Group #11933382 — subgroups

Subgroups — C₅×#10662 #11933382

Order 1280.

320 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1985	C₂	1	1	yes	maximal, characteristic
2	C₅×#1849	C₂	1	1	yes	maximal, characteristic
3	C₅×#1571	C₂	1	2	yes	maximal
4	C₅×#1487	C₂	1	2	yes	maximal
5	C₅×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₅×#431	C₂²	1	2	yes	—
8	C₂×C₅×#108	—	2	2	no	—
9	C₅×#423	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#97	C₂²	1	1	yes	characteristic
12	C₅×#336	—	2	2	no	—
13	C₅×#335	—	2	4	no	—
14	C₅×Q₈⋊C₈	—	2	4	no	—
15	#10662	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₅×Q₈	Q₈	1	1	yes	characteristic
17	C₂×C₅×M₁₆	—	2	2	no	—
18	C₂²×C₅×C₈	—	2	1	no	—
19	C₂²×C₅×C₈	D₄	1	2	yes	—
20	C₄×C₅×Q₈	—	2	2	no	—
21	C₄×C₅×Q₈	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	—	2	1	no	—
23	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₅	—	2	1	no	—
25	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
26	C₅×#108	—	4	4	no	—
27	C₅×C₄⋊C₈	—	2	2	no	—
28	C₅×C₄⋊C₈	—	2	2	no	—
29	C₅×C₄⋊C₈	—	2	2	no	—
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₅×C₂²⋊C₈	—	2	2	no	—
32	C₅×#97	—	2	2	no	—
33	C₄×C₅×C₈	D₄	1	4	yes	—
34	#1985	C₂×C₅	1	1	yes	characteristic
35	#1849	C₂×C₅	1	1	yes	characteristic
36	#1571	C₂×C₅	1	2	yes	—
37	#1487	C₂×C₅	1	2	yes	—
38	#1275	C₂×C₅	1	1	yes	characteristic
39	C₂×C₅×Q₈	—	2	2	no	—
40	C₂×C₅×Q₈	—	4	2	no	—
41	C₂²×C₄×C₅	—	2	1	no	—
42	C₂²×C₄×C₅	—	2	1	no	—
43	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	#41	1	1	yes	characteristic
46	C₅×M₁₆	—	4	4	no	—
47	C₂×C₅×C₈	—	2	2	no	—
48	C₂×C₅×C₈	—	2	2	no	—
49	C₂×C₅×C₈	—	2	4	no	—
50	C₂×C₅×C₈	—	4	1	no	—
51	C₂×C₅×C₈	—	4	2	no	—
52	C₂×C₅×C₈	—	4	2	no	—
53	C₂×C₅×C₈	Q₁₆	1	4	yes	—
54	C₅×C₄⋊C₄	—	2	2	no	—
55	C₅×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₅	—	4	1	no	—
57	C₄²×C₅	—	4	1	no	—
58	C₄²×C₅	C₂×D₄	1	2	yes	—
59	C₄²×C₅	#41	1	2	yes	—
60	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
61	#431	C₂²×C₅	1	2	yes	—
62	C₂×#108	—	2	2	no	—
63	#423	C₂²×C₅	1	2	yes	—
64	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
65	C₂×#97	C₂²×C₅	1	1	yes	characteristic
66	#336	—	2	2	no	—
67	#335	—	2	4	no	—
68	Q₈⋊C₈	—	2	4	no	—
69	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
70	C₅×Q₈	—	2	4	no	—
71	C₅×Q₈	—	4	2	no	—
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	4	1	no	—
74	C₂×C₄×C₅	—	4	1	no	—
75	C₂×C₄×C₅	—	4	1	no	—
76	C₂×C₄×C₅	—	4	1	no	—
77	C₂×C₄×C₅	—	4	2	no	—
78	C₂×C₄×C₅	—	4	1	no	—
79	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₅	#123	1	2	yes	—
81	C₂×C₄×C₅	#123	1	1	yes	characteristic
82	C₂×C₄×C₅	#127	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
85	C₅×C₈	—	2	4	no	—
86	C₅×C₈	—	4	4	no	—
87	C₅×C₈	—	8	1	no	—
88	C₅×C₈	—	8	2	no	—
89	C₂²×Q₈	C₅×Q₈	1	1	yes	characteristic
90	C₂×M₁₆	—	2	2	no	—
91	C₂²×C₈	—	2	1	no	—
92	C₂²×C₈	C₅×D₄	1	2	yes	—
93	C₄×Q₈	—	2	2	no	—
94	C₄×Q₈	—	2	2	no	—
95	C₂×C₄⋊C₄	—	2	1	no	—
96	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
97	C₂×C₄²	—	2	1	no	—
98	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
99	#108	—	4	4	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₂²⋊C₈	—	2	2	no	—
105	#97	—	2	2	no	—
106	C₄×C₈	C₅×D₄	1	4	yes	—
107	C₂²×C₅	—	2	1	no	—
108	C₂²×C₅	—	2	1	no	—
109	C₂²×C₅	#399	1	1	yes	characteristic
110	C₂²×C₅	#477	1	2	yes	—
111	C₄×C₅	—	2	2	no	—
112	C₄×C₅	—	2	2	no	—
113	C₄×C₅	—	4	2	no	—
114	C₄×C₅	—	8	1	no	—
115	C₄×C₅	#484	1	2	yes	—
116	C₄×C₅	#494	1	1	yes	characteristic
117	C₄×C₅	#496	1	1	yes	characteristic
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
124	C₂²×C₄	C₅×#41	1	1	yes	characteristic
125	M₁₆	—	4	4	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	C₅×Q₁₆	1	4	yes	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄²	—	4	1	no	—
136	C₄²	—	4	1	no	—
137	C₄²	C₂×C₅×D₄	1	2	yes	—
138	C₄²	C₅×#41	1	2	yes	—
139	C₂×C₅	—	2	2	no	—
140	C₂×C₅	#2091	1	1	yes	characteristic
141	C₂×C₅	#2095	1	1	yes	characteristic
142	C₂×C₅	#2098	1	1	yes	characteristic
143	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
144	Q₈	—	2	4	no	—
145	Q₈	—	4	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
154	C₂×C₄	C₅×#123	1	2	yes	—
155	C₂×C₄	C₅×#123	1	1	yes	characteristic
156	C₂×C₄	C₅×#127	1	1	yes	characteristic
157	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
158	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
159	C₈	—	2	4	no	—
160	C₈	—	4	4	no	—
161	C₈	—	8	1	no	—
162	C₈	—	8	2	no	—
163	C₅	#10662	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₅×#399	1	1	yes	characteristic
167	C₂²	C₅×#477	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	1	no	—
172	C₄	C₅×#484	1	2	yes	—
173	C₄	C₅×#494	1	1	yes	characteristic
174	C₄	C₅×#496	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	C₅×#2091	1	1	yes	characteristic
177	C₂	C₅×#2095	1	1	yes	characteristic
178	C₂	C₅×#2098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1985

C₂

yes

maximal, characteristic

C₅×#1849

C₂

yes

maximal, characteristic

C₅×#1571

C₂

yes

maximal

C₅×#1487

C₂

yes

maximal

C₅×#1275

C₂

yes

maximal, characteristic

C₂×C₄×C₅×Q₈

C₂²

yes

characteristic

C₅×#431

C₂²

yes

—

C₂×C₅×#108

—

C₅×#423

C₂²

yes

—

C₂×C₅×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₅×#97

C₂²

yes

characteristic

C₅×#336

—

C₅×#335

—

C₅×Q₈⋊C₈

—

#10662

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×Q₈

Q₈

yes

characteristic

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

—

C₂²×C₅×C₈

D₄

yes

—

C₄×C₅×Q₈

—

C₄×C₅×Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

Q₈

yes

characteristic

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₅×#108

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₅×#97

—

C₄×C₅×C₈

D₄

yes

—

#1985

C₂×C₅

yes

characteristic

#1849

C₂×C₅

yes

characteristic

#1571

C₂×C₅

yes

—

#1487

C₂×C₅

yes

—

#1275

C₂×C₅

yes

characteristic

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅

#41

yes

characteristic

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

Q₁₆

yes

—

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

#431

C₂²×C₅

yes

—

C₂×#108

—

#423

C₂²×C₅

yes

—

C₂×C₄⋊C₈

C₂²×C₅

yes

characteristic

C₂×#97

C₂²×C₅

yes

characteristic

#336

—

#335

—

Q₈⋊C₈

—

C₂³×C₅

C₂²⋊Q₈

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

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

C₂×Q₁₆

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₂²×Q₈

C₅×Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₅×D₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₅×Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³×C₅

yes

characteristic

#108

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₂²⋊C₈

—

105

#97

—

106

C₄×C₈

C₅×D₄

yes

—

107

C₂²×C₅

—

108

C₂²×C₅

—

109

C₂²×C₅

#399

yes

characteristic

110

C₂²×C₅

#477

yes

—

111

C₄×C₅

—

112

C₄×C₅

—

113

C₄×C₅

—

114

C₄×C₅

—

115

C₄×C₅

#484

yes

—

116

C₄×C₅

#494

yes

characteristic

117

C₄×C₅

#496

yes

characteristic

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂×C₅×D₄

yes

characteristic

123

C₂²×C₄

C₂×C₅×Q₈

yes

characteristic

124

C₂²×C₄

C₅×#41

yes

characteristic

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

C₅×Q₁₆

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂×C₅×D₄

yes

—

138

C₄²

C₅×#41

yes

—

139

C₂×C₅

—

140

C₂×C₅

#2091

yes

characteristic

141

C₂×C₅

#2095

yes

characteristic

142

C₂×C₅

#2098

yes

characteristic

143

C₂³

C₅×C₂²⋊Q₈

yes

characteristic

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₅×C₂²⋊Q₈

yes

—

154

C₂×C₄

C₅×#123

yes

—

155

C₂×C₄

C₅×#123

yes

characteristic

156

C₂×C₄

C₅×#127

yes

characteristic

157

C₂×C₄

C₂×C₅×Q₁₆

yes

—

158

C₂×C₄

C₅×AGL(1,ℤ/8)

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₅

#10662

yes

characteristic, direct product factor, Sylow

164

C₂²

—

165

C₂²

—

166

C₂²

C₅×#399

yes

characteristic

167

C₂²

C₅×#477

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

C₅×#484

yes

—

173

C₄

C₅×#494

yes

characteristic

174

C₄

C₅×#496

yes

characteristic

175

C₂

—

176

C₂

C₅×#2091

yes

characteristic

177

C₂

C₅×#2095

yes

characteristic

178

C₂

C₅×#2098

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#399
prime radical	C₂³×C₅	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

subgroup

G/H

center

C₂×C₄×C₅

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅

socle

C₂²×C₅

#399

prime radical

C₂³×C₅

C₂²⋊Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

N	H ≅ G/N	type
15. #10662	C₅	direct (N × H)
163. C₅	#10662	direct (N × H)

H ≅ G/N

type

15. #10662

C₅

direct (N × H)

163. C₅

#10662

direct (N × H)

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

subgroup	index
C₅×#1985	2
C₅×#1849	2
C₅×#1571	2
C₅×#1487	2
C₅×#1275	2
#10662	5

subgroup

index

Subgroups — C₅×#10662 #11933382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#10662 #11933382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10662 #11933382