Group #12014306 — subgroups

Subgroups — #12014306

Order 1280.

358 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 195 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	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
2	#10600542	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic
4	#10598295	C₂	1	4	yes	maximal
5	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#3947	C₄	1	2	yes	—
8	C₂×C₈×Dic₅	C₂²	1	2	yes	—
9	C₂²×#672	C₂²	1	2	yes	—
10	#65221	C₄	1	1	yes	characteristic
11	C₂×#3935	C₄	1	1	yes	characteristic
12	C₅×#346	C₄	1	4	yes	complemented
13	C₄×#672	C₄	1	4	yes	complemented
14	#12592	—	5	1	no	maximal, Sylow
15	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
19	C₂²×#168	C₂×C₄	1	2	yes	—
20	C₂×C₅×C₁₆	—	2	4	no	—
21	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
24	#3963	C₈	1	4	yes	—
25	#3947	—	2	4	no	—
26	C₈×Dic₅	—	2	4	no	—
27	C₂×#672	—	2	4	no	—
28	C₂×#672	C₂×C₄	1	4	yes	—
29	#3935	C₈	1	4	yes	—
30	C₂×#346	D₅	1	1	yes	characteristic
31	C₂×C₄×C₁₆	—	5	1	no	—
32	#1726	—	5	1	no	—
33	#1374	—	5	4	no	—
34	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
35	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
36	C₂²×Dic₅	C₂×C₈	1	4	yes	—
37	C₂×C₅×C₈	C₄²	1	2	yes	—
38	C₂×C₅×C₈	M₁₆	1	4	yes	—
39	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₅×C₈	#41	1	4	yes	—
42	C₄²×C₅	C₄²	1	2	yes	—
43	C₄²×C₅	C₂×C₈	1	2	yes	—
44	C₄×Dic₅	—	2	2	no	—
45	C₄×Dic₅	—	2	2	no	—
46	C₂×#168	—	2	2	no	—
47	C₂×#168	—	2	2	no	—
48	C₂×#168	C₂×C₈	1	4	yes	—
49	C₅×C₁₆	—	2	4	no	—
50	#672	—	2	4	no	—
51	C₂²×C₁₆	—	5	2	no	—
52	C₂²×C₁₆	C₂×D₅	1	2	yes	—
53	C₂×#97	—	5	1	no	—
54	C₂×#97	—	5	2	no	—
55	C₂×C₄×C₈	—	5	2	no	—
56	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
57	#376	—	5	1	no	—
58	#346	Dic₅	1	4	yes	complemented
59	C₄×C₁₆	—	5	4	no	—
60	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅	—	2	4	no	—
62	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
63	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₅	#117	1	2	yes	—
65	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
67	C₂×Dic₅	—	2	4	no	—
68	C₂×Dic₅	—	2	8	no	—
69	C₅×C₈	#97	1	4	yes	—
70	C₅×C₈	#117	1	2	yes	—
71	C₅×C₈	#117	1	2	yes	—
72	#168	—	2	4	no	—
73	C₂²×C₈	—	5	2	no	—
74	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
75	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
76	C₂×C₄²	—	5	2	no	—
77	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
78	C₂×C₁₆	—	2	4	no	—
79	C₂×C₁₆	—	5	4	no	—
80	C₂×C₁₆	—	10	4	no	—
81	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
82	#97	—	5	4	no	—
83	#97	—	10	4	no	—
84	C₄×C₈	—	10	4	no	—
85	C₄×C₈	C₄×D₅	1	2	yes	—
86	C₄×C₈	C₂×Dic₅	1	2	yes	—
87	#95	—	5	4	no	—
88	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
89	C₂²×C₅	#431	1	2	yes	—
90	C₂²×C₅	#504	1	4	yes	—
91	C₄×C₅	—	2	4	no	—
92	C₄×C₅	#376	1	1	yes	characteristic
93	C₄×C₅	C₂×#97	1	1	yes	characteristic
94	C₄×C₅	#431	1	2	yes	—
95	Dic₅	—	4	4	no	—
96	C₂²×C₄	—	5	4	no	—
97	C₂²×C₄	C₈×D₅	1	2	yes	—
98	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
99	C₂×C₈	—	5	4	no	—
100	C₂×C₈	—	10	2	no	—
101	C₂×C₈	—	10	2	no	—
102	C₂×C₈	#676	1	4	yes	—
103	C₂×C₈	C₄×Dic₅	1	2	yes	—
104	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
105	C₂×C₈	#709	1	4	yes	—
106	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
107	C₄²	—	10	2	no	—
108	C₄²	—	10	2	no	—
109	C₄²	C₈×D₅	1	2	yes	—
110	C₄²	C₄×Dic₅	1	2	yes	—
111	C₁₆	—	2	4	no	—
112	C₁₆	—	10	4	no	—
113	C₂×C₅	#1744	1	1	yes	characteristic
114	C₂×C₅	#2108	1	2	yes	—
115	C₂×C₅	#2164	1	4	yes	—
116	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	10	8	no	—
119	C₂×C₄	—	10	4	no	—
120	C₂×C₄	C₈×Dic₅	1	4	yes	—
121	C₂×C₄	#4018	1	2	yes	—
122	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
123	C₂×C₄	C₂×#676	1	2	yes	—
124	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
125	C₈	—	10	4	no	—
126	C₈	#3947	1	4	yes	—
127	C₈	#4018	1	2	yes	—
128	C₈	#4072	1	2	yes	—
129	C₅	#12592	1	1	yes	characteristic, complemented, Sylow
130	C₂²	#64974	1	2	yes	—
131	C₂²	#65196	1	4	yes	—
132	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
133	C₄	—	2	4	no	—
134	C₄	—	20	4	no	—
135	C₄	#64974	1	2	yes	—
136	C₄	#65221	1	1	yes	characteristic
137	C₄	C₂×#3947	1	1	yes	characteristic
138	C₂	#10599867	1	4	yes	—
139	C₂	#10600543	1	1	yes	characteristic
140	C₂	#10602212	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#346

C₂

yes

maximal, characteristic

#10600542

C₂

yes

maximal, characteristic

C₂×C₄×#672

C₂

yes

maximal, characteristic

#10598295

C₂

yes

maximal

C₂²×C₅×C₁₆

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3947

C₄

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

C₂²×#672

C₂²

yes

—

#65221

C₄

yes

characteristic

C₂×#3935

C₄

yes

characteristic

C₅×#346

C₄

yes

complemented

C₄×#672

C₄

yes

complemented

#12592

—

maximal, Sylow

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

C₂×C₄

yes

—

C₂²×#168

C₂×C₄

yes

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

#3963

C₈

yes

—

#3947

—

C₈×Dic₅

—

C₂×#672

—

C₂×#672

C₂×C₄

yes

—

#3935

C₈

yes

—

C₂×#346

D₅

yes

characteristic

C₂×C₄×C₁₆

—

#1726

—

#1374

—

C₂²×C₄×C₅

C₂×C₈

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

C₂×C₈

yes

—

C₂×C₅×C₈

C₄²

yes

—

C₂×C₅×C₈

M₁₆

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

#41

yes

—

C₄²×C₅

C₄²

yes

—

C₄²×C₅

C₂×C₈

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

—

C₂×#168

—

C₂×#168

C₂×C₈

yes

—

C₅×C₁₆

—

#672

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#376

—

#346

Dic₅

yes

complemented

C₄×C₁₆

—

C₂³×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×C₈

yes

—

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

#97

yes

—

C₅×C₈

#117

yes

—

C₅×C₈

#117

yes

—

#168

—

C₂²×C₈

—

C₂²×C₈

C₄×D₅

yes

characteristic

C₂²×C₈

C₂²×D₅

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₅

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₅

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₄×D₅

yes

—

C₄×C₈

C₂×Dic₅

yes

—

#95

—

C₂²×C₅

C₂×C₄×C₈

yes

characteristic

C₂²×C₅

#431

yes

—

C₂²×C₅

#504

yes

—

C₄×C₅

—

C₄×C₅

#376

yes

characteristic

C₄×C₅

C₂×#97

yes

characteristic

C₄×C₅

#431

yes

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

C₈×D₅

yes

—

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

#676

yes

—

103

C₂×C₈

C₄×Dic₅

yes

—

104

C₂×C₈

C₂×C₄×D₅

yes

characteristic

105

C₂×C₈

#709

yes

—

106

C₂×C₈

C₂²×Dic₅

yes

characteristic

107

C₄²

—

108

C₄²

—

109

C₄²

C₈×D₅

yes

—

110

C₄²

C₄×Dic₅

yes

—

111

C₁₆

—

112

C₁₆

—

113

C₂×C₅

#1744

yes

characteristic

114

C₂×C₅

#2108

yes

—

115

C₂×C₅

#2164

yes

—

116

C₂³

C₂×C₈×D₅

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₈×Dic₅

yes

—

121

C₂×C₄

#4018

yes

—

122

C₂×C₄

C₂×C₈×D₅

yes

characteristic

123

C₂×C₄

C₂×#676

yes

—

124

C₂×C₄

C₂×C₄×Dic₅

yes

characteristic

125

C₈

—

126

C₈

#3947

yes

—

127

C₈

#4018

yes

—

128

C₈

#4072

yes

—

129

C₅

#12592

yes

characteristic, complemented, Sylow

130

C₂²

#64974

yes

—

131

C₂²

#65196

yes

—

132

C₂²

C₂×C₈×Dic₅

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

#64974

yes

—

136

C₄

#65221

yes

characteristic

137

C₄

C₂×#3947

yes

characteristic

138

C₂

#10599867

yes

—

139

C₂

#10600543

yes

characteristic

140

C₂

#10602212

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₂×C₅×#346	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#346	D₅
hypocenter	C₅	#12592
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₅

commutator

C₂²×C₅

C₂×C₄×C₈

Fitting

C₂×C₅×#346

C₂

socle, prime radical

C₂³×C₅

C₂²×C₈

cosocle

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
12. C₅×#346	C₄	semidirect
13. C₄×#672	C₄	semidirect
58. #346	Dic₅	semidirect
129. C₅	#12592	semidirect

H ≅ G/N

type

12. C₅×#346

C₄

semidirect

13. C₄×#672

C₄

semidirect

58. #346

Dic₅

semidirect

129. C₅

#12592

semidirect

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

subgroup	index
C₂×C₅×#346	2
#10600542	2
C₂×C₄×#672	2
#10598295	2
#12592	5

subgroup

index

C₂×C₅×#346

#10600542

C₂×C₄×#672

#10598295

#12592

Subgroups — #12014306

Conjugacy classes of subgroups

Complements

Maximal subgroups