Group #11986201 — subgroups

Subgroups — #11986201

Order 1280.

216 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 103 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×#1451	C₂	1	1	yes	maximal, characteristic
2	#10600537	C₂	1	1	yes	maximal, characteristic
3	C₄×#3944	C₂	1	1	yes	maximal, characteristic
4	C₅×#431	C₄	1	1	yes	characteristic
5	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#3944	—	2	1	no	—
8	C₅×#322	—	2	1	no	—
9	C₅×#322	—	2	1	no	—
10	C₂²⋊#672	—	2	1	no	—
11	C₄⋊#672	C₄	1	2	yes	—
12	#64681	C₄	1	2	yes	—
13	C₄×#672	—	2	1	no	—
14	C₄×#672	—	2	1	no	—
15	#7166	—	5	1	no	maximal, Sylow
16	C₂×C₅×M₁₆	C₈	1	2	yes	—
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₅×C₄⋊C₈	C₈	1	2	yes	—
21	C₅×C₂²⋊C₈	—	2	1	no	—
22	C₅×#97	—	2	1	no	—
23	C₄×C₅×C₈	—	2	1	no	—
24	C₄×C₅×C₈	—	2	1	no	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×C₈	D₄	1	1	yes	characteristic
29	#3944	—	2	4	no	—
30	C₂×#672	—	2	2	no	—
31	C₂×#672	—	2	2	no	—
32	#2162	—	5	1	no	—
33	C₄×M₃₂	—	5	1	no	—
34	#1451	D₅	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
38	C₅×M₁₆	—	4	2	no	—
39	C₂×C₅×C₈	—	2	2	no	—
40	C₂×C₅×C₈	—	2	2	no	—
41	C₂×C₅×C₈	—	2	2	no	—
42	C₂×C₅×C₈	—	2	2	no	—
43	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅	C₄²	1	1	yes	characteristic
48	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₅	M₁₆	1	1	yes	characteristic
51	#672	—	2	4	no	—
52	#672	—	4	2	no	—
53	C₂×M₃₂	—	10	1	no	—
54	#431	Dic₅	1	1	yes	characteristic
55	C₄×M₁₆	Dic₅	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
57	C₄⋊C₁₆	—	5	2	no	—
58	C₂²⋊C₁₆	—	10	1	no	—
59	#346	—	5	2	no	—
60	C₄×C₁₆	—	10	1	no	—
61	C₄×C₁₆	—	10	1	no	—
62	#322	—	2	1	no	—
63	#322	—	2	1	no	—
64	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₅	—	2	1	no	—
66	C₂×C₄×C₅	—	2	1	no	—
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	#95	1	1	yes	characteristic
70	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
71	C₂×C₄×C₅	#97	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
75	C₅×C₈	—	2	1	no	—
76	C₅×C₈	—	2	1	no	—
77	C₅×C₈	—	4	2	no	—
78	C₅×C₈	—	4	2	no	—
79	C₅×C₈	#104	1	2	yes	—
80	C₅×C₈	#108	1	2	yes	—
81	C₂×M₁₆	#168	1	2	yes	—
82	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
83	C₂²×C₈	D₂₀	1	1	yes	characteristic
84	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
85	M₃₂	—	10	4	no	—
86	C₂×C₁₆	—	10	2	no	—
87	C₂×C₁₆	—	10	2	no	—
88	C₄⋊C₈	#168	1	2	yes	—
89	C₂²⋊C₈	—	2	1	no	—
90	#97	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	C₄×D₅	1	1	yes	characteristic
94	C₄×C₈	C₄×D₅	1	1	yes	characteristic
95	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
96	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
97	C₂²×C₅	—	2	1	no	—
98	C₂²×C₅	—	2	1	no	—
99	C₂²×C₅	—	2	1	no	—
100	C₂²×C₅	#336	1	1	yes	characteristic
101	C₄×C₅	—	2	1	no	—
102	C₄×C₅	—	2	1	no	—
103	C₄×C₅	—	2	1	no	—
104	C₄×C₅	#339	1	1	yes	characteristic
105	C₄×C₅	#341	1	1	yes	characteristic
106	C₄×C₅	#349	1	2	yes	—
107	C₄×C₅	#350	1	2	yes	—
108	C₂²×C₄	C₂×#168	1	1	yes	characteristic
109	C₂²×C₄	#681	1	1	yes	characteristic
110	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
111	M₁₆	—	4	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	#683	1	1	yes	characteristic
117	C₂×C₈	#683	1	1	yes	characteristic
118	C₂×C₈	#685	1	1	yes	characteristic
119	C₂×C₈	#685	1	1	yes	characteristic
120	C₄²	C₂×#168	1	1	yes	characteristic
121	C₄²	#681	1	1	yes	characteristic
122	C₄²	C₄×Dic₅	1	1	yes	characteristic
123	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
124	C₁₆	—	10	4	no	—
125	C₁₆	—	20	2	no	—
126	C₂×C₅	—	4	1	no	—
127	C₂×C₅	#1278	1	1	yes	characteristic
128	C₂×C₅	#1377	1	1	yes	characteristic
129	C₂×C₅	#1379	1	1	yes	characteristic
130	C₂³	C₄⋊#168	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₄×#168	1	1	yes	characteristic
136	C₂×C₄	#3935	1	1	yes	characteristic
137	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
140	C₂×C₄	#3963	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	4	2	no	—
144	C₈	—	4	2	no	—
145	C₈	#3954	1	2	yes	—
146	C₈	#3957	1	2	yes	—
147	C₅	#7166	1	1	yes	characteristic, complemented, Sylow
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#64744	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#64773	1	2	yes	—
156	C₄	#64775	1	1	yes	characteristic
157	C₄	#64778	1	1	yes	characteristic
158	C₄	#64781	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#10598155	1	1	yes	characteristic
161	C₂	#10598158	1	1	yes	characteristic
162	C₂	#10598162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1451

C₂

yes

maximal, characteristic

#10600537

C₂

yes

maximal, characteristic

C₄×#3944

C₂

yes

maximal, characteristic

C₅×#431

C₄

yes

characteristic

C₄×C₅×M₁₆

C₄

yes

characteristic

C₂×C₄×C₅×C₈

C₂²

yes

—

—

—

C₂²⋊#672

—

C₄⋊#672

C₄

yes

—

#64681

C₄

yes

—

—

—

—

C₂×C₅×M₁₆

C₈

yes

—

C₂²×C₅×C₈

D₄

yes

characteristic

C₂²×C₅×C₈

Q₈

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₅×C₄⋊C₈

C₈

yes

—

C₅×C₂²⋊C₈

—

C₅×#97

—

C₄×C₅×C₈

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

—

—

—

—

C₄×M₃₂

—

#1451

D₅

yes

characteristic

C₂²×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂×C₈

yes

characteristic

C₂²×C₄×C₅

M₁₆

yes

characteristic

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₄²×C₅

C₄²

yes

characteristic

C₄²×C₅

C₂²⋊C₄

yes

characteristic

C₄²×C₅

C₂×C₈

yes

characteristic

C₄²×C₅

M₁₆

yes

characteristic

#672

—

#672

—

C₂×M₃₂

—

#431

Dic₅

yes

characteristic

C₄×M₁₆

Dic₅

yes

characteristic

C₂×C₄×C₈

C₂×D₅

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

—

#322

—

C₂³×C₅

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#95

yes

characteristic

C₂×C₄×C₅

C₄×C₈

yes

characteristic

C₂×C₄×C₅

#97

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅

C₄⋊C₈

yes

characteristic

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

#104

yes

—

C₅×C₈

#108

yes

—

C₂×M₁₆

#168

yes

—

C₂²×C₈

Dic₁₀

yes

characteristic

C₂²×C₈

D₂₀

yes

characteristic

C₂×C₄²

C₂×Dic₅

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

#168

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅

yes

characteristic

C₄×C₈

C₄×D₅

yes

characteristic

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

100

C₂²×C₅

#336

yes

characteristic

101

C₄×C₅

—

102

C₄×C₅

—

103

C₄×C₅

—

104

C₄×C₅

#339

yes

characteristic

105

C₄×C₅

#341

yes

characteristic

106

C₄×C₅

#349

yes

—

107

C₄×C₅

#350

yes

—

108

C₂²×C₄

C₂×#168

yes

characteristic

109

C₂²×C₄

#681

yes

characteristic

110

C₂²×C₄

C₄⋊Dic₅

yes

characteristic

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

#683

yes

characteristic

117

C₂×C₈

#683

yes

characteristic

118

C₂×C₈

#685

yes

characteristic

119

C₂×C₈

#685

yes

characteristic

120

C₄²

C₂×#168

yes

characteristic

121

C₄²

#681

yes

characteristic

122

C₄²

C₄×Dic₅

yes

characteristic

123

C₄²

C₂²⋊Dic₅

yes

characteristic

124

C₁₆

—

125

C₁₆

—

126

C₂×C₅

—

127

C₂×C₅

#1278

yes

characteristic

128

C₂×C₅

#1377

yes

characteristic

129

C₂×C₅

#1379

yes

characteristic

130

C₂³

C₄⋊#168

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₄×#168

yes

characteristic

136

C₂×C₄

#3935

yes

characteristic

137

C₂×C₄

C₄⋊#168

yes

characteristic

138

C₂×C₄

C₂²⋊#168

yes

characteristic

139

C₂×C₄

C₂²⋊#168

yes

characteristic

140

C₂×C₄

#3963

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

#3954

yes

—

146

C₈

#3957

yes

—

147

C₅

#7166

yes

characteristic, complemented, Sylow

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#64744

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#64773

yes

—

156

C₄

#64775

yes

characteristic

157

C₄

#64778

yes

characteristic

158

C₄

#64781

yes

—

159

C₂

—

160

C₂

#10598155

yes

characteristic

161

C₂

#10598158

yes

characteristic

162

C₂

#10598162

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅
Fitting	C₅×#1451	C₂
socle	C₂²×C₅	#336
prime radical	C₂³×C₅	C₄⋊C₈
cosocle	C₂×C₄×C₅×C₈	C₂²
hypercenter	#1451	D₅
hypocenter	C₅	#7166
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1451

subgroup

G/H

center

C₄²

C₂²⋊Dic₅

commutator

C₂×C₄×C₅

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₅

C₂

C₂²×C₅

C₂³×C₅

C₄⋊C₈

cosocle

C₂×C₄×C₅×C₈

C₂²

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1451

N	H ≅ G/N	type
147. C₅	#7166	semidirect

H ≅ G/N

type

147. C₅

#7166

semidirect

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

subgroup	index
C₅×#1451	2
#10600537	2
C₄×#3944	2
#7166	5

subgroup

index

Subgroups — #11986201

Conjugacy classes of subgroups

Complements

Maximal subgroups