Group #11989752 — subgroups

Subgroups — #11989752

Order 1280.

330 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 131 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	#10601535	C₂	1	2	yes	maximal
2	#10600597	C₂	1	1	yes	maximal, characteristic
3	C₅×#1275	C₂	1	1	yes	maximal, characteristic
4	#10598228	C₂	1	2	yes	maximal
5	#10598151	C₂	1	1	yes	maximal, characteristic
6	C₅×#431	C₂²	1	2	yes	—
7	C₂×C₅×#97	C₂²	1	1	yes	characteristic
8	C₂×#3947	C₄	1	2	yes	—
9	C₂×C₈×Dic₅	C₄	1	2	yes	—
10	#65221	C₂²	1	1	yes	characteristic
11	#65214	C₂²	1	2	yes	—
12	C₂×C₄×#168	C₂²	1	1	yes	characteristic
13	#64675	C₄	1	4	yes	complemented
14	#64674	C₄	1	4	yes	complemented
15	#8140	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
18	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
19	C₂²×#168	C₂×C₄	1	1	yes	characteristic
20	C₂²×#168	C₂×C₄	1	1	yes	characteristic
21	C₅×C₄⋊C₈	—	2	2	no	—
22	C₅×C₂²⋊C₈	—	2	2	no	—
23	C₅×#97	—	2	2	no	—
24	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
25	#3963	—	2	1	no	—
26	#3963	—	2	1	no	—
27	C₂²⋊#168	—	2	2	no	—
28	#3947	—	2	4	no	—
29	C₈×Dic₅	—	2	4	no	—
30	C₄⋊#168	—	2	2	no	—
31	C₄×#168	C₂×C₄	1	4	yes	—
32	C₄×#168	C₂×C₄	1	2	yes	—
33	C₄×#168	C₂×C₄	1	2	yes	—
34	#1726	—	5	1	no	—
35	#1452	—	5	2	no	—
36	#1275	—	5	1	no	—
37	#1275	D₅	1	1	yes	characteristic
38	#1274	—	5	2	no	—
39	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	#41	1	1	yes	characteristic
41	C₂²×C₄×C₅	#41	1	1	yes	characteristic
42	C₂²×Dic₅	—	2	2	no	—
43	C₂×C₅×C₈	—	2	2	no	—
44	C₂×C₅×C₈	—	2	2	no	—
45	C₂×C₅×C₈	—	2	4	no	—
46	C₂×C₅×C₈	M₁₆	1	4	yes	—
47	C₄²×C₅	C₂²×C₄	1	2	yes	—
48	C₄²×C₅	#41	1	2	yes	—
49	C₄×Dic₅	—	2	2	no	—
50	C₄×Dic₅	M₁₆	1	4	yes	—
51	C₂×#168	—	2	2	no	—
52	C₂×#168	—	2	2	no	—
53	C₂×#168	—	2	4	no	—
54	C₂×#168	C₄²	1	1	yes	characteristic
55	C₂×#168	C₄²	1	2	yes	—
56	C₂×#168	C₄²	1	1	yes	characteristic
57	#431	—	5	2	no	—
58	#431	C₂×D₅	1	2	yes	—
59	C₂×#97	—	5	2	no	—
60	C₂×#97	C₂×D₅	1	1	yes	characteristic
61	C₂×C₄×C₈	—	5	1	no	—
62	C₂×C₄×C₈	—	5	2	no	—
63	#376	—	5	1	no	—
64	#322	—	5	4	no	—
65	#322	—	5	4	no	—
66	C₂³×C₅	#117	1	1	yes	characteristic
67	C₂×C₄×C₅	—	4	1	no	—
68	C₂×C₄×C₅	—	4	1	no	—
69	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₅	#117	1	2	yes	—
71	C₂×C₄×C₅	#117	1	1	yes	characteristic
72	C₂×C₄×C₅	#117	1	2	yes	—
73	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
75	C₂×Dic₅	—	2	4	no	—
76	C₂×Dic₅	—	4	2	no	—
77	C₂×Dic₅	—	4	2	no	—
78	C₅×C₈	—	2	4	no	—
79	C₅×C₈	—	4	4	no	—
80	#168	—	2	1	no	—
81	#168	—	2	1	no	—
82	#168	—	2	2	no	—
83	#168	—	4	4	no	—
84	C₂²×C₈	—	5	1	no	—
85	C₂²×C₈	—	5	1	no	—
86	C₂²×C₈	C₄×D₅	1	2	yes	—
87	C₂×C₄²	—	5	2	no	—
88	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	—	10	2	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	C₂²⋊C₈	—	10	2	no	—
93	#97	—	2	2	no	—
94	#97	—	10	4	no	—
95	C₄×C₈	—	5	2	no	—
96	C₄×C₈	—	5	2	no	—
97	C₄×C₈	—	5	4	no	—
98	C₄×C₈	—	10	4	no	—
99	C₄×C₈	C₄×D₅	1	4	yes	—
100	#95	—	10	1	no	—
101	#95	—	10	1	no	—
102	C₂²×C₅	—	2	1	no	—
103	C₂²×C₅	—	2	1	no	—
104	C₂²×C₅	#376	1	1	yes	characteristic
105	C₂²×C₅	#432	1	2	yes	—
106	C₄×C₅	—	2	2	no	—
107	C₄×C₅	—	2	2	no	—
108	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
109	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
110	C₄×C₅	#432	1	2	yes	—
111	Dic₅	—	4	4	no	—
112	C₂²×C₄	—	10	2	no	—
113	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
114	C₂²×C₄	#711	1	1	yes	characteristic
115	C₂²×C₄	#713	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	5	1	no	—
120	C₂×C₈	—	5	1	no	—
121	C₂×C₈	—	5	2	no	—
122	C₂×C₈	—	10	2	no	—
123	C₂×C₈	—	10	2	no	—
124	C₂×C₈	—	10	4	no	—
125	C₂×C₈	#676	1	4	yes	—
126	C₄²	—	5	4	no	—
127	C₄²	—	10	2	no	—
128	C₄²	C₂×C₄×D₅	1	2	yes	—
129	C₄²	#711	1	2	yes	—
130	C₂×C₅	—	2	2	no	—
131	C₂×C₅	#1745	1	1	yes	characteristic
132	C₂×C₅	#1747	1	1	yes	characteristic
133	C₂×C₅	#1747	1	1	yes	characteristic
134	C₂³	#4038	1	1	yes	characteristic
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	10	4	no	—
138	C₂×C₄	—	20	2	no	—
139	C₂×C₄	—	20	2	no	—
140	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
141	C₂×C₄	#4023	1	2	yes	—
142	C₂×C₄	#4023	1	1	yes	characteristic
143	C₂×C₄	#4038	1	2	yes	—
144	C₂×C₄	C₂×#676	1	2	yes	—
145	C₂×C₄	D₅×M₁₆	1	2	yes	—
146	C₈	—	2	4	no	—
147	C₈	—	4	4	no	—
148	C₈	—	10	1	no	—
149	C₈	—	10	1	no	—
150	C₈	—	10	2	no	—
151	C₈	—	20	4	no	—
152	C₅	#8140	1	1	yes	characteristic, complemented, Sylow
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#64937	1	1	yes	characteristic
156	C₂²	#65124	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	20	4	no	—
160	C₄	C₄×#676	1	1	yes	characteristic
161	C₄	#64996	1	1	yes	characteristic
162	C₄	#65008	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#10599390	1	1	yes	characteristic
165	C₂	#10599490	1	1	yes	characteristic
166	C₂	#10599510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10601535

C₂

yes

maximal

#10600597

C₂

yes

maximal, characteristic

C₅×#1275

C₂

yes

maximal, characteristic

#10598228

C₂

yes

maximal

#10598151

C₂

yes

maximal, characteristic

C₅×#431

C₂²

yes

—

C₂×C₅×#97

C₂²

yes

characteristic

C₂×#3947

C₄

yes

—

C₂×C₈×Dic₅

C₄

yes

—

#65221

C₂²

yes

characteristic

#65214

C₂²

yes

—

C₂×C₄×#168

C₂²

yes

characteristic

#64675

C₄

yes

complemented

#64674

C₄

yes

complemented

#8140

—

maximal, Sylow

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄×Dic₅

C₂×C₄

yes

—

C₂²×#168

C₂×C₄

yes

characteristic

C₂²×#168

C₂×C₄

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₅×#97

—

C₄×C₅×C₈

C₂×C₄

yes

—

#3963

—

#3963

—

C₂²⋊#168

—

#3947

—

C₈×Dic₅

—

C₄⋊#168

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

—

—

—

D₅

yes

characteristic

#1274

—

C₂²×C₄×C₅

C₂²×C₄

yes

C₂²×C₄×C₅

yes

C₂²×C₄×C₅

yes

C₂²×Dic₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

M₁₆

yes

—

C₄²×C₅

C₂²×C₄

yes

—

C₄²×C₅

#41

yes

—

C₄×Dic₅

—

C₄×Dic₅

M₁₆

yes

—

—

—

—

C₄²

yes

characteristic

C₂×#168

C₄²

yes

—

C₂×#168

C₄²

yes

characteristic

#431

—

#431

C₂×D₅

yes

—

C₂×#97

—

C₂×#97

C₂×D₅

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

#322

—

#322

—

C₂³×C₅

#117

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×C₄×C₅

#117

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

C₅×C₈

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₅

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅

yes

—

100

#95

—

101

#95

—

102

C₂²×C₅

—

103

C₂²×C₅

—

104

C₂²×C₅

#376

yes

characteristic

105

C₂²×C₅

#432

yes

—

106

C₄×C₅

—

107

C₄×C₅

—

108

C₄×C₅

C₄×M₁₆

yes

characteristic

109

C₄×C₅

C₄×M₁₆

yes

characteristic

110

C₄×C₅

#432

yes

—

111

Dic₅

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

114

C₂²×C₄

#711

yes

characteristic

115

C₂²×C₄

#713

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

#676

yes

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂×C₄×D₅

yes

—

129

C₄²

#711

yes

—

130

C₂×C₅

—

131

C₂×C₅

#1745

yes

characteristic

132

C₂×C₅

#1747

yes

characteristic

133

C₂×C₅

#1747

yes

characteristic

134

C₂³

#4038

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₄²×D₅

yes

characteristic

141

C₂×C₄

#4023

yes

—

142

C₂×C₄

#4023

yes

characteristic

143

C₂×C₄

#4038

yes

—

144

C₂×C₄

C₂×#676

yes

—

145

C₂×C₄

D₅×M₁₆

yes

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₅

#8140

yes

characteristic, complemented, Sylow

153

C₂²

—

154

C₂²

—

155

C₂²

#64937

yes

characteristic

156

C₂²

#65124

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

C₄×#676

yes

characteristic

161

C₄

#64996

yes

characteristic

162

C₄

#65008

yes

—

163

C₂

—

164

C₂

#10599390

yes

characteristic

165

C₂

#10599490

yes

characteristic

166

C₂

#10599510

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1275	C₂
socle	C₂²×C₅	#376
prime radical	C₂³×C₅	#117
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1275	D₅
hypocenter	C₅	#8140
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #1275

subgroup

G/H

center

C₂×C₄

C₄²×D₅

commutator

C₂×C₄×C₅

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₅

C₁ ◁ C₂×C₄ ◁ #1275

N	H ≅ G/N	type
13. #64675	C₄	semidirect
14. #64674	C₄	semidirect
152. C₅	#8140	semidirect

H ≅ G/N

type

13. #64675

C₄

semidirect

14. #64674

C₄

semidirect

152. C₅

#8140

semidirect

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

subgroup	index
#10601535	2
#10600597	2
C₅×#1275	2
#10598228	2
#10598151	2
#8140	5

subgroup

index

Subgroups — #11989752

Conjugacy classes of subgroups

Complements

Maximal subgroups