Group #11935566 — subgroups

Subgroups — C₅×#12846 #11935566

Order 1280.

478 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 174 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	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₅×#2112	C₂	1	2	yes	maximal
3	C₅×#1377	C₂	1	4	yes	maximal, complemented
4	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₂²	1	4	yes	—
8	C₂×C₅×#97	—	2	4	no	—
9	C₂×C₅×#97	C₄	1	2	yes	—
10	C₂×C₅×#97	C₂²	1	2	yes	—
11	C₅×C₂²⋊C₁₆	—	2	4	no	—
12	#12846	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₄×C₅	C₈	1	2	yes	—
14	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₅×C₈	—	2	2	no	—
16	C₂²×C₅×C₈	—	2	4	no	—
17	C₂²×C₅×C₈	—	2	4	no	—
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₅×C₈	D₄	1	4	yes	—
21	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₅	—	2	2	no	—
23	C₂×C₄²×C₅	—	2	2	no	—
24	C₂×C₄²×C₅	C₈	1	2	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
26	C₅×M₃₂	—	2	8	no	—
27	C₂×C₅×C₁₆	—	2	4	no	—
28	C₅×#97	—	2	4	no	—
29	C₅×#97	—	2	4	no	—
30	C₂²×#97	C₂×C₅	1	1	yes	characteristic
31	#2112	C₂×C₅	1	2	yes	—
32	#1377	C₂×C₅	1	4	yes	complemented
33	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₅	—	2	2	no	—
35	C₂²×C₄×C₅	—	2	2	no	—
36	C₂²×C₄×C₅	—	2	2	no	—
37	C₂²×C₄×C₅	—	2	2	no	—
38	C₂²×C₄×C₅	—	4	4	no	—
39	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₅	M₁₆	1	2	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	—	2	8	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	—	2	2	no	—
48	C₂×C₅×C₈	—	2	8	no	—
49	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
50	C₂×C₅×C₈	M₁₆	1	4	yes	—
51	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
52	C₂×C₅×C₈	#41	1	4	yes	—
53	C₄²×C₅	—	2	4	no	—
54	C₄²×C₅	—	2	2	no	—
55	C₄²×C₅	—	2	2	no	—
56	C₅×C₁₆	—	4	4	no	—
57	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
58	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
59	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
60	C₂×M₃₂	C₂²×C₅	1	4	yes	—
61	C₂×#97	—	2	4	no	—
62	C₂×#97	C₄×C₅	1	2	yes	—
63	C₂×#97	C₂²×C₅	1	2	yes	—
64	C₂²⋊C₁₆	—	2	4	no	—
65	C₂³×C₅	—	2	4	no	—
66	C₂³×C₅	—	2	2	no	—
67	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
68	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	—	2	2	no	—
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	2	4	no	—
73	C₂×C₄×C₅	—	2	4	no	—
74	C₂×C₄×C₅	—	4	4	no	—
75	C₂×C₄×C₅	—	4	4	no	—
76	C₂×C₄×C₅	—	4	4	no	—
77	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅	#117	1	2	yes	—
80	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
82	C₅×C₈	—	2	2	no	—
83	C₅×C₈	—	2	4	no	—
84	C₅×C₈	#123	1	4	yes	—
85	C₂³×C₄	C₅×C₈	1	2	yes	—
86	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
87	C₂²×C₈	—	2	2	no	—
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	—	2	4	no	—
90	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
91	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
92	C₂²×C₈	C₅×D₄	1	4	yes	—
93	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
94	C₂×C₄²	—	2	2	no	—
95	C₂×C₄²	—	2	2	no	—
96	C₂×C₄²	C₅×C₈	1	2	yes	—
97	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
98	M₃₂	—	2	8	no	—
99	C₂×C₁₆	—	2	4	no	—
100	#97	—	2	4	no	—
101	#97	—	2	4	no	—
102	C₂²×C₅	—	2	8	no	—
103	C₂²×C₅	—	2	2	no	—
104	C₂²×C₅	—	2	2	no	—
105	C₂²×C₅	—	2	2	no	—
106	C₂²×C₅	#349	1	4	yes	—
107	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₂²×C₅	#431	1	2	yes	—
109	C₄×C₅	—	2	2	no	—
110	C₄×C₅	—	4	4	no	—
111	C₄×C₅	#381	1	1	yes	characteristic
112	C₄×C₅	#407	1	1	yes	characteristic
113	C₄×C₅	#432	1	2	yes	—
114	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	4	4	no	—
120	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
121	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
122	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
123	C₂²×C₄	C₅×M₁₆	1	2	yes	—
124	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
125	C₂×C₈	—	2	8	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
131	C₂×C₈	C₅×M₁₆	1	4	yes	—
132	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
133	C₂×C₈	C₅×#41	1	4	yes	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	4	no	—
137	C₁₆	—	4	4	no	—
138	C₂×C₅	—	2	4	no	—
139	C₂×C₅	—	2	2	no	—
140	C₂×C₅	#1817	1	1	yes	characteristic
141	C₂×C₅	C₂×#349	1	2	yes	—
142	C₂³	—	2	4	no	—
143	C₂³	—	2	2	no	—
144	C₂³	C₅×C₂²⋊C₈	1	2	yes	—
145	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
155	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄	C₅×#117	1	2	yes	—
157	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
158	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	4	no	—
161	C₈	C₅×#123	1	4	yes	—
162	C₅	#12846	1	1	yes	characteristic, direct product factor, Sylow
163	C₂²	—	2	8	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	C₅×#349	1	4	yes	—
168	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
169	C₂²	C₅×#431	1	2	yes	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	C₅×#381	1	1	yes	characteristic
173	C₄	C₅×#407	1	1	yes	characteristic
174	C₄	C₅×#432	1	2	yes	—
175	C₂	—	2	4	no	—
176	C₂	—	2	2	no	—
177	C₂	C₅×#1817	1	1	yes	characteristic
178	C₂	C₂×C₅×#349	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅×#97

C₂

yes

maximal, characteristic

C₅×#2112

C₂

yes

maximal

C₅×#1377

C₂

yes

maximal, complemented

C₂³×C₅×C₈

C₄

yes

characteristic

C₂³×C₅×C₈

C₂²

yes

characteristic

C₂²×C₄²×C₅

C₄

yes

characteristic

C₂×C₅×M₃₂

C₂²

yes

—

C₂×C₅×#97

—

C₂×C₅×#97

C₄

yes

—

C₂×C₅×#97

C₂²

yes

—

C₅×C₂²⋊C₁₆

—

#12846

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₄×C₅

C₈

yes

—

C₂³×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×C₅×C₈

—

C₂²×C₅×C₈

—

C₂²×C₅×C₈

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂²×C₅×C₈

D₄

yes

—

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×C₄²×C₅

—

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₈

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

—

C₅×M₃₂

—

C₂×C₅×C₁₆

—

C₅×#97

—

C₅×#97

—

C₂²×#97

C₂×C₅

yes

characteristic

#2112

C₂×C₅

yes

—

#1377

C₂×C₅

yes

complemented

C₂⁴×C₅

C₂×C₈

yes

characteristic

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂×C₈

yes

characteristic

C₂²×C₄×C₅

C₂×C₈

yes

—

C₂²×C₄×C₅

M₁₆

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

M₁₆

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

#41

yes

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₅×C₁₆

—

C₂³×C₈

C₄×C₅

yes

characteristic

C₂³×C₈

C₂²×C₅

yes

characteristic

C₂²×C₄²

C₄×C₅

yes

characteristic

C₂×M₃₂

C₂²×C₅

yes

—

C₂×#97

—

C₂×#97

C₄×C₅

yes

—

C₂×#97

C₂²×C₅

yes

—

C₂²⋊C₁₆

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂²⋊C₈

yes

—

C₂³×C₅

C₂²×C₈

yes

characteristic

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

C₂²⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₅×C₈

—

C₅×C₈

—

C₅×C₈

#123

yes

—

C₂³×C₄

C₅×C₈

yes

—

C₂³×C₄

C₂×C₄×C₅

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₅

yes

characteristic

C₂²×C₈

C₂×C₄×C₅

yes

—

C₂²×C₈

C₅×D₄

yes

—

C₂²×C₈

C₂³×C₅

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₅×C₈

yes

—

C₂×C₄²

C₂×C₄×C₅

yes

—

M₃₂

—

C₂×C₁₆

—

100

#97

—

101

#97

—

102

C₂²×C₅

—

103

C₂²×C₅

—

104

C₂²×C₅

—

105

C₂²×C₅

—

106

C₂²×C₅

#349

yes

—

107

C₂²×C₅

C₂×C₂²⋊C₈

yes

characteristic

108

C₂²×C₅

#431

yes

—

109

C₄×C₅

—

110

C₄×C₅

—

111

C₄×C₅

#381

yes

characteristic

112

C₄×C₅

#407

yes

characteristic

113

C₄×C₅

#432

yes

—

114

C₂⁴

C₂×C₅×C₈

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₅×C₂²⋊C₄

yes

—

121

C₂²×C₄

C₂×C₅×C₈

yes

characteristic

122

C₂²×C₄

C₂×C₅×C₈

yes

—

123

C₂²×C₄

C₅×M₁₆

yes

—

124

C₂²×C₄

C₂²×C₄×C₅

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₅×C₂²⋊C₄

yes

—

131

C₂×C₈

C₅×M₁₆

yes

—

132

C₂×C₈

C₂×C₅×D₄

yes

—

133

C₂×C₈

C₅×#41

yes

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₁₆

—

138

C₂×C₅

—

139

C₂×C₅

—

140

C₂×C₅

#1817

yes

characteristic

141

C₂×C₅

C₂×#349

yes

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₅×C₂²⋊C₈

yes

—

145

C₂³

C₂²×C₅×C₈

yes

characteristic

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₅×C₂²⋊C₈

yes

—

155

C₂×C₄

C₂×C₅×C₂²⋊C₄

yes

characteristic

156

C₂×C₄

C₅×#117

yes

—

157

C₂×C₄

C₂×C₅×M₁₆

yes

—

158

C₂×C₄

C₂×C₅×M₁₆

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

C₅×#123

yes

—

162

C₅

#12846

yes

characteristic, direct product factor, Sylow

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

C₅×#349

yes

—

168

C₂²

C₂×C₅×C₂²⋊C₈

yes

characteristic

169

C₂²

C₅×#431

yes

—

170

C₄

—

171

C₄

—

172

C₄

C₅×#381

yes

characteristic

173

C₄

C₅×#407

yes

characteristic

174

C₄

C₅×#432

yes

—

175

C₂

—

176

C₂

—

177

C₂

C₅×#1817

yes

characteristic

178

C₂

C₂×C₅×#349

yes

—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

subgroup

G/H

center

C₂×C₄×C₅

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₅×C₈

Frattini, cosocle

C₂²×C₈

C₂³×C₅

socle

C₂²×C₅

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₅

C₂×C₈

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

N	H ≅ G/N	type
3. C₅×#1377	C₂	semidirect
12. #12846	C₅	direct (N × H)
32. #1377	C₂×C₅	semidirect
162. C₅	#12846	direct (N × H)

H ≅ G/N

type

3. C₅×#1377

C₂

semidirect

12. #12846

C₅

direct (N × H)

32. #1377

C₂×C₅

semidirect

162. C₅

#12846

direct (N × H)

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

subgroup	index
C₂²×C₅×#97	2
C₅×#2112	2
C₅×#1377	2
#12846	5

subgroup

index

C₂²×C₅×#97

C₅×#2112

C₅×#1377

#12846

Subgroups — C₅×#12846 #11935566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#12846 #11935566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12846 #11935566