Group #12004078 — subgroups

Subgroups — #12004078

Order 1280.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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	#10602597	C₂	1	2	yes	maximal
3	#10600230	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598205	C₂	1	2	yes	maximal, 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	C₂²	1	2	yes	—
12	#64767	—	2	2	no	—
13	#64764	—	2	2	no	—
14	#64744	—	2	2	no	—
15	#10629	—	5	1	no	maximal, Sylow
16	C₂×C₅×#41	—	2	1	no	—
17	C₂×C₅×M₁₆	—	2	1	no	—
18	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
19	C₅×#127	D₄	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	—	2	2	no	—
34	#1963	D₅	1	1	yes	characteristic
35	#1832	—	5	1	no	—
36	#1540	—	5	2	no	—
37	#1301	—	5	2	no	—
38	#1274	—	5	1	no	—
39	C₅×#41	—	4	2	no	—
40	C₅×#41	—	4	2	no	—
41	C₂×C₅×Q₈	—	2	1	no	—
42	C₂×C₅×D₄	—	2	1	no	—
43	C₂×C₅×D₄	—	2	2	no	—
44	C₂×C₅×D₄	—	4	1	no	—
45	C₂²×C₄×C₅	—	2	2	no	—
46	C₂²×C₄×C₅	—	4	1	no	—
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	#41	1	2	yes	—
49	C₅×M₁₆	—	4	2	no	—
50	C₂×C₅×C₈	—	4	1	no	—
51	C₅×C₄⋊C₄	—	2	2	no	—
52	C₅×C₂²⋊C₄	—	4	2	no	—
53	C₄²×C₅	—	2	2	no	—
54	C₄²×C₅	—	2	2	no	—
55	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅	#41	1	1	yes	characteristic
57	C₄²×C₅	#41	1	2	yes	—
58	C₂×#168	—	4	2	no	—
59	C₂×#168	—	4	2	no	—
60	C₂×#168	—	4	2	no	—
61	C₂×#168	—	4	2	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×#104	—	2	2	no	—
67	#376	C₂×D₅	1	1	yes	characteristic
68	#336	—	10	2	no	—
69	#332	—	10	2	no	—
70	#331	—	10	2	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	2	no	—
75	C₅×D₄	—	8	2	no	—
76	C₅×D₄	—	8	2	no	—
77	C₂×C₄×C₅	—	4	2	no	—
78	C₂×C₄×C₅	—	4	4	no	—
79	C₂×C₄×C₅	—	4	1	no	—
80	C₂×C₄×C₅	—	4	2	no	—
81	C₂×C₄×C₅	—	8	1	no	—
82	C₂×C₄×C₅	—	8	1	no	—
83	C₂×C₄×C₅	#121	1	1	yes	characteristic
84	C₂×C₄×C₅	#123	1	2	yes	—
85	C₂×C₄×C₅	#124	1	1	yes	characteristic
86	C₂×C₄×C₅	#126	1	2	yes	—
87	C₂×C₄×C₅	#135	1	2	yes	—
88	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
89	C₅×C₈	—	8	1	no	—
90	#168	—	8	2	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	D₂₀	1	1	yes	characteristic
96	#127	D₂₀	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	#104	—	4	4	no	—
107	C₂²⋊C₈	—	10	2	no	—
108	C₄×C₈	—	10	2	no	—
109	#95	—	2	2	no	—
110	C₂²×C₅	—	2	2	no	—
111	C₂²×C₅	—	8	1	no	—
112	C₂²×C₅	—	8	2	no	—
113	C₂²×C₅	#397	1	1	yes	characteristic
114	C₂²×C₅	#482	1	2	yes	—
115	C₄×C₅	—	2	2	no	—
116	C₄×C₅	—	2	2	no	—
117	C₄×C₅	—	8	2	no	—
118	C₄×C₅	—	8	1	no	—
119	C₄×C₅	#463	1	2	yes	—
120	C₄×C₅	#488	1	1	yes	characteristic
121	C₄×C₅	#490	1	1	yes	characteristic
122	#41	—	4	2	no	—
123	#41	—	4	2	no	—
124	C₂×Q₈	—	2	1	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	—	4	1	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	4	1	no	—
130	C₂²×C₄	#711	1	2	yes	—
131	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
132	M₁₆	—	4	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	20	2	no	—
135	C₂×C₈	—	20	2	no	—
136	C₂×C₈	—	20	2	no	—
137	C₂×C₈	—	20	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	4	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	C₂×D₂₀	1	1	yes	characteristic
143	C₄²	#709	1	1	yes	characteristic
144	C₄²	#711	1	2	yes	—
145	C₂×C₅	—	2	2	no	—
146	C₂×C₅	—	16	1	no	—
147	C₂×C₅	#2073	1	1	yes	characteristic
148	C₂×C₅	#2080	1	2	yes	—
149	C₂³	—	4	1	no	—
150	C₂³	#4081	1	1	yes	characteristic
151	Q₈	—	4	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	8	2	no	—
154	D₄	—	8	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	#4025	1	2	yes	—
162	C₂×C₄	#4032	1	2	yes	—
163	C₂×C₄	#4076	1	1	yes	characteristic
164	C₂×C₄	#4078	1	2	yes	—
165	C₂×C₄	#4082	1	1	yes	characteristic
166	C₂×C₄	#4096	1	2	yes	—
167	C₈	—	8	1	no	—
168	C₈	—	40	2	no	—
169	C₈	—	40	2	no	—
170	C₅	#10629	1	1	yes	characteristic, complemented, Sylow
171	C₂²	—	2	2	no	—
172	C₂²	—	8	2	no	—
173	C₂²	—	8	1	no	—
174	C₂²	#65242	1	1	yes	characteristic
175	C₂²	#65324	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	8	2	no	—
179	C₄	—	8	1	no	—
180	C₄	#65305	1	2	yes	—
181	C₄	#65345	1	1	yes	characteristic
182	C₄	#65360	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	16	1	no	—
185	C₂	#10600434	1	1	yes	characteristic
186	C₂	#10601197	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1963

C₂

yes

maximal, characteristic

#10602597

C₂

yes

maximal

#10600230

C₂

yes

maximal, characteristic, complemented

#10598205

C₂

yes

maximal, 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

C₂²

yes

—

—

—

—

—

C₂×C₅×#41

—

C₂×C₅×M₁₆

—

C₅×C₄⋊Q₈

D₄

yes

characteristic

C₅×#127

D₄

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

—

—

—

—

—

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

—

C₄²×C₅

C₂×D₄

yes

C₄²×C₅

yes

C₄²×C₅

yes

—

—

—

—

—

C₂×D₅

yes

characteristic

#431

—

#423

C₂×D₅

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#104

—

#376

C₂×D₅

yes

—

—

—

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

#121

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

#124

yes

characteristic

C₂×C₄×C₅

#126

yes

—

C₂×C₄×C₅

#135

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₅×C₈

—

#168

—

#168

—

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

D₂₀

yes

characteristic

#127

D₂₀

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

#104

—

107

C₂²⋊C₈

—

108

C₄×C₈

—

109

#95

—

110

C₂²×C₅

—

111

C₂²×C₅

—

112

C₂²×C₅

—

113

C₂²×C₅

#397

yes

characteristic

114

C₂²×C₅

#482

yes

—

115

C₄×C₅

—

116

C₄×C₅

—

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

#463

yes

—

120

C₄×C₅

#488

yes

characteristic

121

C₄×C₅

#490

yes

characteristic

122

#41

—

123

#41

—

124

C₂×Q₈

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

#711

yes

—

131

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

132

M₁₆

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₄⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

C₂×D₂₀

yes

characteristic

143

C₄²

#709

yes

characteristic

144

C₄²

#711

yes

—

145

C₂×C₅

—

146

C₂×C₅

—

147

C₂×C₅

#2073

yes

characteristic

148

C₂×C₅

#2080

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

#4025

yes

—

162

C₂×C₄

#4032

yes

—

163

C₂×C₄

#4076

yes

characteristic

164

C₂×C₄

#4078

yes

—

165

C₂×C₄

#4082

yes

characteristic

166

C₂×C₄

#4096

yes

—

167

C₈

—

168

C₈

—

169

C₈

—

170

C₅

#10629

yes

characteristic, complemented, Sylow

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#65242

yes

characteristic

175

C₂²

#65324

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#65305

yes

—

181

C₄

#65345

yes

characteristic

182

C₄

#65360

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

#10600434

yes

characteristic

186

C₂

#10601197

yes

—

	subgroup	G/H
center	C₂×C₄	#4082
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1963	C₂
socle	C₂²×C₅	#397
prime radical	C₅×#532	C₂²
hypercenter	#1963	D₅
hypocenter	C₅	#10629
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

subgroup

G/H

center

C₂×C₄

#4082

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. #10600230	C₂	semidirect
4. #10598205	C₂	semidirect
5. #10598105	C₂	semidirect
170. C₅	#10629	semidirect

H ≅ G/N

type

3. #10600230

C₂

semidirect

4. #10598205

C₂

semidirect

5. #10598105

C₂

semidirect

170. C₅

#10629

semidirect

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

subgroup	index
C₅×#1963	2
#10602597	2
#10600230	2
#10598205	2
#10598105	2
#10629	5

subgroup

index

Subgroups — #12004078

Conjugacy classes of subgroups

Complements

Maximal subgroups