Group #13018751 — subgroups

Subgroups — #13018751

Order 1280.

262 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 80 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	#10603857	C₂	1	1	yes	maximal, characteristic
2	#10603846	C₂	1	1	yes	maximal, characteristic
3	#10598840	C₂	1	1	yes	maximal, characteristic
4	#10598527	C₂	1	4	yes	maximal
5	#65689	C₂²	1	2	yes	—
6	#65682	C₂²	1	2	yes	—
7	#64974	C₂²	1	2	yes	—
8	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
9	#64896	—	2	2	no	—
10	#64893	—	2	2	no	—
11	C₅×C₈²	C₄	1	1	yes	characteristic, complemented
12	#64674	C₄	1	2	yes	complemented
13	C₈×#168	C₄	1	1	yes	characteristic, complemented
14	#8791	—	5	1	no	maximal, Sylow
15	C₂×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
16	C₂×#700	—	2	2	no	—
17	C₂×C₈×D₅	D₄	1	2	yes	—
18	C₄²×D₅	C₂³	1	1	yes	characteristic
19	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	4	no	—
20	#4004	—	2	4	no	—
21	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	#3952	—	2	2	no	—
24	#3946	—	2	2	no	—
25	C₈×Dic₅	Q₈	1	2	yes	—
26	C₄×#168	C₂×C₄	1	2	yes	—
27	C₄×#168	C₂×C₄	1	1	yes	characteristic
28	#1837	—	5	1	no	—
29	#1835	—	5	1	no	—
30	#1450	—	5	1	no	—
31	#1286	—	5	4	no	—
32	C₂²×AGL(1,𝔽₅)	—	4	2	no	—
33	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×D₅	#41	1	2	yes	—
35	C₂×#170	—	4	2	no	—
36	C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
37	#700	—	4	4	no	—
38	C₂×C₅×C₈	—	2	2	no	—
39	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
40	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
42	C₄×Dic₅	#41	1	2	yes	—
43	C₂×#168	—	2	2	no	—
44	C₂×#168	C₄⋊C₄	1	2	yes	—
45	C₈×D₅	—	2	4	no	—
46	#431	—	5	2	no	—
47	#423	—	5	2	no	—
48	C₂×C₄×C₈	—	5	1	no	—
49	#382	—	5	2	no	—
50	#341	—	10	2	no	—
51	#340	—	10	2	no	—
52	#322	—	5	2	no	—
53	C₈²	—	5	1	no	—
54	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
55	C₂²×D₅	#124	1	1	yes	characteristic
56	C₂×AGL(1,𝔽₅)	—	8	4	no	—
57	C₂×AGL(1,𝔽₅)	—	8	2	no	—
58	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅	#117	1	2	yes	—
60	C₂×Dic₅	#124	1	1	yes	characteristic
61	C₂×Dic₅	#125	1	2	yes	—
62	C₄×D₅	—	4	2	no	—
63	C₄×D₅	#135	1	4	yes	—
64	#170	—	8	2	no	—
65	C₅×C₈	—	2	2	no	—
66	C₅×C₈	—	4	2	no	—
67	#168	—	2	2	no	—
68	#168	—	4	2	no	—
69	C₂×M₁₆	—	10	2	no	—
70	C₂²×C₈	—	5	2	no	—
71	C₂×C₄⋊C₄	—	10	2	no	—
72	C₂×C₄²	—	5	1	no	—
73	C₄⋊C₈	—	10	2	no	—
74	C₄⋊C₈	—	10	4	no	—
75	C₂²⋊C₈	—	10	2	no	—
76	C₄×C₈	—	5	2	no	—
77	C₄×C₈	—	5	1	no	—
78	C₄×C₈	—	5	2	no	—
79	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
80	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
81	#95	—	10	4	no	—
82	C₂²×C₅	#382	1	1	yes	characteristic
83	C₂×D₅	—	2	2	no	—
84	C₂×D₅	#488	1	2	yes	—
85	AGL(1,𝔽₅)	—	16	2	no	—
86	C₄×C₅	—	2	2	no	—
87	C₄×C₅	#427	1	2	yes	—
88	Dic₅	—	2	2	no	—
89	Dic₅	#488	1	2	yes	—
90	C₂²×C₄	—	5	1	no	—
91	C₂²×C₄	—	5	2	no	—
92	C₂²×C₄	—	20	2	no	—
93	M₁₆	—	20	4	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	5	2	no	—
96	C₂×C₈	—	10	4	no	—
97	C₂×C₈	—	10	2	no	—
98	C₂×C₈	—	20	2	no	—
99	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
100	C₄⋊C₄	—	20	4	no	—
101	C₄²	—	5	1	no	—
102	C₄²	—	5	2	no	—
103	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
104	C₂×C₅	#1833	1	2	yes	—
105	C₂×C₅	#1835	1	1	yes	characteristic
106	D₅	—	2	2	no	—
107	C₂³	—	5	1	no	—
108	C₂×C₄	—	5	1	no	—
109	C₂×C₄	—	5	4	no	—
110	C₂×C₄	—	5	2	no	—
111	C₂×C₄	—	20	2	no	—
112	C₂×C₄	—	40	2	no	—
113	C₂×C₄	—	40	4	no	—
114	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂×C₄	#4130	1	2	yes	—
116	C₈	—	2	2	no	—
117	C₈	—	4	2	no	—
118	C₈	—	10	2	no	—
119	C₈	—	20	2	no	—
120	C₈	—	40	2	no	—
121	C₅	#8791	1	1	yes	characteristic, complemented, Sylow
122	C₂²	—	5	2	no	—
123	C₂²	—	10	2	no	—
124	C₂²	#65689	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	5	2	no	—
127	C₄	—	10	2	no	—
128	C₄	—	80	2	no	—
129	C₄	#65721	1	2	yes	—
130	C₂	—	10	2	no	—
131	C₂	#10603846	1	1	yes	characteristic
132	C₂	#10603852	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10603857

C₂

yes

maximal, characteristic

#10603846

C₂

yes

maximal, characteristic

#10598840

C₂

yes

maximal, characteristic

#10598527

C₂

yes

maximal

#65689

C₂²

yes

—

#65682

C₂²

yes

—

#64974

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64896

—

#64893

—

C₅×C₈²

C₄

yes

characteristic, complemented

#64674

C₄

yes

complemented

C₈×#168

C₄

yes

characteristic, complemented

#8791

—

maximal, Sylow

C₂×C₄⋊AGL(1,𝔽₅)

—

C₂×#700

—

C₂×C₈×D₅

D₄

yes

—

C₄²×D₅

C₂³

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₄

—

#4004

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3952

—

#3946

—

C₈×Dic₅

Q₈

yes

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

—

—

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

C₂×D₄

yes

characteristic

C₂×C₄×D₅

#41

yes

—

C₂×#170

—

C₄⋊AGL(1,𝔽₅)

—

#700

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂×Q₈

yes

characteristic

C₄×Dic₅

#41

yes

—

C₂×#168

—

C₂×#168

C₄⋊C₄

yes

—

C₈×D₅

—

#431

—

#423

—

C₂×C₄×C₈

—

—

—

—

—

C₈²

—

C₈²

AGL(1,𝔽₅)

yes

characteristic, complemented

C₂²×D₅

yes

—

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×Dic₅

#124

yes

characteristic

C₂×Dic₅

#125

yes

—

C₄×D₅

—

C₄×D₅

#135

yes

—

#170

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

#95

—

C₂²×C₅

#382

yes

characteristic

C₂×D₅

—

C₂×D₅

#488

yes

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

#427

yes

—

Dic₅

—

Dic₅

#488

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

104

C₂×C₅

#1833

yes

—

105

C₂×C₅

#1835

yes

characteristic

106

D₅

—

107

C₂³

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

115

C₂×C₄

#4130

yes

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₅

#8791

yes

characteristic, complemented, Sylow

122

C₂²

—

123

C₂²

—

124

C₂²

#65689

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#65721

yes

—

130

C₂

—

131

C₂

#10603846

yes

characteristic

132

C₂

#10603852

yes

—

	subgroup	G/H
center	C₂²	#65689
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₈²	C₄
socle	C₂²×C₅	#382
prime radical	C₂²×D₅	#124
cosocle	C₄²×D₅	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	#8791
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#65689

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₅×C₈²

C₄

C₂²×C₅

C₂²×D₅

C₄²×D₅

C₂³

C₈²

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
11. C₅×C₈²	C₄	semidirect
12. #64674	C₄	semidirect
13. C₈×#168	C₄	semidirect
54. C₈²	AGL(1,𝔽₅)	semidirect
121. C₅	#8791	semidirect

H ≅ G/N

type

11. C₅×C₈²

C₄

semidirect

12. #64674

C₄

semidirect

13. C₈×#168

C₄

semidirect

54. C₈²

AGL(1,𝔽₅)

semidirect

121. C₅

#8791

semidirect

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

subgroup	index
#10603857	2
#10603846	2
#10598840	2
#10598527	2
#8791	5

subgroup

index

Subgroups — #13018751

Conjugacy classes of subgroups

Complements

Maximal subgroups