Group #11989720 — subgroups

Subgroups — #11989720

Order 1280.

520 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 151 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	#10610769	C₂	1	3	yes	maximal
2	C₅×#1309	C₂	1	1	yes	maximal, characteristic, complemented
3	#10598203	C₂	1	3	yes	maximal, complemented
4	C₂×C₄²×D₅	C₂²	1	1	yes	characteristic
5	C₅×#423	C₂²	1	3	yes	—
6	#65415	C₄	1	6	yes	—
7	M₁₆×Dic₅	C₄	1	6	yes	—
8	#65213	C₂²	1	3	yes	—
9	#65124	—	2	6	no	—
10	#65122	—	2	6	no	—
11	#8130	—	5	1	no	maximal, Sylow
12	C₂²×C₄×D₅	C₂×C₄	1	3	yes	—
13	C₂×C₅×M₁₆	C₂×C₄	1	6	yes	—
14	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
15	C₂×C₄×Dic₅	C₂×C₄	1	3	yes	—
16	C₂×#681	C₂×C₄	1	6	yes	—
17	C₄²×D₅	D₄	1	6	yes	—
18	C₄²×D₅	Q₈	1	2	yes	—
19	C₅×C₄⋊C₈	—	2	6	no	—
20	#3952	—	2	12	no	—
21	#3947	—	2	6	no	—
22	C₈×Dic₅	—	2	6	no	—
23	C₄⋊#168	—	2	6	no	—
24	#2920	—	5	3	no	—
25	#1309	—	5	3	no	—
26	#1309	D₅	1	1	yes	characteristic, complemented
27	C₂³×D₅	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
29	C₂²×Dic₅	C₄²	1	3	yes	—
30	C₂×C₄×D₅	—	4	3	no	—
31	C₂×C₄×D₅	—	4	3	no	—
32	C₂×C₄×D₅	C₂²⋊C₄	1	6	yes	—
33	C₂×C₄×D₅	C₄⋊C₄	1	6	yes	—
34	C₅×M₁₆	—	4	6	no	—
35	C₂×C₅×C₈	—	2	6	no	—
36	C₄²×C₅	C₂×D₄	1	3	yes	—
37	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
38	C₄×Dic₅	C₂²⋊C₄	1	3	yes	—
39	C₄×Dic₅	C₂²⋊C₄	1	3	yes	—
40	C₄×Dic₅	C₄⋊C₄	1	6	yes	—
41	#681	—	4	6	no	—
42	C₂×#168	—	2	6	no	—
43	C₂²×C₄²	—	5	1	no	—
44	#432	—	10	6	no	—
45	#431	—	10	6	no	—
46	#423	—	5	3	no	—
47	#423	C₂×D₅	1	3	yes	—
48	#407	—	5	6	no	—
49	C₄×M₁₆	—	5	6	no	—
50	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
51	C₂²×D₅	—	4	3	no	—
52	C₂²×D₅	#95	1	2	yes	—
53	C₂×C₄×C₅	—	2	3	no	—
54	C₂×C₄×C₅	—	2	3	no	—
55	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
56	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
57	C₂×Dic₅	—	4	3	no	—
58	C₂×Dic₅	#95	1	6	yes	—
59	C₄×D₅	—	2	6	no	—
60	C₄×D₅	—	2	6	no	—
61	C₄×D₅	—	4	6	no	—
62	C₅×C₈	—	4	6	no	—
63	#168	—	4	6	no	—
64	C₂³×C₄	—	5	3	no	—
65	C₂×M₁₆	—	5	6	no	—
66	C₂×M₁₆	C₄×D₅	1	6	yes	—
67	C₂×C₄²	—	5	3	no	—
68	C₂×C₄²	—	5	6	no	—
69	C₂×C₄²	—	5	2	no	—
70	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
71	C₄⋊C₈	—	2	6	no	—
72	C₄⋊C₈	—	10	6	no	—
73	C₂²⋊C₈	—	10	12	no	—
74	#97	—	10	6	no	—
75	C₄×C₈	—	10	6	no	—
76	C₂²×C₅	—	2	3	no	—
77	C₂²×C₅	C₂×#95	1	1	yes	characteristic
78	C₂×D₅	—	2	6	no	—
79	C₂×D₅	—	4	4	no	—
80	C₄×C₅	—	2	3	no	—
81	C₄×C₅	#413	1	3	yes	—
82	C₄×C₅	#413	1	3	yes	—
83	Dic₅	—	2	6	no	—
84	C₂⁴	—	5	1	no	—
85	C₂²×C₄	—	5	3	no	—
86	C₂²×C₄	—	5	6	no	—
87	C₂²×C₄	—	5	6	no	—
88	C₂²×C₄	—	20	3	no	—
89	C₂²×C₄	—	20	3	no	—
90	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
91	M₁₆	—	4	6	no	—
92	M₁₆	—	20	6	no	—
93	C₂×C₈	—	2	6	no	—
94	C₂×C₈	—	10	6	no	—
95	C₄²	—	5	3	no	—
96	C₄²	—	5	3	no	—
97	C₄²	—	5	6	no	—
98	C₄²	D₄×D₅	1	3	yes	—
99	C₄²	Q₈×D₅	1	1	yes	characteristic
100	C₂×C₅	—	4	1	no	—
101	C₂×C₅	#1742	1	3	yes	—
102	D₅	—	4	2	no	—
103	C₂³	—	5	2	no	—
104	C₂³	—	20	3	no	—
105	C₂³	C₄²×D₅	1	1	yes	characteristic
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	5	6	no	—
109	C₂×C₄	—	10	6	no	—
110	C₂×C₄	—	10	6	no	—
111	C₂×C₄	—	20	6	no	—
112	C₂×C₄	—	20	3	no	—
113	C₂×C₄	D₅×C₂²⋊C₄	1	3	yes	—
114	C₂×C₄	D₅×C₄⋊C₄	1	3	yes	—
115	C₈	—	4	6	no	—
116	C₈	—	20	6	no	—
117	C₅	#8130	1	1	yes	characteristic, complemented, Sylow
118	C₂²	—	2	3	no	—
119	C₂²	—	10	6	no	—
120	C₂²	—	20	4	no	—
121	C₂²	D₅×#95	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	10	6	no	—
124	C₄	#65034	1	3	yes	—
125	C₄	#65040	1	3	yes	—
126	C₂	—	4	1	no	—
127	C₂	—	20	2	no	—
128	C₂	#10599782	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10610769

C₂

yes

maximal

C₅×#1309

C₂

yes

maximal, characteristic, complemented

#10598203

C₂

yes

maximal, complemented

C₂×C₄²×D₅

C₂²

yes

characteristic

C₅×#423

C₂²

yes

—

#65415

C₄

yes

—

M₁₆×Dic₅

C₄

yes

—

#65213

C₂²

yes

—

—

—

—

C₂²×C₄×D₅

C₂×C₄

yes

—

C₂×C₅×M₁₆

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄×Dic₅

C₂×C₄

yes

—

C₂×#681

C₂×C₄

yes

—

C₄²×D₅

D₄

yes

—

C₄²×D₅

Q₈

yes

—

C₅×C₄⋊C₈

—

#3952

—

#3947

—

C₈×Dic₅

—

—

—

—

D₅

yes

characteristic, complemented

C₂³×D₅

C₄²

yes

characteristic

C₂²×C₄×C₅

C₂²×C₄

yes

—

C₂²×Dic₅

C₄²

yes

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂²⋊C₄

yes

—

C₂×C₄×D₅

C₄⋊C₄

yes

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×Q₈

yes

characteristic

C₄×Dic₅

C₂²⋊C₄

yes

—

C₄×Dic₅

C₂²⋊C₄

yes

—

C₄×Dic₅

C₄⋊C₄

yes

—

#681

—

C₂×#168

—

C₂²×C₄²

—

—

—

—

C₂×D₅

yes

—

#407

—

C₄×M₁₆

—

C₂³×C₅

C₂×C₄²

yes

characteristic

C₂²×D₅

—

C₂²×D₅

#95

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×Dic₅

—

C₂×Dic₅

#95

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

#168

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₅

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

C₂×#95

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#413

yes

—

C₄×C₅

#413

yes

—

Dic₅

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₅

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

D₄×D₅

yes

—

C₄²

Q₈×D₅

yes

characteristic

100

C₂×C₅

—

101

C₂×C₅

#1742

yes

—

102

D₅

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₄²×D₅

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

D₅×C₂²⋊C₄

yes

—

114

C₂×C₄

D₅×C₄⋊C₄

yes

—

115

C₈

—

116

C₈

—

117

C₅

#8130

yes

characteristic, complemented, Sylow

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

D₅×#95

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

#65034

yes

—

125

C₄

#65040

yes

—

126

C₂

—

127

C₂

—

128

C₂

#10599782

yes

—

	subgroup	G/H
center	C₂²	D₅×#95
commutator	C₂³×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1309	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂³×D₅	C₄²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1309	D₅
hypocenter	C₅	#8130
LCS	G ▷ C₂³×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

D₅×#95

commutator

C₂³×C₅

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₅×#1309

C₂

socle

C₂²×C₅

C₂×#95

prime radical

C₂³×D₅

C₄²

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂³×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
2. C₅×#1309	C₂	semidirect
3. #10598203	C₂	semidirect
26. #1309	D₅	semidirect
117. C₅	#8130	semidirect

H ≅ G/N

type

2. C₅×#1309

C₂

semidirect

3. #10598203

C₂

semidirect

26. #1309

D₅

semidirect

117. C₅

#8130

semidirect

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

subgroup	index
#10610769	2
C₅×#1309	2
#10598203	2
#8130	5

subgroup

index

Subgroups — #11989720

Conjugacy classes of subgroups

Complements

Maximal subgroups