Subgroups — Dic₁₃⋊SL(2,𝔽₃) #11928331

Order 1248.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈⋊(C₁₃⋊C₃)	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	Q₈⋊(C₁₃⋊C₃)	C₄	1	1	yes	characteristic, complemented
4	C₂×Dic₁₃⋊C₃	—	4	1	no	maximal
5	C₂×Q₈×C₁₃	C₂×C₃	1	1	yes	characteristic
6	C₄⋊Dic₁₃	—	3	1	no	—
7	C₄×Dic₁₃	—	3	1	no	—
8	C₂²×C₁₃⋊C₃	—	4	1	no	—
9	Dic₁₃⋊C₃	—	4	2	no	—
10	Q₈×C₁₃	—	3	1	no	—
11	Q₈×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented
12	C₂×C₄×C₁₃	—	3	1	no	—
13	C₂×Dic₁₃	—	3	1	no	—
14	C₂×Dic₁₃	A₄	1	1	yes	characteristic
15	C₄×SL(2,𝔽₃)	—	13	1	no	maximal, Hall
16	C₂×C₁₃⋊C₃	—	4	1	no	—
17	C₂×C₁₃⋊C₃	—	4	1	no	—
18	C₂×C₁₃⋊C₃	—	4	1	no	—
19	C₂²×C₁₃	C₂×A₄	1	1	yes	characteristic
20	C₄×C₁₃	—	3	1	no	—
21	C₄×C₁₃	—	3	1	no	—
22	Dic₁₃	—	6	1	no	—
23	Dic₁₃	SL(2,𝔽₃)	1	2	yes	complemented
24	C₂×SL(2,𝔽₃)	—	13	1	no	—
25	C₁₃⋊C₃	—	4	1	no	Hall
26	C₄×Q₈	—	13	1	no	Sylow
27	C₂×C₁₃	C₄×A₄	1	1	yes	characteristic
28	C₂×C₁₃	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
29	C₂×C₁₃	#41⋊C₃	1	1	yes	characteristic
30	C₂×C₃×C₄	—	52	1	no	—
31	SL(2,𝔽₃)	—	13	1	no	—
32	C₂×Q₈	D₁₃⋊C₃	1	1	yes	characteristic
33	C₄⋊C₄	—	39	1	no	—
34	C₄²	—	39	1	no	—
35	C₁₃	C₄×SL(2,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₃	—	52	1	no	—
37	C₃×C₄	—	52	2	no	—
38	Q₈	—	3	1	no	—
39	Q₈	Dic₁₃⋊C₃	1	1	yes	characteristic, complemented
40	C₂×C₄	—	3	1	no	—
41	C₂×C₄	—	13	1	no	—
42	C₂×C₄	—	39	1	no	—
43	C₂×C₃	—	52	1	no	—
44	C₂×C₃	—	52	1	no	—
45	C₂×C₃	—	52	1	no	—
46	C₂²	D₁₃⋊A₄	1	1	yes	characteristic
47	C₄	—	3	1	no	—
48	C₄	—	3	1	no	—
49	C₄	—	13	2	no	—
50	C₄	—	78	1	no	—
51	C₃	—	52	1	no	Sylow
52	C₂	#10597713	1	1	yes	characteristic
53	C₂	D₁₃⋊SL(2,𝔽₃)	1	1	yes	characteristic
54	C₂	Dic₁₃⋊A₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₁₃⋊A₄
commutator, hypocenter	Q₈×C₁₃	C₃×C₄
Fitting, cosocle	C₂×Q₈×C₁₃	C₂×C₃
socle	C₂²×C₁₃	C₂×A₄
prime radical	C₂×Q₈⋊(C₁₃⋊C₃)	C₂
LCS	G ▷ Q₈×C₁₃
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₁₃	C₃	semidirect
3. Q₈⋊(C₁₃⋊C₃)	C₄	semidirect
11. Q₈×C₁₃	C₃×C₄	semidirect
23. Dic₁₃	SL(2,𝔽₃)	semidirect
35. C₁₃	C₄×SL(2,𝔽₃)	semidirect
39. Q₈	Dic₁₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈⋊(C₁₃⋊C₃)	2
Q₈×Dic₁₃	3
C₂×Dic₁₃⋊C₃	4
C₄×SL(2,𝔽₃)	13

Subgroups — Dic13⋊SL(2,𝔽3) #11928331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₃⋊SL(2,𝔽₃) #11928331