Subgroups — SL(2,𝔽₃)×D₁₅ #10624076

Order 720.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₁₅	C₃	1	1	yes	maximal, characteristic, complemented
3	D₅×SL(2,𝔽₃)	—	3	1	no	maximal
4	C₂×C₃×D₁₅	—	4	1	no	maximal
5	D₃×SL(2,𝔽₃)	—	5	1	no	maximal, Hall
6	C₃×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
7	C₄×D₁₅	—	3	1	no	—
8	Dic₃₀	—	3	1	no	—
9	C₅×SL(2,𝔽₃)	—	2	1	no	—
10	C₅×SL(2,𝔽₃)	D₃	1	1	yes	characteristic, complemented
11	C₂×C₃²×C₅	—	4	1	no	—
12	C₃×D₁₅	—	4	2	no	—
13	Q₈×D₅	—	3	1	no	Hall
14	C₃×SL(2,𝔽₃)	D₅	1	1	yes	characteristic, complemented
15	C₂×D₁₅	A₄	1	1	yes	characteristic
16	C₂×C₃×D₅	—	12	1	no	—
17	C₃×C₄×C₅	—	3	1	no	—
18	Dic₁₅	—	3	1	no	—
19	D₃×Q₈	—	5	1	no	—
20	C₂×SL(2,𝔽₃)	—	15	1	no	—
21	C₃²×C₅	—	4	1	no	Hall
22	C₅×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
23	C₄×D₅	—	9	1	no	—
24	Dic₁₀	—	9	1	no	—
25	C₂×C₃×D₃	—	20	1	no	—
26	C₂×C₃×C₅	—	4	1	no	—
27	C₂×C₃×C₅	—	8	1	no	—
28	C₂×C₃×C₅	C₂×A₄	1	1	yes	characteristic
29	D₁₅	SL(2,𝔽₃)	1	2	yes	direct product factor
30	C₃×D₅	—	12	2	no	—
31	C₃×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
32	C₄×D₃	—	15	1	no	—
33	Dic₆	—	15	1	no	—
34	SL(2,𝔽₃)	—	2	1	no	—
35	SL(2,𝔽₃)	D₁₅	1	1	yes	characteristic, direct product factor
36	C₂×D₅	—	3	1	no	—
37	C₄×C₅	—	3	1	no	—
38	Dic₅	—	9	1	no	—
39	C₂×C₃²	—	4	1	no	—
40	C₃×D₃	—	20	2	no	—
41	C₂×Q₈	—	15	1	no	Sylow
42	C₃×C₅	—	4	1	no	—
43	C₃×C₅	—	8	1	no	—
44	C₃×C₅	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
45	C₂²×C₃	—	60	1	no	—
46	C₂×D₃	—	5	1	no	—
47	C₃×C₄	—	3	1	no	—
48	Dic₃	—	15	1	no	—
49	C₂×C₅	D₃×A₄	1	1	yes	characteristic
50	D₅	—	3	2	no	—
51	C₃²	—	4	1	no	Sylow
52	Q₈	—	45	1	no	—
53	Q₈	C₃×D₁₅	1	1	yes	characteristic, complemented
54	C₂×C₄	—	45	1	no	—
55	C₂×C₃	—	4	1	no	—
56	C₂×C₃	—	8	1	no	—
57	C₂×C₃	—	60	2	no	—
58	C₂×C₃	D₅×A₄	1	1	yes	characteristic
59	D₃	—	5	2	no	—
60	C₅	D₃×SL(2,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	15	1	no	—
62	C₄	—	3	1	no	—
63	C₄	—	45	1	no	—
64	C₃	—	4	1	no	—
65	C₃	—	8	1	no	—
66	C₃	D₅×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
67	C₂	—	15	2	no	—
68	C₂	A₄×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄×D₁₅
commutator, Fitting, cosocle, hypocenter	C₃×C₅×Q₈	C₂×C₃
socle	C₂×C₃×C₅	C₂×A₄
LCS	G ▷ C₃×C₅×Q₈
UCS	C₁ ◁ C₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×C₅×SL(2,𝔽₃)	C₂	semidirect
2. Q₈×D₁₅	C₃	semidirect
6. C₃×C₅×Q₈	C₂×C₃	semidirect
10. C₅×SL(2,𝔽₃)	D₃	semidirect
14. C₃×SL(2,𝔽₃)	D₅	semidirect
22. C₅×Q₈	C₃×D₃	semidirect
29. D₁₅	SL(2,𝔽₃)	direct (N × H)
31. C₃×Q₈	C₃×D₅	semidirect
35. SL(2,𝔽₃)	D₁₅	direct (N × H)
44. C₃×C₅	C₂×SL(2,𝔽₃)	semidirect
53. Q₈	C₃×D₁₅	semidirect
60. C₅	D₃×SL(2,𝔽₃)	semidirect
66. C₃	D₅×SL(2,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×SL(2,𝔽₃)	2
Q₈×D₁₅	3
D₅×SL(2,𝔽₃)	3
C₂×C₃×D₁₅	4
D₃×SL(2,𝔽₃)	5

Subgroups — SL(2,𝔽3)×D15 #10624076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SL(2,𝔽₃)×D₁₅ #10624076