C₂×C₅	—	2	2	no	—
143	C₂×C₅	—	2	1	no	—
144	C₂×C₅	#2076	1	1	yes	characteristic
145	D₅	—	8	1	no	—
146	C₂³	—	10	1	no	—
147	C₂³	#4065	1	1	yes	characteristic
148	Q₈	—	10	2	no	—
149	D₄	—	10	2	no	—
150	D₄	—	20	2	no	—
151	C₂×C₄	—	10	1	no	—
152	C₂×C₄	—	20	1	no	—
153	C₂×C₄	—	20	1	no	—
154	C₂×C₄	—	40	2	no	—
155	C₂×C₄	#4040	1	2	yes	—
156	C₂×C₄	#4057	1	2	yes	—
157	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
158	C₂×C₄	#4084	1	1	yes	characteristic
159	C₈	—	8	2	no	—
160	C₈	—	20	1	no	—
161	C₈	#4044	1	2	yes	—
162	C₈	#4075	1	1	yes	characteristic
163	C₈	#4093	1	1	yes	characteristic
164	C₅	#13499	1	1	yes	characteristic, complemented, Sylow
165	C₂²	—	4	1	no	—
166	C₂²	—	20	2	no	—
167	C₂²	—	20	1	no	—
168	C₂²	#65174	1	2	yes	—
169	C₂²	#65477	1	1	yes	characteristic
170	C₄	—	20	1	no	—
171	C₄	—	80	2	no	—
172	C₄	#65173	1	2	yes	—
173	C₄	#65282	1	1	yes	characteristic
174	C₄	#65448	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	40	1	no	—
178	C₂	#10601430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2161

C₂

yes

maximal, characteristic, complemented

#10602225

C₂

yes

maximal, characteristic

#10601512

C₂

yes

maximal

#10598309

C₂

yes

maximal, characteristic, complemented

#10598220

C₂

yes

maximal, complemented

C₂×#4047

C₂²

yes

characteristic

C₂×C₅×M₃₂

C₂²

yes

characteristic

C₂×C₅×#108

C₂²

yes

—

#65428

—

#65395

C₂²

yes

—

C₂²×#672

C₂²

yes

—

—

—

—

D₄

yes

characteristic

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

C₂³

yes

characteristic

#4072

—

C₂×#681

D₄

yes

—

—

—

C₂×C₈×D₅

—

C₅×M₃₂

—

C₂×C₅×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

D₅

yes

characteristic, complemented

—

—

—

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₅

—

#709

—

#709

—

C₂×D₂₀

Q₁₆

yes

characteristic

C₂×C₄×D₅

—

C₂×Dic₁₀

Q₁₆

yes

characteristic

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

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

#681

—

C₂×#168

—

#676

—

C₈×D₅

—

C₈×D₅

—

C₅×C₁₆

—

#672

—

C₂×#131

—

C₂×M₃₂

C₂×D₅

yes

characteristic

C₂²×C₁₆

—

C₂×#108

C₂×D₅

yes

—

#427

—

#421

—

#368

—

C₂²⋊C₁₆

—

#344

—

C₂³×C₅

#135

yes

characteristic

C₂²×D₅

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

yes

C₅×C₈

yes

C₅×C₈

yes

—

#168

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₅

yes

characteristic

#117

—

M₃₂

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

C₂²⋊D₅

yes

—

103

#108

—

104

#108

—

105

#107

—

106

#106

—

107

#104

—

108

C₂²×C₅

—

109

C₂²×C₅

#467

yes

characteristic

110

C₂²×C₅

#482

yes

—

111

C₂×D₅

—

112

C₂×D₅

—

113

C₄×C₅

#397

yes

characteristic

114

C₄×C₅

#469

yes

characteristic

115

C₄×C₅

#484

yes

—

116

Dic₅

—

117

Dic₅

—

118

#41

—

119

#41

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

D₄×D₅

yes

characteristic

125

M₁₆

—

126

M₁₆

—

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

#709

yes

—

134

C₂×C₈

#711

yes

characteristic

135

C₂×C₈

#713

yes

—

136

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₄²

—

140

C₁₆

—

141

C₁₆

—

142

C₂×C₅

—

143

C₂×C₅

—

144

C₂×C₅

#2076

yes

characteristic

145

D₅

—

146

C₂³

—

147

C₂³

#4065

yes

characteristic

148

Q₈

—

149

D₄

—

150

D₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

#4040

yes

—

156

C₂×C₄

#4057

yes

—

157

C₂×C₄

D₅×Q₁₆

yes

characteristic

158

C₂×C₄

#4084

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

#4044

yes

—

162

C₈

#4075

yes

characteristic

163

C₈

#4093

yes

characteristic

164

C₅

#13499

yes

characteristic, complemented, Sylow

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#65174

yes

—

169

C₂²

#65477

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

#65173

yes

—

173

C₄

#65282

yes

characteristic

174

C₄

#65448

yes

characteristic

175

C₂

—

176

C₂

—

177

C₂

—

178

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	C₂×#709	D₄
hypercenter	#2161	D₅
hypocenter	C₅	#13499
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

D₄

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. #10598309	C₂	semidirect
5. #10598220	C₂	semidirect
34. #2161	D₅	semidirect
164. C₅	#13499	semidirect

H ≅ G/N

type

1. C₅×#2161

C₂

semidirect

4. #10598309

C₂

semidirect

5. #10598220

C₂

semidirect

34. #2161

D₅

semidirect

164. C₅

#13499

semidirect

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

subgroup	index
C₅×#2161	2
#10602225	2
#10601512	2
#10598309	2
#10598220	2
#13499	5

subgroup

index

Subgroups — #12019196

Conjugacy classes of subgroups

Complements

Maximal subgroups