Group #11986321 — subgroups

Subgroups — #11986321

Order 1280.

212 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2162	C₂	1	1	yes	maximal, characteristic
2	#10600536	C₂	1	1	yes	maximal, characteristic
3	C₈×#681	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
5	C₂×#3944	C₄	1	2	yes	—
6	#65214	C₄	1	1	yes	characteristic
7	C₄×#681	C₄	1	1	yes	characteristic
8	C₅×C₄⋊C₁₆	—	2	1	no	—
9	C₅×C₂²⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₅×C₁₆	—	2	1	no	—
11	C₄⋊#672	—	2	1	no	—
12	C₄⋊#672	—	2	1	no	—
13	#64674	—	2	1	no	—
14	C₈×#168	—	2	1	no	—
15	#7212	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×#681	—	2	1	no	—
20	C₂×C₅×C₁₆	—	2	2	no	—
21	C₄×C₅×C₈	D₄	1	1	yes	characteristic
22	C₄×C₅×C₈	D₄	1	1	yes	characteristic
23	C₄×C₅×C₈	D₄	1	1	yes	characteristic
24	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
25	C₂²⋊#168	—	2	1	no	—
26	#3944	—	4	2	no	—
27	C₂×#672	—	2	2	no	—
28	C₄⋊#168	C₈	1	2	yes	—
29	#3935	C₈	1	2	yes	—
30	C₄×#168	—	2	1	no	—
31	C₄×#168	—	2	1	no	—
32	#2162	D₅	1	1	yes	characteristic
33	#2150	—	5	1	no	—
34	C₈×M₁₆	—	5	1	no	—
35	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
38	C₂×C₅×C₈	—	2	1	no	—
39	C₂×C₅×C₈	—	2	1	no	—
40	C₂×C₅×C₈	—	4	1	no	—
41	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
49	#681	—	2	4	no	—
50	C₂×#168	—	2	2	no	—
51	C₂×#168	—	2	2	no	—
52	C₅×C₁₆	—	4	2	no	—
53	#672	—	4	2	no	—
54	C₂×M₃₂	—	5	2	no	—
55	#431	—	5	1	no	—
56	C₄×M₁₆	—	5	1	no	—
57	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
58	C₄⋊C₁₆	—	2	1	no	—
59	C₄⋊C₁₆	—	10	1	no	—
60	C₄⋊C₁₆	—	10	1	no	—
61	C₂²⋊C₁₆	Dic₅	1	2	yes	—
62	C₄×C₁₆	—	2	1	no	—
63	#322	—	10	1	no	—
64	C₈²	—	10	1	no	—
65	C₂³×C₅	#97	1	1	yes	characteristic
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	#95	1	1	yes	characteristic
71	C₂×C₄×C₅	C₄×C₈	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	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
76	C₅×C₈	—	2	1	no	—
77	C₅×C₈	—	2	1	no	—
78	C₅×C₈	—	2	1	no	—
79	C₅×C₈	#100	1	1	yes	characteristic
80	C₅×C₈	#101	1	1	yes	characteristic
81	#168	—	2	4	no	—
82	#168	—	4	2	no	—
83	C₂×M₁₆	—	10	1	no	—
84	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
85	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
86	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
87	M₃₂	—	20	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	10	2	no	—
90	C₄⋊C₈	—	5	2	no	—
91	C₂²⋊C₈	—	10	1	no	—
92	#97	—	5	2	no	—
93	C₄×C₈	—	10	1	no	—
94	C₄×C₈	—	10	1	no	—
95	C₄×C₈	Dic₁₀	1	1	yes	characteristic
96	C₄×C₈	D₂₀	1	1	yes	characteristic
97	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
98	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
99	C₂²×C₅	—	2	1	no	—
100	C₂²×C₅	—	2	1	no	—
101	C₂²×C₅	—	2	1	no	—
102	C₂²×C₅	#336	1	1	yes	characteristic
103	C₄×C₅	—	2	1	no	—
104	C₄×C₅	—	2	1	no	—
105	C₄×C₅	—	2	1	no	—
106	C₄×C₅	#343	1	1	yes	characteristic
107	C₄×C₅	#344	1	1	yes	characteristic
108	C₄×C₅	#350	1	2	yes	—
109	C₄×C₅	#364	1	2	yes	—
110	C₂²×C₄	#676	1	1	yes	characteristic
111	C₂²×C₄	#676	1	1	yes	characteristic
112	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
113	M₁₆	—	10	4	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	10	2	no	—
118	C₂×C₈	—	10	2	no	—
119	C₂×C₈	#683	1	1	yes	characteristic
120	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
121	C₂×C₈	#685	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
123	C₄²	C₈×D₅	1	1	yes	characteristic
124	C₄²	C₈×D₅	1	1	yes	characteristic
125	C₄²	#683	1	1	yes	characteristic
126	C₄²	#685	1	1	yes	characteristic
127	C₁₆	—	4	2	no	—
128	C₁₆	—	20	2	no	—
129	C₂×C₅	—	4	1	no	—
130	C₂×C₅	#1283	1	1	yes	characteristic
131	C₂×C₅	#1376	1	1	yes	characteristic
132	C₂×C₅	#1379	1	1	yes	characteristic
133	C₂³	#3947	1	1	yes	characteristic
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	C₈×Dic₅	1	1	yes	characteristic
139	C₂×C₄	#3946	1	1	yes	characteristic
140	C₂×C₄	#3946	1	1	yes	characteristic
141	C₂×C₄	#3952	1	1	yes	characteristic
142	C₂×C₄	#3952	1	1	yes	characteristic
143	C₂×C₄	#3963	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	10	4	no	—
148	C₈	—	20	2	no	—
149	C₈	#3955	1	1	yes	characteristic
150	C₈	#3956	1	1	yes	characteristic
151	C₅	#7212	1	1	yes	characteristic, complemented, Sylow
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#64770	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#64728	1	2	yes	—
160	C₄	#64731	1	2	yes	—
161	C₄	#64753	1	1	yes	characteristic
162	C₄	#64774	1	1	yes	characteristic
163	C₂	—	4	1	no	—
164	C₂	#10598235	1	1	yes	characteristic
165	C₂	#10598296	1	1	yes	characteristic
166	C₂	#10598301	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2162

C₂

yes

maximal, characteristic

#10600536

C₂

yes

maximal, characteristic

C₈×#681

C₂

yes

maximal, characteristic

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3944

C₄

yes

—

#65214

C₄

yes

characteristic

C₄×#681

C₄

yes

characteristic

C₅×C₄⋊C₁₆

—

C₅×C₂²⋊C₁₆

C₄

yes

—

C₄×C₅×C₁₆

—

—

—

—

—

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×#681

—

C₂×C₅×C₁₆

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

Q₈

yes

characteristic

C₂²⋊#168

—

#3944

—

C₂×#672

—

C₄⋊#168

C₈

yes

—

#3935

C₈

yes

—

C₄×#168

—

C₄×#168

—

#2162

D₅

yes

characteristic

#2150

—

C₈×M₁₆

—

C₂²×C₄×C₅

C₄²

yes

characteristic

C₂²×C₄×C₅

M₁₆

yes

characteristic

C₂²×C₄×C₅

M₁₆

yes

characteristic

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

C₂²⋊C₄

yes

characteristic

C₄²×C₅

C₄⋊C₄

yes

characteristic

C₄²×C₅

C₂×C₈

yes

characteristic

C₄²×C₅

C₂×C₈

yes

—

—

—

C₅×C₁₆

—

#672

—

C₂×M₃₂

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

Dic₅

yes

—

C₄×C₁₆

—

#322

—

C₈²

—

C₂³×C₅

#97

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

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅

C₄⋊C₈

yes

characteristic

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

yes

C₅×C₈

yes

—

—

C₂×M₁₆

—

C₂²×C₈

C₄×D₅

yes

characteristic

C₂²×C₈

C₂×Dic₅

yes

characteristic

C₂×C₄²

C₄×D₅

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₀

yes

characteristic

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₂²×C₅

—

100

C₂²×C₅

—

101

C₂²×C₅

—

102

C₂²×C₅

#336

yes

characteristic

103

C₄×C₅

—

104

C₄×C₅

—

105

C₄×C₅

—

106

C₄×C₅

#343

yes

characteristic

107

C₄×C₅

#344

yes

characteristic

108

C₄×C₅

#350

yes

—

109

C₄×C₅

#364

yes

—

110

C₂²×C₄

#676

yes

characteristic

111

C₂²×C₄

#676

yes

characteristic

112

C₂²×C₄

C₄×Dic₅

yes

characteristic

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

#683

yes

characteristic

120

C₂×C₈

C₄⋊Dic₅

yes

characteristic

121

C₂×C₈

#685

yes

characteristic

122

C₂×C₈

C₂²⋊Dic₅

yes

characteristic

123

C₄²

C₈×D₅

yes

characteristic

124

C₄²

C₈×D₅

yes

characteristic

125

C₄²

#683

yes

characteristic

126

C₄²

#685

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₂×C₅

—

130

C₂×C₅

#1283

yes

characteristic

131

C₂×C₅

#1376

yes

characteristic

132

C₂×C₅

#1379

yes

characteristic

133

C₂³

#3947

yes

characteristic

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₈×Dic₅

yes

characteristic

139

C₂×C₄

#3946

yes

characteristic

140

C₂×C₄

#3946

yes

characteristic

141

C₂×C₄

#3952

yes

characteristic

142

C₂×C₄

#3952

yes

characteristic

143

C₂×C₄

#3963

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

#3955

yes

characteristic

150

C₈

#3956

yes

characteristic

151

C₅

#7212

yes

characteristic, complemented, Sylow

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#64770

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#64728

yes

—

160

C₄

#64731

yes

—

161

C₄

#64753

yes

characteristic

162

C₄

#64774

yes

characteristic

163

C₂

—

164

C₂

#10598235

yes

characteristic

165

C₂

#10598296

yes

characteristic

166

C₂

#10598301

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#685
commutator	C₂×C₄×C₅	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅
Fitting	C₅×#2162	C₂
socle	C₂²×C₅	#336
prime radical	C₂³×C₅	#97
cosocle	C₂×C₄×C₅×C₈	C₂²
hypercenter	#2162	D₅
hypocenter	C₅	#7212
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

subgroup

G/H

center

C₂×C₈

#685

commutator

C₂×C₄×C₅

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄×C₅×C₈

C₂²

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

N	H ≅ G/N	type
151. C₅	#7212	semidirect

H ≅ G/N

type

151. C₅

#7212

semidirect

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

subgroup	index
C₅×#2162	2
#10600536	2
C₈×#681	2
#7212	5

subgroup

index

Subgroups — #11986321

Conjugacy classes of subgroups

Complements

Maximal subgroups