Group #11935560 — subgroups

Subgroups — C₅×#12840 #11935560

Order 1280.

354 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 154 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×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#368	C₂	1	2	yes	maximal, complemented
3	C₅×#1813	C₂	1	2	yes	maximal
4	C₅×#1385	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#131	C₂²	1	1	yes	characteristic
6	C₅×#504	—	2	4	no	—
7	C₂×C₅×M₃₂	C₄	1	2	yes	—
8	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₄	1	2	yes	—
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₅×#108	C₂²	1	4	yes	—
12	C₅×#413	—	2	2	no	—
13	C₅×#368	—	2	4	no	—
14	C₅×#338	—	2	2	no	—
15	#12840	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₅×#41	D₄	1	1	yes	characteristic
17	C₅×#131	—	2	2	no	—
18	C₅×#131	—	2	2	no	—
19	C₂×C₅×M₁₆	—	2	4	no	—
20	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
21	C₂×C₅×M₁₆	Q₈	1	2	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₅×M₃₂	—	2	2	no	—
25	C₅×M₃₂	—	2	4	no	—
26	C₂×C₅×C₁₆	—	2	4	no	—
27	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×C₁₆	D₄	1	4	yes	—
31	C₅×#108	—	2	4	no	—
32	C₅×#108	—	2	4	no	—
33	C₅×#101	—	4	2	no	—
34	C₅×#100	—	4	2	no	—
35	C₂×#504	C₂×C₅	1	1	yes	characteristic
36	C₂×#368	C₂×C₅	1	2	yes	complemented
37	#1813	C₂×C₅	1	2	yes	—
38	#1385	C₂×C₅	1	2	yes	complemented
39	C₅×#41	—	2	2	no	—
40	C₅×#41	—	2	2	no	—
41	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
42	C₂×C₅×D₄	D₈	1	1	yes	characteristic
43	C₂×C₅×D₄	Q₁₆	1	2	yes	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₅×M₁₆	—	2	2	no	—
47	C₅×M₁₆	—	2	4	no	—
48	C₅×M₁₆	—	4	4	no	—
49	C₅×M₁₆	—	4	4	no	—
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	—	4	4	no	—
53	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
56	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₅×C₈	#41	1	2	yes	—
58	C₅×C₁₆	—	2	2	no	—
59	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
60	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
61	C₂×#131	C₂²×C₅	1	1	yes	characteristic
62	#504	—	2	4	no	—
63	C₂×M₃₂	C₄×C₅	1	2	yes	—
64	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
65	C₂²×C₁₆	C₄×C₅	1	2	yes	—
66	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
67	C₂×#108	C₂²×C₅	1	4	yes	—
68	#413	—	2	2	no	—
69	#368	—	2	4	no	—
70	#338	—	2	2	no	—
71	C₂³×C₅	#107	1	2	yes	—
72	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
73	C₅×Q₈	—	2	2	no	—
74	C₅×D₄	—	2	4	no	—
75	C₅×D₄	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	#107	1	2	yes	—
79	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	#121	1	2	yes	—
81	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅	#135	1	2	yes	—
83	C₅×C₈	—	2	2	no	—
84	C₅×C₈	—	8	4	no	—
85	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₅×C₈	#117	1	1	yes	characteristic
87	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
88	C₂×#41	C₅×D₄	1	1	yes	characteristic
89	#131	—	2	2	no	—
90	#131	—	2	2	no	—
91	C₂×M₁₆	—	2	4	no	—
92	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
93	C₂×M₁₆	C₅×Q₈	1	2	yes	—
94	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
95	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	2	4	no	—
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
100	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂×C₁₆	C₅×D₄	1	4	yes	—
103	#108	—	2	4	no	—
104	#108	—	2	4	no	—
105	#101	—	4	2	no	—
106	#100	—	4	2	no	—
107	C₂²×C₅	—	2	2	no	—
108	C₂²×C₅	—	4	1	no	—
109	C₂²×C₅	—	4	2	no	—
110	C₂²×C₅	C₂×#107	1	1	yes	characteristic
111	C₂²×C₅	#467	1	2	yes	—
112	C₄×C₅	—	2	2	no	—
113	C₄×C₅	#380	1	1	yes	characteristic
114	C₄×C₅	#427	1	1	yes	characteristic
115	C₄×C₅	#466	1	2	yes	—
116	#41	—	2	2	no	—
117	#41	—	2	2	no	—
118	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
119	C₂×D₄	C₅×D₈	1	1	yes	characteristic
120	C₂×D₄	C₅×Q₁₆	1	2	yes	—
121	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
122	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
123	M₁₆	—	2	2	no	—
124	M₁₆	—	2	4	no	—
125	M₁₆	—	4	4	no	—
126	M₁₆	—	4	4	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	4	no	—
130	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
131	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
132	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
133	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
134	C₂×C₈	C₅×#41	1	2	yes	—
135	C₁₆	—	2	2	no	—
136	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
137	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
138	C₂×C₅	—	2	1	no	—
139	C₂×C₅	—	2	2	no	—
140	C₂×C₅	—	4	2	no	—
141	C₂×C₅	#1808	1	1	yes	characteristic
142	C₂³	C₅×#107	1	2	yes	—
143	C₂³	C₂×C₅×Q₁₆	1	1	yes	characteristic
144	Q₈	—	2	2	no	—
145	D₄	—	2	4	no	—
146	D₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₅×#107	1	2	yes	—
150	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄	C₅×#121	1	2	yes	—
152	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
153	C₂×C₄	C₅×#135	1	2	yes	—
154	C₈	—	2	2	no	—
155	C₈	—	8	4	no	—
156	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
157	C₈	C₅×#117	1	1	yes	characteristic
158	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
159	C₅	#12840	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²	—	2	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	C₂×C₅×#107	1	1	yes	characteristic
164	C₂²	C₅×#467	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	C₅×#380	1	1	yes	characteristic
167	C₄	C₅×#427	1	1	yes	characteristic
168	C₄	C₅×#466	1	2	yes	—
169	C₂	—	2	1	no	—
170	C₂	—	2	2	no	—
171	C₂	—	4	2	no	—
172	C₂	C₅×#1808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#504

C₂

yes

maximal, characteristic

C₂×C₅×#368

C₂

yes

maximal, complemented

C₅×#1813

C₂

yes

maximal

C₅×#1385

C₂

yes

maximal, complemented

C₂×C₅×#131

C₂²

yes

characteristic

C₅×#504

—

C₂×C₅×M₃₂

C₄

yes

—

C₂×C₅×M₃₂

C₂²

yes

characteristic

C₂²×C₅×C₁₆

C₄

yes

—

C₂²×C₅×C₁₆

C₂²

yes

characteristic

C₂×C₅×#108

C₂²

yes

—

—

—

—

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅×#41

D₄

yes

characteristic

C₅×#131

—

C₅×#131

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

D₄

yes

characteristic

C₂×C₅×M₁₆

Q₈

yes

—

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂²×C₅×C₈

C₂³

yes

characteristic

C₅×M₃₂

—

C₅×M₃₂

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

D₄

yes

—

—

—

—

—

C₂×C₅

yes

characteristic

C₂×#368

C₂×C₅

yes

complemented

#1813

C₂×C₅

yes

—

#1385

C₂×C₅

yes

complemented

C₅×#41

—

C₅×#41

—

C₂×C₅×Q₈

D₈

yes

characteristic

C₂×C₅×D₄

D₈

yes

characteristic

C₂×C₅×D₄

Q₁₆

yes

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

C₂×Q₈

yes

characteristic

C₂×C₅×C₈

#41

yes

—

C₅×C₁₆

—

C₅×C₁₆

C₂²⋊C₄

yes

—

C₅×C₁₆

C₂²⋊C₄

yes

—

C₂×#131

C₂²×C₅

yes

characteristic

#504

—

C₂×M₃₂

C₄×C₅

yes

—

C₂×M₃₂

C₂²×C₅

yes

characteristic

C₂²×C₁₆

C₄×C₅

yes

—

C₂²×C₁₆

C₂²×C₅

yes

characteristic

C₂×#108

C₂²×C₅

yes

—

#413

—

#368

—

#338

—

C₂³×C₅

#107

yes

—

C₂³×C₅

C₂×Q₁₆

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#107

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

#135

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₅×C₈

#117

yes

characteristic

C₅×C₈

C₂²⋊Q₈

yes

—

C₂×#41

C₅×D₄

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₅×D₄

yes

characteristic

C₂×M₁₆

C₅×Q₈

yes

—

C₂²×C₈

C₂×C₄×C₅

yes

—

C₂²×C₈

C₂³×C₅

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₅

yes

—

100

C₂×C₁₆

C₂×C₄×C₅

yes

characteristic

101

C₂×C₁₆

C₂×C₄×C₅

yes

characteristic

102

C₂×C₁₆

C₅×D₄

yes

—

103

#108

—

104

#108

—

105

#101

—

106

#100

—

107

C₂²×C₅

—

108

C₂²×C₅

—

109

C₂²×C₅

—

110

C₂²×C₅

C₂×#107

yes

characteristic

111

C₂²×C₅

#467

yes

—

112

C₄×C₅

—

113

C₄×C₅

#380

yes

characteristic

114

C₄×C₅

#427

yes

characteristic

115

C₄×C₅

#466

yes

—

116

#41

—

117

#41

—

118

C₂×Q₈

C₅×D₈

yes

characteristic

119

C₂×D₄

C₅×D₈

yes

characteristic

120

C₂×D₄

C₅×Q₁₆

yes

—

121

C₂²×C₄

C₅×C₄⋊C₄

yes

—

122

C₂²×C₄

C₂×C₅×D₄

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₅×C₄⋊C₄

yes

—

131

C₂×C₈

C₂²×C₄×C₅

yes

characteristic

132

C₂×C₈

C₂×C₅×D₄

yes

—

133

C₂×C₈

C₂×C₅×Q₈

yes

characteristic

134

C₂×C₈

C₅×#41

yes

—

135

C₁₆

—

136

C₁₆

C₅×C₂²⋊C₄

yes

—

137

C₁₆

C₅×C₂²⋊C₄

yes

—

138

C₂×C₅

—

139

C₂×C₅

—

140

C₂×C₅

—

141

C₂×C₅

#1808

yes

characteristic

142

C₂³

C₅×#107

yes

—

143

C₂³

C₂×C₅×Q₁₆

yes

characteristic

144

Q₈

—

145

D₄

—

146

D₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

C₅×#107

yes

—

150

C₂×C₄

C₂×C₅×C₄⋊C₄

yes

characteristic

151

C₂×C₄

C₅×#121

yes

—

152

C₂×C₄

C₂×C₅×D₈

yes

characteristic

153

C₂×C₄

C₅×#135

yes

—

154

C₈

—

155

C₈

—

156

C₈

C₂×C₅×C₂²⋊C₄

yes

characteristic

157

C₈

C₅×#117

yes

characteristic

158

C₈

C₅×C₂²⋊Q₈

yes

—

159

C₅

#12840

yes

characteristic, direct product factor, Sylow

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

C₂×C₅×#107

yes

characteristic

164

C₂²

C₅×#467

yes

—

165

C₄

—

166

C₄

C₅×#380

yes

characteristic

167

C₄

C₅×#427

yes

characteristic

168

C₄

C₅×#466

yes

—

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

C₅×#1808

yes

characteristic

	subgroup	G/H
center	C₄×C₅	#427
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1808
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₂×C₅×#131 ◁ G

subgroup

G/H

center

C₄×C₅

#427

commutator

C₂×C₈

C₂²×C₄×C₅

Frattini, cosocle

C₂²×C₈

C₂³×C₅

socle

C₂×C₅

#1808

prime radical

C₂×C₅×#41

D₄

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₂×C₅×#131 ◁ G

N	H ≅ G/N	type
2. C₂×C₅×#368	C₂	semidirect
4. C₅×#1385	C₂	semidirect
15. #12840	C₅	direct (N × H)
36. C₂×#368	C₂×C₅	semidirect
38. #1385	C₂×C₅	semidirect
159. C₅	#12840	direct (N × H)

H ≅ G/N

type

2. C₂×C₅×#368

C₂

semidirect

4. C₅×#1385

C₂

semidirect

15. #12840

C₅

direct (N × H)

36. C₂×#368

C₂×C₅

semidirect

38. #1385

C₂×C₅

semidirect

159. C₅

#12840

direct (N × H)

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

subgroup	index
C₂×C₅×#504	2
C₂×C₅×#368	2
C₅×#1813	2
C₅×#1385	2
#12840	5

subgroup

index

C₂×C₅×#504

C₂×C₅×#368

C₅×#1813

C₅×#1385

#12840

Subgroups — C₅×#12840 #11935560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#12840 #11935560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12840 #11935560