Group #11931341 — subgroups

Subgroups — C₅×#8621 #11931341

Order 1280.

362 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 142 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	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1568	C₂	1	2	yes	maximal
3	C₅×#1467	C₂	1	2	yes	maximal
4	C₅×#1290	C₂	1	2	yes	maximal
5	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
6	C₅×C₈×Q₈	—	2	2	no	—
7	C₅×C₈×Q₈	—	2	2	no	—
8	C₅×#432	C₂²	1	4	yes	—
9	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#335	—	2	2	no	—
14	C₅×#334	—	2	2	no	—
15	C₅×#324	—	2	4	no	—
16	#8621	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×Q₈	Q₈	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₅×Q₈	—	2	2	no	—
22	C₄×C₅×Q₈	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
24	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₅×C₄⋊C₈	—	2	4	no	—
28	C₅×C₄⋊C₈	—	2	2	no	—
29	C₅×C₄⋊C₈	—	2	4	no	—
30	C₅×C₂²⋊C₈	—	2	4	no	—
31	C₅×#97	—	2	4	no	—
32	C₄×C₅×C₈	—	2	2	no	—
33	C₄×C₅×C₈	—	2	2	no	—
34	C₄×C₅×C₈	D₄	1	2	yes	—
35	C₄×C₅×C₈	D₄	1	2	yes	—
36	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
37	#1568	C₂×C₅	1	2	yes	—
38	#1467	C₂×C₅	1	2	yes	—
39	#1290	C₂×C₅	1	2	yes	—
40	C₂×C₅×Q₈	—	2	4	no	—
41	C₂×C₅×Q₈	—	2	2	no	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₅×C₈	—	2	2	no	—
48	C₂×C₅×C₈	—	2	2	no	—
49	C₂×C₅×C₈	—	4	1	no	—
50	C₂×C₅×C₈	—	4	2	no	—
51	C₂×C₅×C₈	—	4	4	no	—
52	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
54	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₅×C₄⋊C₄	—	2	4	no	—
56	C₅×C₄⋊C₄	—	2	2	no	—
57	C₄²×C₅	—	2	2	no	—
58	C₄²×C₅	—	2	2	no	—
59	C₄²×C₅	C₂×D₄	1	2	yes	—
60	C₄²×C₅	#41	1	2	yes	—
61	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
62	C₈×Q₈	—	2	2	no	—
63	C₈×Q₈	—	2	2	no	—
64	#432	C₂²×C₅	1	4	yes	—
65	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
66	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
67	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
68	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
69	#335	—	2	2	no	—
70	#334	—	2	2	no	—
71	#324	—	2	4	no	—
72	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₅×Q₈	—	2	8	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	1	no	—
78	C₂×C₄×C₅	—	2	1	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	4	1	no	—
81	C₂×C₄×C₅	—	4	2	no	—
82	C₂×C₄×C₅	#108	1	4	yes	—
83	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅	#117	1	1	yes	characteristic
85	C₂×C₄×C₅	#121	1	2	yes	—
86	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
87	C₅×C₈	—	2	1	no	—
88	C₅×C₈	—	2	2	no	—
89	C₅×C₈	—	4	2	no	—
90	C₅×C₈	—	8	4	no	—
91	C₅×C₈	#101	1	2	yes	—
92	C₂²×Q₈	C₅×Q₈	1	1	yes	characteristic
93	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
94	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
95	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
96	C₄×Q₈	—	2	2	no	—
97	C₄×Q₈	—	2	2	no	—
98	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
99	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
100	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
101	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
102	C₄⋊C₈	—	2	4	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	4	no	—
105	C₂²⋊C₈	—	2	4	no	—
106	#97	—	2	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	C₄×C₈	C₅×D₄	1	2	yes	—
110	C₄×C₈	C₅×D₄	1	2	yes	—
111	C₂²×C₅	—	2	1	no	—
112	C₂²×C₅	—	2	1	no	—
113	C₂²×C₅	#380	1	1	yes	characteristic
114	C₂²×C₅	C₂×#108	1	2	yes	—
115	C₄×C₅	—	2	1	no	—
116	C₄×C₅	—	2	4	no	—
117	C₄×C₅	—	2	2	no	—
118	C₄×C₅	—	2	2	no	—
119	C₄×C₅	—	4	2	no	—
120	C₄×C₅	C₂×#101	1	1	yes	characteristic
121	C₄×C₅	#413	1	1	yes	characteristic
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
125	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
126	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
136	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	C₂×C₅×D₄	1	2	yes	—
142	C₄²	C₅×#41	1	2	yes	—
143	C₂×C₅	—	2	2	no	—
144	C₂×C₅	#1801	1	1	yes	characteristic
145	C₂×C₅	#1810	1	1	yes	characteristic
146	C₂×C₅	#1812	1	1	yes	characteristic
147	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
148	Q₈	—	2	8	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	C₅×#108	1	4	yes	—
158	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₅×#117	1	1	yes	characteristic
160	C₂×C₄	C₅×#121	1	2	yes	—
161	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
162	C₈	—	2	1	no	—
163	C₈	—	2	2	no	—
164	C₈	—	4	2	no	—
165	C₈	—	8	4	no	—
166	C₈	C₅×#101	1	2	yes	—
167	C₅	#8621	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	C₅×#380	1	1	yes	characteristic
171	C₂²	C₂×C₅×#108	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	4	no	—
175	C₄	—	2	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×C₅×#101	1	1	yes	characteristic
178	C₄	C₅×#413	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	C₅×#1801	1	1	yes	characteristic
181	C₂	C₅×#1810	1	1	yes	characteristic
182	C₂	C₅×#1812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×C₈×Q₈

C₂

yes

maximal, characteristic

C₅×#1568

C₂

yes

maximal

C₅×#1467

C₂

yes

maximal

C₅×#1290

C₂

yes

maximal

C₂×C₄×C₅×Q₈

C₂²

yes

characteristic

C₅×C₈×Q₈

—

C₅×C₈×Q₈

—

C₅×#432

C₂²

yes

—

C₂×C₅×C₄⋊C₈

C₄

yes

—

C₂×C₅×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₅×C₈

C₄

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

—

—

—

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×Q₈

Q₈

yes

characteristic

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×Q₈

—

C₄×C₅×Q₈

—

C₂×C₅×C₄⋊C₄

D₄

yes

—

C₂×C₅×C₄⋊C₄

Q₈

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₅×#97

—

C₄×C₅×C₈

—

C₄×C₅×C₈

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

D₄

yes

—

C₂×C₈×Q₈

C₂×C₅

yes

characteristic

#1568

C₂×C₅

yes

—

#1467

C₂×C₅

yes

—

#1290

C₂×C₅

yes

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×Q₈

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

#41

yes

—

C₂×C₄×Q₈

C₂²×C₅

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#432

C₂²×C₅

yes

—

C₂×C₄⋊C₈

C₄×C₅

yes

—

C₂×C₄⋊C₈

C₂²×C₅

yes

characteristic

C₂×C₄×C₈

C₄×C₅

yes

—

C₂×C₄×C₈

C₂²×C₅

yes

—

—

—

C₂³×C₅

C₂×C₄⋊C₄

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#108

yes

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

#101

yes

—

C₂²×Q₈

C₅×Q₈

yes

characteristic

C₂²×C₈

C₂×C₄×C₅

yes

characteristic

C₂²×C₈

C₂×C₄×C₅

yes

—

C₂²×C₈

C₂×C₄×C₅

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₅×D₄

yes

—

C₂×C₄⋊C₄

C₅×Q₈

yes

characteristic

100

C₂×C₄²

C₂×C₄×C₅

yes

—

101

C₂×C₄²

C₂³×C₅

yes

characteristic

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

—

106

#97

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₅×D₄

yes

—

110

C₄×C₈

C₅×D₄

yes

—

111

C₂²×C₅

—

112

C₂²×C₅

—

113

C₂²×C₅

#380

yes

characteristic

114

C₂²×C₅

C₂×#108

yes

—

115

C₄×C₅

—

116

C₄×C₅

—

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

—

120

C₄×C₅

C₂×#101

yes

characteristic

121

C₄×C₅

#413

yes

characteristic

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂²×C₄

C₅×C₄⋊C₄

yes

—

125

C₂²×C₄

C₅×C₄⋊C₄

yes

—

126

C₂²×C₄

C₂²×C₄×C₅

yes

characteristic

127

C₂²×C₄

C₂×C₅×D₄

yes

characteristic

128

C₂²×C₄

C₂×C₅×Q₈

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

C₅×C₂²⋊C₄

yes

characteristic

135

C₂×C₈

C₅×C₂²⋊C₄

yes

—

136

C₂×C₈

C₅×C₂²⋊C₄

yes

characteristic

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

—

140

C₄²

—

141

C₄²

C₂×C₅×D₄

yes

—

142

C₄²

C₅×#41

yes

—

143

C₂×C₅

—

144

C₂×C₅

#1801

yes

characteristic

145

C₂×C₅

#1810

yes

characteristic

146

C₂×C₅

#1812

yes

characteristic

147

C₂³

C₂×C₅×C₄⋊C₄

yes

characteristic

148

Q₈

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₅×#108

yes

—

158

C₂×C₄

C₂×C₅×C₂²⋊C₄

yes

characteristic

159

C₂×C₄

C₅×#117

yes

characteristic

160

C₂×C₄

C₅×#121

yes

—

161

C₂×C₄

C₅×C₂²⋊Q₈

yes

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

C₅×#101

yes

—

167

C₅

#8621

yes

characteristic, direct product factor, Sylow

168

C₂²

—

169

C₂²

—

170

C₂²

C₅×#380

yes

characteristic

171

C₂²

C₂×C₅×#108

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

C₂×C₅×#101

yes

characteristic

178

C₄

C₅×#413

yes

characteristic

179

C₂

—

180

C₂

C₅×#1801

yes

characteristic

181

C₂

C₅×#1810

yes

characteristic

182

C₂

C₅×#1812

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

subgroup

G/H

center, socle

C₂²×C₅

#380

commutator

C₂²×C₄

C₂²×C₄×C₅

Frattini, cosocle

C₂×C₄²

C₂³×C₅

prime radical

C₂³×C₅

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

N	H ≅ G/N	type
16. #8621	C₅	direct (N × H)
167. C₅	#8621	direct (N × H)

H ≅ G/N

type

16. #8621

C₅

direct (N × H)

167. C₅

#8621

direct (N × H)

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

subgroup	index
C₂×C₅×C₈×Q₈	2
C₅×#1568	2
C₅×#1467	2
C₅×#1290	2
#8621	5

subgroup

index

C₂×C₅×C₈×Q₈

Subgroups — C₅×#8621 #11931341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#8621 #11931341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8621 #11931341