Group #13018814 — subgroups

Subgroups — #13018814

Order 1280.

354 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 136 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	#10603858	C₂	1	2	yes	maximal
2	#10603817	C₂	1	4	yes	maximal
3	C₈²×D₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#3995	—	2	4	no	—
5	#65681	C₂²	1	4	yes	—
6	C₄×C₈×D₅	C₂²	1	2	yes	—
7	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
8	#64883	—	2	8	no	—
9	C₅×C₈²	C₄	1	1	yes	characteristic
10	C₈×#168	C₄	1	2	yes	—
11	C₈×#168	C₄	1	1	yes	characteristic
12	#8822	—	5	1	no	maximal, Sylow
13	C₂×#700	—	2	4	no	—
14	C₂×C₈×D₅	D₄	1	2	yes	—
15	C₂×C₈×D₅	Q₈	1	4	yes	—
16	C₄²×D₅	C₂³	1	1	yes	characteristic
17	C₄⋊#170	—	2	8	no	—
18	#3995	—	4	8	no	—
19	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₈×Dic₅	D₄	1	4	yes	—
22	C₈×Dic₅	Q₈	1	2	yes	—
23	C₄×#168	C₂×C₄	1	1	yes	characteristic
24	C₄×#168	C₂×C₄	1	2	yes	—
25	#1848	—	5	2	no	—
26	#1566	—	5	4	no	—
27	C₂×C₈²	—	5	1	no	—
28	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
30	C₂×#170	—	4	4	no	—
31	#700	—	4	8	no	—
32	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
33	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
34	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₄×Dic₅	C₂×D₄	1	2	yes	—
36	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
37	C₂×#168	C₄⋊C₄	1	2	yes	—
38	C₂×#168	C₄⋊C₄	1	4	yes	—
39	C₈×D₅	—	2	8	no	—
40	C₈×D₅	SD₁₆	1	8	yes	—
41	C₂×#108	—	10	4	no	—
42	#423	—	5	4	no	—
43	C₂×C₄×C₈	—	5	2	no	—
44	C₂×C₄×C₈	—	5	1	no	—
45	#334	—	10	8	no	—
46	C₈²	—	5	1	no	—
47	C₈²	—	5	2	no	—
48	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₂²×D₅	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
52	C₂×Dic₅	#127	1	1	yes	characteristic
53	C₂×Dic₅	C₄⋊Q₈	1	2	yes	—
54	C₄×D₅	—	2	4	no	—
55	C₄×D₅	C₂×SD₁₆	1	4	yes	—
56	#170	—	8	4	no	—
57	C₅×C₈	—	2	4	no	—
58	C₅×C₈	#106	1	4	yes	—
59	#168	#106	1	4	yes	—
60	#168	#108	1	8	yes	complemented
61	C₂×M₁₆	—	10	4	no	—
62	C₂²×C₈	—	5	2	no	—
63	C₂²×C₈	—	5	4	no	—
64	C₂×C₄²	—	5	1	no	—
65	#108	—	20	8	no	—
66	C₄⋊C₈	—	10	8	no	—
67	C₄×C₈	—	5	4	no	—
68	C₄×C₈	—	5	2	no	—
69	C₄×C₈	—	5	2	no	—
70	C₄×C₈	—	5	1	no	—
71	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
72	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₂²×C₅	#384	1	1	yes	characteristic
74	C₂×D₅	—	2	2	no	—
75	C₂×D₅	#498	1	2	yes	—
76	C₄×C₅	C₂×#106	1	2	yes	—
77	C₄×C₅	C₂×#108	1	4	yes	—
78	Dic₅	—	2	2	no	—
79	Dic₅	#492	1	2	yes	—
80	C₂²×C₄	—	5	1	no	—
81	C₂²×C₄	—	5	2	no	—
82	M₁₆	—	20	8	no	—
83	C₂×C₈	—	5	2	no	—
84	C₂×C₈	—	5	4	no	—
85	C₂×C₈	—	5	8	no	—
86	C₂×C₈	—	10	8	no	—
87	C₂×C₈	—	20	4	no	—
88	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
89	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
90	C₄²	—	5	2	no	—
91	C₄²	—	5	1	no	—
92	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₂×C₅	#1844	1	1	yes	characteristic
94	C₂×C₅	#1848	1	2	yes	—
95	D₅	—	2	2	no	—
96	C₂³	—	5	1	no	—
97	C₂×C₄	—	5	1	no	—
98	C₂×C₄	—	5	4	no	—
99	C₂×C₄	—	5	2	no	—
100	C₂×C₄	—	10	4	no	—
101	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₈	—	5	4	no	—
105	C₈	—	5	8	no	—
106	C₈	—	40	4	no	—
107	C₈	#3993	1	4	yes	—
108	C₅	#8822	1	1	yes	characteristic, complemented, Sylow
109	C₂²	—	5	2	no	—
110	C₂²	—	10	2	no	—
111	C₂²	#65688	1	1	yes	characteristic
112	C₄	—	5	2	no	—
113	C₄	—	10	2	no	—
114	C₄	C₂×#3993	1	2	yes	—
115	C₄	D₅⋊#108	1	4	yes	—
116	C₂	—	10	2	no	—
117	C₂	#10603847	1	1	yes	characteristic
118	C₂	#10603860	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10603858

C₂

yes

maximal

#10603817

C₂

yes

maximal

C₈²×D₅

C₂

yes

maximal, characteristic

C₂×#3995

—

#65681

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64883

—

C₅×C₈²

C₄

yes

characteristic

C₈×#168

C₄

yes

—

C₈×#168

C₄

yes

—

—

C₂×C₈×D₅

D₄

yes

—

C₂×C₈×D₅

Q₈

yes

—

C₄²×D₅

C₂³

yes

characteristic

C₄⋊#170

—

#3995

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

C₄×#168

C₂×C₄

yes

characteristic

C₄×#168

C₂×C₄

yes

—

#1848

—

#1566

—

C₂×C₈²

—

C₂×C₄×D₅

C₂×D₄

yes

characteristic

C₂×C₄×D₅

C₂×Q₈

yes

—

C₂×#170

—

#700

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

—

C₄×Dic₅

C₂×Q₈

yes

characteristic

C₂×#168

C₄⋊C₄

yes

—

C₂×#168

C₄⋊C₄

yes

—

C₈×D₅

—

C₈×D₅

SD₁₆

yes

—

C₂×#108

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#334

—

C₈²

—

C₈²

—

C₈²

AGL(1,𝔽₅)

yes

characteristic

C₂²×D₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×Dic₅

#127

yes

characteristic

C₂×Dic₅

C₄⋊Q₈

yes

—

C₄×D₅

—

C₄×D₅

C₂×SD₁₆

yes

—

#170

—

C₅×C₈

—

C₅×C₈

#106

yes

—

#168

#106

yes

—

#168

#108

yes

complemented

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄×C₈

yes

C₂²×C₅

yes

C₂×D₅

—

C₂×D₅

#498

yes

—

C₄×C₅

C₂×#106

yes

—

C₄×C₅

C₂×#108

yes

—

Dic₅

—

Dic₅

#492

yes

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×AGL(1,𝔽₅)

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

102

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

#3993

yes

—

108

C₅

#8822

yes

characteristic, complemented, Sylow

109

C₂²

—

110

C₂²

—

111

C₂²

#65688

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

C₂×#3993

yes

—

115

C₄

D₅⋊#108

yes

—

116

C₂

—

117

C₂

#10603847

yes

characteristic

118

C₂

#10603860

yes

—

	subgroup	G/H
center	C₂²	#65688
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₈²	C₄
socle	C₂²×C₅	#384
prime radical	C₂²×D₅	C₄⋊Q₈
cosocle	C₄²×D₅	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	#8822
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#65688

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₅×C₈²

C₄

socle

C₂²×C₅

#384

prime radical

C₂²×D₅

C₄⋊Q₈

cosocle

C₄²×D₅

C₂³

C₈²

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
60. #168	#108	semidirect
108. C₅	#8822	semidirect

H ≅ G/N

type

60. #168

#108

semidirect

108. C₅

#8822

semidirect

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

subgroup	index
#10603858	2
#10603817	2
C₈²×D₅	2
#8822	5

subgroup

index

#10603858

#10603817

C₈²×D₅

#8822

Subgroups — #13018814

Conjugacy classes of subgroups

Complements

Maximal subgroups