Group #11930359 — subgroups

Subgroups — C₅×#7639 #11930359

Order 1280.

468 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 138 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	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1745	C₂	1	2	yes	maximal
3	C₅×#1272	C₂	1	4	yes	maximal
4	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₅×Q₈	C₄	1	2	yes	—
6	C₂×C₅×#97	—	2	2	no	—
7	C₅×#391	C₂²	1	2	yes	—
8	C₅×#389	—	2	2	no	—
9	C₅×#384	C₄	1	1	yes	characteristic
10	C₄×C₅×C₄⋊C₄	—	2	2	no	—
11	C₄³×C₅	C₂²	1	1	yes	characteristic
12	C₅×#336	—	2	2	no	—
13	C₅×#336	C₂²	1	4	yes	—
14	#7639	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
16	C₂²×C₅×C₈	—	2	4	no	—
17	C₅×C₄⋊Q₈	—	2	4	no	—
18	C₄×C₅×Q₈	—	4	4	no	—
19	C₂×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₅×C₄⋊C₄	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	—	2	2	no	—
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	D₄	1	2	yes	—
27	C₂×C₄²×C₅	D₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₅×#97	—	2	8	no	—
30	C₅×#95	—	2	4	no	—
31	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
32	#1745	C₂×C₅	1	2	yes	—
33	#1272	C₂×C₅	1	4	yes	—
34	C₂×C₅×Q₈	—	2	2	no	—
35	C₂×C₅×Q₈	—	4	2	no	—
36	C₂×C₅×Q₈	—	4	2	no	—
37	C₂×C₅×Q₈	M₁₆	1	4	yes	—
38	C₂²×C₄×C₅	—	2	2	no	—
39	C₂²×C₄×C₅	—	2	2	no	—
40	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
45	C₂×C₅×C₈	—	2	8	no	—
46	C₂×C₅×C₈	—	4	4	no	—
47	C₂×C₅×C₈	—	4	4	no	—
48	C₅×C₄⋊C₄	—	2	8	no	—
49	C₅×C₄⋊C₄	—	4	2	no	—
50	C₅×C₄⋊C₄	—	4	4	no	—
51	C₅×C₄⋊C₄	—	4	4	no	—
52	C₄²×C₅	—	2	2	no	—
53	C₄²×C₅	—	2	4	no	—
54	C₄²×C₅	—	2	2	no	—
55	C₄²×C₅	—	2	4	no	—
56	C₄²×C₅	—	4	1	no	—
57	C₄²×C₅	—	4	4	no	—
58	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
59	C₂×C₄×Q₈	C₄×C₅	1	2	yes	—
60	C₂×#97	—	2	2	no	—
61	#391	C₂²×C₅	1	2	yes	—
62	#389	—	2	2	no	—
63	#384	C₄×C₅	1	1	yes	characteristic
64	C₄×C₄⋊C₄	—	2	2	no	—
65	C₄³	C₂²×C₅	1	1	yes	characteristic
66	#336	—	2	2	no	—
67	#336	C₂²×C₅	1	4	yes	—
68	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₅×Q₈	—	4	4	no	—
70	C₅×Q₈	—	4	4	no	—
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	1	no	—
74	C₂×C₄×C₅	—	2	1	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	4	no	—
79	C₂×C₄×C₅	—	2	1	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	4	2	no	—
82	C₂×C₄×C₅	—	4	2	no	—
83	C₂×C₄×C₅	—	4	2	no	—
84	C₂×C₄×C₅	—	4	4	no	—
85	C₂×C₄×C₅	#100	1	2	yes	—
86	C₂×C₄×C₅	#104	1	4	yes	—
87	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₅	#137	1	4	yes	—
89	C₅×C₈	—	4	8	no	—
90	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
91	C₂²×C₈	—	2	4	no	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₄×Q₈	—	4	4	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
98	C₂×C₄²	—	2	2	no	—
99	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂×C₄²	C₅×D₄	1	2	yes	—
102	C₂×C₄²	C₅×D₄	1	2	yes	—
103	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
104	#97	—	2	8	no	—
105	#95	—	2	4	no	—
106	C₂²×C₅	#407	1	1	yes	characteristic
107	C₂²×C₅	#410	1	1	yes	characteristic
108	C₂²×C₅	C₂×#100	1	1	yes	characteristic
109	C₂²×C₅	#419	1	2	yes	—
110	C₂²×C₅	C₂×#104	1	2	yes	—
111	C₄×C₅	—	2	2	no	—
112	C₄×C₅	—	2	2	no	—
113	C₄×C₅	—	2	4	no	—
114	C₄×C₅	—	2	4	no	—
115	C₄×C₅	—	4	4	no	—
116	C₄×C₅	—	4	1	no	—
117	C₄×C₅	—	8	2	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂×Q₈	—	4	2	no	—
121	C₂×Q₈	C₅×M₁₆	1	4	yes	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
127	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	—	4	4	no	—
131	C₂×C₈	—	4	4	no	—
132	C₄⋊C₄	—	2	8	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	4	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	4	1	no	—
141	C₄²	—	4	4	no	—
142	C₂×C₅	#1466	1	1	yes	characteristic
143	C₂×C₅	#1491	1	2	yes	—
144	C₂×C₅	#1506	1	2	yes	—
145	C₂×C₅	#1525	1	1	yes	characteristic
146	C₂×C₅	#1558	1	1	yes	characteristic
147	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
148	Q₈	—	4	4	no	—
149	Q₈	—	4	4	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	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	C₅×#100	1	2	yes	—
165	C₂×C₄	C₅×#104	1	4	yes	—
166	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
167	C₂×C₄	C₅×#137	1	4	yes	—
168	C₈	—	4	8	no	—
169	C₅	#7639	1	1	yes	characteristic, direct product factor, Sylow
170	C₂²	C₅×#407	1	1	yes	characteristic
171	C₂²	C₅×#410	1	1	yes	characteristic
172	C₂²	C₂×C₅×#100	1	1	yes	characteristic
173	C₂²	C₅×#419	1	2	yes	—
174	C₂²	C₂×C₅×#104	1	2	yes	—
175	C₄	—	2	4	no	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	4	no	—
179	C₄	—	4	1	no	—
180	C₄	—	4	4	no	—
181	C₄	—	8	2	no	—
182	C₂	C₅×#1466	1	1	yes	characteristic
183	C₂	C₅×#1491	1	2	yes	—
184	C₂	C₅×#1506	1	2	yes	—
185	C₂	C₅×#1525	1	1	yes	characteristic
186	C₂	C₅×#1558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₅×#1745

C₂

yes

maximal

C₅×#1272

C₂

yes

maximal

C₂×C₅×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄×C₅×Q₈

C₄

yes

—

C₂×C₅×#97

—

C₅×#391

C₂²

yes

—

C₅×#389

—

C₅×#384

C₄

yes

characteristic

C₄×C₅×C₄⋊C₄

—

C₄³×C₅

C₂²

yes

characteristic

C₅×#336

—

C₅×#336

C₂²

yes

—

#7639

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×Q₈

C₂×C₄

yes

—

C₂²×C₅×C₈

—

C₅×C₄⋊Q₈

—

C₄×C₅×Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₅×#97

—

C₅×#95

—

C₄×C₄⋊Q₈

C₂×C₅

yes

characteristic

#1745

C₂×C₅

yes

—

#1272

C₂×C₅

yes

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

M₁₆

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₂×C₄⋊Q₈

C₄×C₅

yes

characteristic

C₂×C₄×Q₈

C₄×C₅

yes

—

C₂×#97

—

#391

C₂²×C₅

yes

—

#389

—

#384

C₄×C₅

yes

characteristic

C₄×C₄⋊C₄

—

C₄³

C₂²×C₅

yes

characteristic

#336

—

#336

C₂²×C₅

yes

—

C₂³×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₅×Q₈

—

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

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#100

yes

—

C₂×C₄×C₅

#104

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

#137

yes

—

C₅×C₈

—

C₂²×Q₈

C₂×C₄×C₅

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄×C₅

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₅

yes

characteristic

100

C₂×C₄²

C₂×C₄×C₅

yes

characteristic

101

C₂×C₄²

C₅×D₄

yes

—

102

C₂×C₄²

C₅×D₄

yes

—

103

C₂×C₄²

C₂³×C₅

yes

characteristic

104

#97

—

105

#95

—

106

C₂²×C₅

#407

yes

characteristic

107

C₂²×C₅

#410

yes

characteristic

108

C₂²×C₅

C₂×#100

yes

characteristic

109

C₂²×C₅

#419

yes

—

110

C₂²×C₅

C₂×#104

yes

—

111

C₄×C₅

—

112

C₄×C₅

—

113

C₄×C₅

—

114

C₄×C₅

—

115

C₄×C₅

—

116

C₄×C₅

—

117

C₄×C₅

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

C₅×M₁₆

yes

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₅×C₂²⋊C₄

yes

characteristic

125

C₂²×C₄

C₅×C₂²⋊C₄

yes

characteristic

126

C₂²×C₄

C₅×C₂²⋊C₄

yes

—

127

C₂²×C₄

C₂²×C₄×C₅

yes

characteristic

128

C₂²×C₄

C₂×C₅×D₄

yes

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₂×C₅

#1466

yes

characteristic

143

C₂×C₅

#1491

yes

—

144

C₂×C₅

#1506

yes

—

145

C₂×C₅

#1525

yes

characteristic

146

C₂×C₅

#1558

yes

characteristic

147

C₂³

C₂×C₅×C₂²⋊C₄

yes

characteristic

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₅×#100

yes

—

165

C₂×C₄

C₅×#104

yes

—

166

C₂×C₄

C₂×C₅×M₁₆

yes

—

167

C₂×C₄

C₅×#137

yes

—

168

C₈

—

169

C₅

#7639

yes

characteristic, direct product factor, Sylow

170

C₂²

C₅×#407

yes

characteristic

171

C₂²

C₅×#410

yes

characteristic

172

C₂²

C₂×C₅×#100

yes

characteristic

173

C₂²

C₅×#419

yes

—

174

C₂²

C₂×C₅×#104

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₂

C₅×#1466

yes

characteristic

183

C₂

C₅×#1491

yes

—

184

C₂

C₅×#1506

yes

—

185

C₂

C₅×#1525

yes

characteristic

186

C₂

C₅×#1558

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅

C₂×C₂²⋊C₄

commutator

C₂²×C₄

C₂²×C₄×C₅

Frattini, cosocle

C₂×C₄²

C₂³×C₅

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

N	H ≅ G/N	type
14. #7639	C₅	direct (N × H)
169. C₅	#7639	direct (N × H)

H ≅ G/N

type

14. #7639

C₅

direct (N × H)

169. C₅

#7639

direct (N × H)

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₅×#1745	2
C₅×#1272	2
#7639	5

subgroup

index

C₄×C₅×C₄⋊Q₈

C₅×#1745

C₅×#1272

#7639

Subgroups — C₅×#7639 #11930359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#7639 #11930359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7639 #11930359