Group #12004313 — subgroups

Subgroups — #12004313

Order 1280.

354 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 99 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×#1963	C₂	1	1	yes	maximal, characteristic
2	#10602593	C₂	1	2	yes	maximal
3	#10601286	C₂	1	2	yes	maximal, complemented
4	#10600232	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598105	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#532	C₂²	1	1	yes	characteristic
7	C₅×#423	C₂²	1	1	yes	characteristic
8	C₂×C₅×#104	—	2	2	no	—
9	C₅×#376	C₂²	1	1	yes	characteristic
10	#65214	C₂²	1	2	yes	—
11	C₂×C₄⋊#168	—	2	2	no	—
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	D₄⋊#168	—	2	4	no	—
14	#64700	—	2	4	no	—
15	#10666	—	5	1	no	maximal, Sylow
16	C₂×C₅×#41	—	2	1	no	—
17	C₂×C₅×M₁₆	—	2	1	no	—
18	C₅×C₄⋊Q₈	Q₈	1	1	yes	characteristic
19	C₅×#127	Q₈	1	1	yes	characteristic
20	C₅×#124	—	2	1	no	—
21	C₅×#121	—	2	2	no	—
22	C₄×C₅×D₄	—	2	2	no	—
23	C₂×C₄²×C₅	D₄	1	2	yes	—
24	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
25	C₂²×#168	—	2	2	no	—
26	C₅×C₄⋊C₈	—	2	2	no	—
27	C₅×#104	—	4	4	no	—
28	C₅×#95	—	2	2	no	—
29	C₂²⋊#168	—	2	2	no	—
30	C₄⋊#168	—	2	2	no	—
31	C₄⋊#168	—	2	2	no	—
32	C₄⋊#168	—	2	2	no	—
33	C₄⋊#168	—	4	4	no	—
34	C₄×#168	D₄	1	4	yes	—
35	#1963	D₅	1	1	yes	characteristic
36	#1843	—	5	1	no	—
37	#1570	—	5	2	no	—
38	#1480	—	5	2	no	—
39	#1274	—	5	1	no	—
40	C₅×#41	—	4	2	no	—
41	C₅×#41	—	4	2	no	—
42	C₂×C₅×Q₈	—	2	1	no	—
43	C₂×C₅×D₄	—	2	2	no	—
44	C₂×C₅×D₄	—	2	1	no	—
45	C₂×C₅×D₄	—	4	1	no	—
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	—	4	1	no	—
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	#41	1	2	yes	—
50	C₅×M₁₆	—	4	2	no	—
51	C₂×C₅×C₈	—	4	1	no	—
52	C₅×C₄⋊C₄	—	2	2	no	—
53	C₅×C₂²⋊C₄	—	4	2	no	—
54	C₄²×C₅	—	2	2	no	—
55	C₄²×C₅	SD₁₆	1	4	yes	—
56	C₄²×C₅	C₂×D₄	1	2	yes	—
57	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₅	#41	1	1	yes	characteristic
59	C₂×#168	—	2	4	no	—
60	C₂×#168	—	4	2	no	—
61	C₂×#168	—	4	4	no	—
62	#532	C₂×D₅	1	1	yes	characteristic
63	#431	—	5	2	no	—
64	#423	C₂×D₅	1	1	yes	characteristic
65	C₂×C₄⋊C₈	—	5	2	no	—
66	C₂×C₄⋊C₈	—	10	2	no	—
67	C₂×#104	—	2	2	no	—
68	#376	C₂×D₅	1	1	yes	characteristic
69	#334	—	10	4	no	—
70	D₄⋊C₈	—	10	4	no	—
71	C₂³×C₅	—	4	1	no	—
72	C₂³×C₅	#123	1	1	yes	characteristic
73	C₅×Q₈	—	4	2	no	—
74	C₅×D₄	—	4	4	no	—
75	C₅×D₄	—	4	2	no	—
76	C₅×D₄	—	8	2	no	—
77	C₂×C₄×C₅	—	2	4	no	—
78	C₂×C₄×C₅	—	4	2	no	—
79	C₂×C₄×C₅	—	4	4	no	—
80	C₂×C₄×C₅	—	4	1	no	—
81	C₂×C₄×C₅	—	8	1	no	—
82	C₂×C₄×C₅	—	8	1	no	—
83	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅	#123	1	2	yes	—
86	C₂×C₄×C₅	#127	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
89	C₅×C₈	—	8	1	no	—
90	#168	—	4	4	no	—
91	#168	—	8	2	no	—
92	C₂×#41	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂²×C₈	—	10	2	no	—
95	C₄⋊Q₈	Dic₁₀	1	1	yes	characteristic
96	#127	Dic₁₀	1	1	yes	characteristic
97	#124	—	2	1	no	—
98	#121	—	2	2	no	—
99	C₄×D₄	—	2	2	no	—
100	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
101	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	10	2	no	—
104	C₄⋊C₈	—	10	2	no	—
105	C₄⋊C₈	—	10	2	no	—
106	C₄⋊C₈	—	20	4	no	—
107	#104	—	4	4	no	—
108	C₂²⋊C₈	—	10	2	no	—
109	C₄×C₈	—	5	4	no	—
110	#95	—	2	2	no	—
111	C₂²×C₅	—	2	2	no	—
112	C₂²×C₅	—	8	1	no	—
113	C₂²×C₅	—	8	2	no	—
114	C₂²×C₅	#399	1	1	yes	characteristic
115	C₂²×C₅	#481	1	2	yes	—
116	C₄×C₅	—	2	2	no	—
117	C₄×C₅	—	2	2	no	—
118	C₄×C₅	—	4	4	no	—
119	C₄×C₅	—	8	1	no	—
120	C₄×C₅	#476	1	2	yes	—
121	C₄×C₅	#492	1	1	yes	characteristic
122	C₄×C₅	#496	1	1	yes	characteristic
123	#41	—	4	2	no	—
124	#41	—	4	2	no	—
125	C₂×Q₈	—	2	1	no	—
126	C₂×D₄	—	2	2	no	—
127	C₂×D₄	—	2	1	no	—
128	C₂×D₄	—	4	1	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	4	1	no	—
131	C₂²×C₄	#711	1	2	yes	—
132	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
133	M₁₆	—	4	2	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	10	4	no	—
136	C₂×C₈	—	20	2	no	—
137	C₂×C₈	—	20	4	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	4	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	#687	1	4	yes	—
142	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
143	C₄²	#709	1	1	yes	characteristic
144	C₄²	D₄×D₅	1	2	yes	—
145	C₂×C₅	—	2	2	no	—
146	C₂×C₅	—	16	1	no	—
147	C₂×C₅	#2094	1	1	yes	characteristic
148	C₂×C₅	#2098	1	2	yes	—
149	C₂³	—	4	1	no	—
150	C₂³	#4081	1	1	yes	characteristic
151	Q₈	—	4	2	no	—
152	D₄	—	4	4	no	—
153	D₄	—	4	2	no	—
154	D₄	—	8	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	#4024	1	2	yes	—
162	C₂×C₄	#4029	1	2	yes	—
163	C₂×C₄	#4070	1	1	yes	characteristic
164	C₂×C₄	C₂×#687	1	2	yes	—
165	C₂×C₄	#4086	1	1	yes	characteristic
166	C₂×C₄	#4095	1	2	yes	—
167	C₈	—	8	1	no	—
168	C₈	—	20	4	no	—
169	C₈	—	40	2	no	—
170	C₅	#10666	1	1	yes	characteristic, complemented, Sylow
171	C₂²	—	2	2	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	#65238	1	1	yes	characteristic
175	C₂²	#65323	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	8	1	no	—
180	C₄	#65297	1	2	yes	—
181	C₄	#65347	1	1	yes	characteristic
182	C₄	#65365	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	16	1	no	—
185	C₂	#10600449	1	1	yes	characteristic
186	C₂	#10601198	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1963

C₂

yes

maximal, characteristic

#10602593

C₂

yes

maximal

#10601286

C₂

yes

maximal, complemented

#10600232

C₂

yes

maximal, characteristic, complemented

#10598105

C₂

yes

maximal, characteristic, complemented

C₅×#532

C₂²

yes

characteristic

C₅×#423

C₂²

yes

characteristic

C₂×C₅×#104

—

C₅×#376

C₂²

yes

characteristic

#65214

C₂²

yes

—

C₂×C₄⋊#168

—

C₂×C₄⋊#168

C₂²

yes

—

—

—

—

C₂×C₅×#41

—

C₂×C₅×M₁₆

—

C₅×C₄⋊Q₈

Q₈

yes

characteristic

C₅×#127

Q₈

yes

characteristic

C₅×#124

—

C₅×#121

—

C₄×C₅×D₄

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂²×#168

—

C₅×C₄⋊C₈

—

C₅×#104

—

C₅×#95

—

C₂²⋊#168

—

—

—

—

—

D₄

yes

—

#1963

D₅

yes

—

—

—

—

C₅×#41

—

C₅×#41

—

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

SD₁₆

yes

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×Q₈

yes

C₄²×C₅

yes

—

—

—

C₂×D₅

yes

characteristic

#431

—

#423

C₂×D₅

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#104

—

#376

C₂×D₅

yes

characteristic

#334

—

D₄⋊C₈

—

C₂³×C₅

—

C₂³×C₅

#123

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₅×C₈

—

#168

—

#168

—

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₁₀

yes

characteristic

#127

Dic₁₀

yes

characteristic

#124

—

#121

—

C₄×D₄

—

100

C₂×C₄²

C₂²⋊D₅

yes

—

101

C₂×C₄²

C₂²×D₅

yes

characteristic

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

#104

—

108

C₂²⋊C₈

—

109

C₄×C₈

—

110

#95

—

111

C₂²×C₅

—

112

C₂²×C₅

—

113

C₂²×C₅

—

114

C₂²×C₅

#399

yes

characteristic

115

C₂²×C₅

#481

yes

—

116

C₄×C₅

—

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

—

120

C₄×C₅

#476

yes

—

121

C₄×C₅

#492

yes

characteristic

122

C₄×C₅

#496

yes

characteristic

123

#41

—

124

#41

—

125

C₂×Q₈

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

#711

yes

—

132

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

133

M₁₆

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₄⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

#687

yes

—

142

C₄²

C₂×Dic₁₀

yes

characteristic

143

C₄²

#709

yes

characteristic

144

C₄²

D₄×D₅

yes

—

145

C₂×C₅

—

146

C₂×C₅

—

147

C₂×C₅

#2094

yes

characteristic

148

C₂×C₅

#2098

yes

—

149

C₂³

—

150

C₂³

#4081

yes

characteristic

151

Q₈

—

152

D₄

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

#4024

yes

—

162

C₂×C₄

#4029

yes

—

163

C₂×C₄

#4070

yes

characteristic

164

C₂×C₄

C₂×#687

yes

—

165

C₂×C₄

#4086

yes

characteristic

166

C₂×C₄

#4095

yes

—

167

C₈

—

168

C₈

—

169

C₈

—

170

C₅

#10666

yes

characteristic, complemented, Sylow

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#65238

yes

characteristic

175

C₂²

#65323

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#65297

yes

—

181

C₄

#65347

yes

characteristic

182

C₄

#65365

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

#10600449

yes

characteristic

186

C₂

#10601198

yes

—

	subgroup	G/H
center	C₂×C₄	#4086
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1963	C₂
socle	C₂²×C₅	#399
prime radical	C₅×#532	C₂²
hypercenter	#1963	D₅
hypocenter	C₅	#10666
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

subgroup

G/H

center

C₂×C₄

#4086

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂²

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

N	H ≅ G/N	type
3. #10601286	C₂	semidirect
4. #10600232	C₂	semidirect
5. #10598105	C₂	semidirect
170. C₅	#10666	semidirect

H ≅ G/N

type

3. #10601286

C₂

semidirect

4. #10600232

C₂

semidirect

5. #10598105

C₂

semidirect

170. C₅

#10666

semidirect

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

subgroup	index
C₅×#1963	2
#10602593	2
#10601286	2
#10600232	2
#10598105	2
#10666	5

subgroup

index

Subgroups — #12004313

Conjugacy classes of subgroups

Complements

Maximal subgroups