Group #72228161 — subgroups

Subgroups — D₅×SL(2,𝔽₉) #72228161

Order 7200.

152 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 10 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	C₅×SL(2,𝔽₉)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SL(2,𝔽₉)	—	5	1	no	maximal
3	D₅×SL(2,𝔽₅)	—	6	2	no	maximal
4	D₅×#622	—	10	1	no	maximal
5	SL(2,𝔽₉)	D₅	1	1	yes	characteristic, direct product factor
6	C₅×SL(2,𝔽₅)	—	6	2	no	—
7	D₅×#226	—	15	2	no	maximal
8	D₅×C₃⋊Dic₃	—	10	1	no	—
9	#67126	—	10	1	no	—
10	C₅×#622	—	10	1	no	—
11	D₅×SL(2,𝔽₃)	—	15	2	no	—
12	C₅⋊#226	—	15	2	no	—
13	C₅×#226	—	15	2	no	—
14	C₂×SL(2,𝔽₅)	—	30	2	no	—
15	D₅×Dic₅	—	36	1	no	—
16	C₂×C₃²×D₅	—	10	1	no	—
17	C₃⋊Dic₁₅	—	10	1	no	—
18	C₅×C₃⋊Dic₃	—	10	1	no	—
19	D₅×Q₁₆	—	45	1	no	—
20	C₂×#622	—	50	1	no	—
21	C₅×SL(2,𝔽₃)	—	15	2	no	—
22	D₅×Dic₃	—	60	2	no	—
23	SL(2,𝔽₅)	—	6	2	no	—
24	C₂×C₅×D₅	—	36	1	no	—
25	#1039	—	36	1	no	—
26	C₅×Dic₅	—	36	1	no	—
27	C₂×#226	—	75	2	no	—
28	C₂×C₃²×C₅	—	10	1	no	—
29	C₃²×D₅	—	10	2	no	—
30	Q₈×D₅	—	15	2	no	—
31	C₅×Q₁₆	—	45	1	no	—
32	#689	—	45	2	no	—
33	Dic₂₀	—	45	1	no	—
34	C₈×D₅	—	45	1	no	—
35	C₂×C₃⋊Dic₃	—	50	1	no	—
36	#622	—	10	1	no	—
37	#622	—	50	1	no	—
38	C₂×C₃×D₅	—	20	2	no	—
39	Dic₁₅	—	60	2	no	—
40	C₅×Dic₃	—	60	2	no	—
41	C₂×C₅²	—	36	1	no	—
42	C₅×D₅	—	36	2	no	—
43	C₂×SL(2,𝔽₃)	—	75	2	no	—
44	#226	—	15	2	no	—
45	#226	—	75	2	no	—
46	C₃²×C₅	—	10	1	no	—
47	C₅×Q₈	—	15	2	no	—
48	C₂×Dic₅	—	180	1	no	—
49	C₄×D₅	—	45	1	no	—
50	Dic₁₀	—	45	2	no	—
51	C₅×C₈	—	45	1	no	—
52	#168	—	45	1	no	—
53	C₂²×C₃²	—	50	1	no	—
54	C₃⋊Dic₃	—	10	1	no	—
55	C₃⋊Dic₃	—	50	1	no	—
56	C₂×Q₁₆	—	225	1	no	Sylow
57	C₂×C₃×C₅	—	20	2	no	—
58	C₃×D₅	—	20	4	no	—
59	C₅²	—	36	1	no	Sylow
60	C₂×Dic₃	—	300	2	no	—
61	SL(2,𝔽₃)	—	15	2	no	—
62	C₂²×C₅	—	180	1	no	—
63	C₂×D₅	A₆	1	1	yes	maximal normal, characteristic
64	C₄×C₅	—	45	1	no	—
65	Dic₅	—	36	1	no	—
66	Dic₅	—	45	1	no	—
67	Dic₅	—	360	2	no	—
68	Dic₅	—	180	1	no	—
69	C₂×C₃²	—	10	1	no	—
70	C₂×C₃²	—	50	2	no	—
71	C₂×Q₈	—	75	2	no	—
72	Q₁₆	—	45	1	no	—
73	Q₁₆	—	225	1	no	—
74	Q₁₆	—	225	2	no	—
75	C₂×C₈	—	225	1	no	—
76	C₃×C₅	—	20	2	no	—
77	C₂²×C₃	—	100	2	no	—
78	Dic₃	—	300	2	no	—
79	Dic₃	—	60	2	no	—
80	C₂×C₅	—	36	1	no	—
81	C₂×C₅	—	72	2	no	—
82	C₂×C₅	—	180	2	no	—
83	C₂×C₅	C₂×A₆	1	1	yes	characteristic
84	D₅	SL(2,𝔽₉)	1	2	yes	direct product factor
85	C₃²	—	10	1	no	Sylow
86	Q₈	—	15	2	no	—
87	Q₈	—	225	2	no	—
88	C₂×C₄	—	225	1	no	—
89	C₈	—	45	1	no	—
90	C₈	—	225	1	no	—
91	C₂×C₃	—	20	2	no	—
92	C₂×C₃	—	100	4	no	—
93	C₅	—	36	1	no	—
94	C₅	—	72	2	no	—
95	C₅	C₂×SL(2,𝔽₉)	1	1	yes	characteristic, complemented
96	C₂²	—	5	1	no	—
97	C₄	—	45	1	no	—
98	C₄	—	225	1	no	—
99	C₃	—	20	2	no	—
100	C₂	—	5	2	no	—
101	C₂	D₅×A₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×SL(2,𝔽₉)

C₂

yes

maximal, characteristic, complemented

C₂×SL(2,𝔽₉)

—

maximal

D₅×SL(2,𝔽₅)

—

—

D₅

yes

characteristic, direct product factor

C₅×SL(2,𝔽₅)

—

D₅×#226

—

maximal

D₅×C₃⋊Dic₃

—

#67126

—

C₅×#622

—

D₅×SL(2,𝔽₃)

—

C₅⋊#226

—

C₅×#226

—

C₂×SL(2,𝔽₅)

—

D₅×Dic₅

—

C₂×C₃²×D₅

—

C₃⋊Dic₁₅

—

C₅×C₃⋊Dic₃

—

D₅×Q₁₆

—

C₂×#622

—

C₅×SL(2,𝔽₃)

—

D₅×Dic₃

—

SL(2,𝔽₅)

—

C₂×C₅×D₅

—

#1039

—

C₅×Dic₅

—

C₂×#226

—

C₂×C₃²×C₅

—

C₃²×D₅

—

Q₈×D₅

—

C₅×Q₁₆

—

#689

—

Dic₂₀

—

C₈×D₅

—

C₂×C₃⋊Dic₃

—

#622

—

#622

—

C₂×C₃×D₅

—

Dic₁₅

—

C₅×Dic₃

—

C₂×C₅²

—

C₅×D₅

—

C₂×SL(2,𝔽₃)

—

#226

—

#226

—

C₃²×C₅

—

C₅×Q₈

—

C₂×Dic₅

—

180

—

C₄×D₅

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₂²×C₃²

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

225

Sylow

C₂×C₃×C₅

—

C₃×D₅

—

C₅²

—

Sylow

C₂×Dic₃

—

300

—

SL(2,𝔽₃)

—

C₂²×C₅

—

180

—

C₂×D₅

A₆

yes

maximal normal, characteristic

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

360

—

Dic₅

—

180

—

C₂×C₃²

—

C₂×C₃²

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

225

—

Q₁₆

—

225

—

C₂×C₈

—

225

—

C₃×C₅

—

C₂²×C₃

—

100

—

Dic₃

—

300

—

Dic₃

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

180

—

C₂×C₅

C₂×A₆

yes

characteristic

D₅

SL(2,𝔽₉)

yes

direct product factor

C₃²

—

Sylow

Q₈

—

Q₈

—

225

—

C₂×C₄

—

225

—

C₈

—

C₈

—

225

—

C₂×C₃

—

C₂×C₃

—

100

—

C₅

—

C₅

—

C₅

C₂×SL(2,𝔽₉)

yes

characteristic, complemented

C₂²

—

C₄

—

C₄

—

225

—

C₃

—

100

C₂

—

101

C₂

D₅×A₆

yes

characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×A₆
commutator, hypocenter	C₅×SL(2,𝔽₉)	C₂
Fitting, socle, cosocle	C₂×C₅	C₂×A₆
LCS	G ▷ C₅×SL(2,𝔽₉)
UCS	C₁ ◁ C₂

subgroup

G/H

center, Frattini, hypercenter

C₂

D₅×A₆

commutator, hypocenter

C₅×SL(2,𝔽₉)

C₂

Fitting, socle, cosocle

C₂×C₅

C₂×A₆

LCS

G ▷ C₅×SL(2,𝔽₉)

UCS

C₁ ◁ C₂

N	H ≅ G/N	type
1. C₅×SL(2,𝔽₉)	C₂	semidirect
5. SL(2,𝔽₉)	D₅	direct (N × H)
84. D₅	SL(2,𝔽₉)	direct (N × H)
95. C₅	C₂×SL(2,𝔽₉)	semidirect

H ≅ G/N

type

1. C₅×SL(2,𝔽₉)

C₂

semidirect

5. SL(2,𝔽₉)

D₅

direct (N × H)

84. D₅

SL(2,𝔽₉)

direct (N × H)

95. C₅

C₂×SL(2,𝔽₉)

semidirect

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

subgroup	index
C₅×SL(2,𝔽₉)	2
C₂×SL(2,𝔽₉)	5
D₅×SL(2,𝔽₅)	6
D₅×#622	10
D₅×#226	15

subgroup

index

C₅×SL(2,𝔽₉)

C₂×SL(2,𝔽₉)

D₅×SL(2,𝔽₅)

D₅×#622

D₅×#226

Subgroups — D₅×SL(2,𝔽₉) #72228161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D5×SL(2,𝔽9) #72228161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×SL(2,𝔽₉) #72228161