Group #11987618 — subgroups

Subgroups — #11987618

Order 1280.

384 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 119 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×#125	C₂	1	1	yes	maximal, characteristic
2	#10600186	C₂	1	2	yes	maximal
3	#10598129	C₂	1	4	yes	maximal
4	C₂×C₅×#125	C₄	1	1	yes	characteristic
5	C₅×#389	C₂²	1	2	yes	—
6	C₅×#387	—	2	2	no	—
7	C₅×#382	C₄	1	1	yes	characteristic
8	C₄×C₅×C₄⋊C₄	—	2	2	no	—
9	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	#64744	—	2	2	no	—
12	#64744	—	2	2	no	—
13	#64744	C₂²	1	4	yes	—
14	#7650	—	5	1	no	maximal, Sylow
15	C₅×#125	—	2	4	no	—
16	C₂×C₅×C₄⋊C₄	—	2	2	no	—
17	C₂×C₅×C₄⋊C₄	—	2	2	no	—
18	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
19	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅	—	2	2	no	—
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	D₄	1	2	yes	—
24	C₂×C₄²×C₅	D₄	1	2	yes	—
25	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
26	C₂²×#168	—	2	4	no	—
27	C₅×#95	—	2	2	no	—
28	C₅×#95	—	2	2	no	—
29	C₅×#95	—	2	2	no	—
30	C₄×#125	D₅	1	1	yes	characteristic
31	#1839	—	5	2	no	—
32	#1271	—	5	4	no	—
33	C₂²×C₄×C₅	—	2	2	no	—
34	C₂²×C₄×C₅	—	2	2	no	—
35	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
40	C₅×C₄⋊C₄	—	2	8	no	—
41	C₅×C₄⋊C₄	—	2	2	no	—
42	C₅×C₄⋊C₄	—	4	2	no	—
43	C₅×C₄⋊C₄	—	4	4	no	—
44	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
45	C₄²×C₅	—	2	2	no	—
46	C₄²×C₅	—	2	2	no	—
47	C₄²×C₅	—	2	4	no	—
48	C₄²×C₅	—	4	2	no	—
49	C₄²×C₅	—	4	1	no	—
50	C₄²×C₅	—	4	4	no	—
51	C₂×#168	—	4	4	no	—
52	C₂×#168	—	4	4	no	—
53	C₂×#168	—	4	4	no	—
54	C₂×#125	Dic₅	1	1	yes	characteristic
55	#389	C₂×D₅	1	2	yes	—
56	#387	—	2	2	no	—
57	#382	Dic₅	1	1	yes	characteristic
58	C₄×C₄⋊C₄	—	2	2	no	—
59	C₄×C₄⋊C₄	Dic₅	1	2	yes	—
60	C₄³	C₂×D₅	1	1	yes	characteristic
61	#336	—	5	4	no	—
62	#336	—	10	2	no	—
63	#336	—	10	2	no	—
64	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅	—	2	2	no	—
66	C₂×C₄×C₅	—	2	1	no	—
67	C₂×C₄×C₅	—	2	1	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	—	2	2	no	—
71	C₂×C₄×C₅	—	2	4	no	—
72	C₂×C₄×C₅	—	2	1	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	4	2	no	—
77	C₂×C₄×C₅	—	4	2	no	—
78	C₂×C₄×C₅	—	4	2	no	—
79	C₂×C₄×C₅	—	4	2	no	—
80	C₂×C₄×C₅	—	4	2	no	—
81	C₂×C₄×C₅	#101	1	2	yes	—
82	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₅	#135	1	4	yes	—
84	#168	—	8	4	no	—
85	C₂²×C₈	—	10	4	no	—
86	#125	—	2	4	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	2	2	no	—
89	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
90	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
91	C₂×C₄²	—	2	2	no	—
92	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
93	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
95	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
96	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
97	#95	—	2	2	no	—
98	#95	—	2	2	no	—
99	#95	—	2	2	no	—
100	C₂²×C₅	#407	1	1	yes	characteristic
101	C₂²×C₅	#410	1	1	yes	characteristic
102	C₂²×C₅	C₂×#101	1	1	yes	characteristic
103	C₂²×C₅	#416	1	2	yes	—
104	C₂²×C₅	#421	1	2	yes	—
105	C₄×C₅	—	2	2	no	—
106	C₄×C₅	—	2	4	no	—
107	C₄×C₅	—	2	2	no	—
108	C₄×C₅	—	4	1	no	—
109	C₄×C₅	—	4	2	no	—
110	C₄×C₅	—	4	4	no	—
111	C₄×C₅	—	8	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
116	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
117	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
118	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
119	C₂×C₈	—	20	4	no	—
120	C₂×C₈	—	20	4	no	—
121	C₂×C₈	—	20	4	no	—
122	C₄⋊C₄	—	2	8	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	4	no	—
126	C₄⋊C₄	#681	1	4	yes	—
127	C₄²	—	2	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	2	no	—
130	C₄²	—	4	2	no	—
131	C₄²	—	4	1	no	—
132	C₄²	—	4	4	no	—
133	C₂×C₅	#1467	1	1	yes	characteristic
134	C₂×C₅	#1488	1	2	yes	—
135	C₂×C₅	#1521	1	2	yes	—
136	C₂×C₅	#1534	1	1	yes	characteristic
137	C₂×C₅	#1544	1	1	yes	characteristic
138	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	#3968	1	2	yes	—
156	C₂×C₄	C₂×#681	1	2	yes	—
157	C₂×C₄	#4096	1	4	yes	—
158	C₈	—	40	4	no	—
159	C₅	#7650	1	1	yes	characteristic, complemented, Sylow
160	C₂²	#65496	1	1	yes	characteristic
161	C₂²	C₂×#3968	1	1	yes	characteristic
162	C₂²	#65522	1	2	yes	—
163	C₂²	#65525	1	2	yes	—
164	C₂²	#65529	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	4	no	—
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	4	no	—
171	C₄	—	8	2	no	—
172	C₂	#10602418	1	1	yes	characteristic
173	C₂	#10602433	1	2	yes	—
174	C₂	#10602529	1	1	yes	characteristic
175	C₂	#10602550	1	2	yes	—
176	C₂	#10602622	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅×#125

C₂

yes

maximal, characteristic

#10600186

C₂

yes

maximal

#10598129

C₂

yes

maximal

C₂×C₅×#125

C₄

yes

characteristic

C₅×#389

C₂²

yes

—

C₅×#387

—

C₅×#382

C₄

yes

characteristic

C₄×C₅×C₄⋊C₄

—

C₄×C₅×C₄⋊C₄

C₄

yes

—

C₄³×C₅

C₂²

yes

—

—

C₂²

yes

—

#7650

—

maximal, Sylow

C₅×#125

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

C₂×C₄

yes

—

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

C₂²×#168

—

C₅×#95

—

C₅×#95

—

C₅×#95

—

C₄×#125

D₅

yes

characteristic

#1839

—

#1271

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

M₁₆

yes

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

—

—

—

Dic₅

yes

characteristic

#389

C₂×D₅

yes

—

#387

—

#382

Dic₅

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₅

yes

—

C₄³

C₂×D₅

yes

—

—

—

C₂³×C₅

C₂×C₂²⋊C₄

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

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#101

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

#135

yes

—

#168

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₅

yes

—

C₂×C₄⋊C₄

C₂×Dic₅

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₅

yes

characteristic

C₂×C₄²

C₂×Dic₅

yes

characteristic

C₂×C₄²

C₂²⋊D₅

yes

—

C₂×C₄²

C₂²⋊D₅

yes

—

C₂×C₄²

C₂²×D₅

yes

—

—

—

100

C₂²×C₅

#407

yes

characteristic

101

C₂²×C₅

#410

yes

characteristic

102

C₂²×C₅

C₂×#101

yes

characteristic

103

C₂²×C₅

#416

yes

—

104

C₂²×C₅

#421

yes

—

105

C₄×C₅

—

106

C₄×C₅

—

107

C₄×C₅

—

108

C₄×C₅

—

109

C₄×C₅

—

110

C₄×C₅

—

111

C₄×C₅

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²⋊Dic₅

yes

characteristic

115

C₂²×C₄

C₂²⋊Dic₅

yes

—

116

C₂²×C₄

C₂²⋊Dic₅

yes

characteristic

117

C₂²×C₄

C₂²×Dic₅

yes

characteristic

118

C₂²×C₄

C₂×C₂²⋊D₅

yes

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

#681

yes

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₂×C₅

#1467

yes

characteristic

134

C₂×C₅

#1488

yes

—

135

C₂×C₅

#1521

yes

—

136

C₂×C₅

#1534

yes

characteristic

137

C₂×C₅

#1544

yes

characteristic

138

C₂³

C₂×C₂²⋊Dic₅

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

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

—

155

C₂×C₄

#3968

yes

—

156

C₂×C₄

C₂×#681

yes

—

157

C₂×C₄

#4096

yes

—

158

C₈

—

159

C₅

#7650

yes

characteristic, complemented, Sylow

160

C₂²

#65496

yes

characteristic

161

C₂²

C₂×#3968

yes

characteristic

162

C₂²

#65522

yes

—

163

C₂²

#65525

yes

—

164

C₂²

#65529

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₂

#10602418

yes

characteristic

173

C₂

#10602433

yes

—

174

C₂

#10602529

yes

characteristic

175

C₂

#10602550

yes

—

176

C₂

#10602622

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×#125	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×#125	D₅
hypocenter	C₅	#7650
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₄×C₅×#125

C₂

socle, prime radical

C₂³×C₅

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
159. C₅	#7650	semidirect

H ≅ G/N

type

159. C₅

#7650

semidirect

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

subgroup	index
C₄×C₅×#125	2
#10600186	2
#10598129	2
#7650	5

subgroup

index

C₄×C₅×#125

#10600186

#10598129

#7650

Subgroups — #11987618

Conjugacy classes of subgroups

Complements

Maximal subgroups