Subgroups — C₂²×C₁₃×SL(2,𝔽₃) #11929255

Order 1248.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×SL(2,𝔽₃)	C₂	1	3	yes	maximal, direct product factor
2	C₂²×Q₈×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂³×C₃×C₁₃	—	4	1	no	maximal
4	C₁₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, direct product factor
5	C₂×Q₈×C₁₃	—	3	3	no	—
6	C₂×Q₈×C₁₃	C₂×C₃	1	3	yes	complemented
7	C₂²×C₄×C₁₃	—	3	1	no	—
8	C₂²×C₃×C₁₃	—	4	4	no	—
9	C₂²×C₃×C₁₃	—	4	3	no	—
10	C₂³×C₁₃	A₄	1	1	yes	characteristic
11	Q₈×C₁₃	—	3	2	no	—
12	Q₈×C₁₃	—	3	3	no	—
13	Q₈×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented
14	C₂×C₄×C₁₃	—	3	3	no	—
15	C₂×C₄×C₁₃	—	3	3	no	—
16	C₂²×SL(2,𝔽₃)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₃×C₁₃	—	4	6	no	—
18	C₂×C₃×C₁₃	—	4	1	no	—
19	C₂²×C₁₃	SL(2,𝔽₃)	1	4	yes	direct product factor
20	C₂²×C₁₃	C₂×A₄	1	3	yes	—
21	C₄×C₁₃	—	3	3	no	—
22	C₄×C₁₃	—	3	1	no	—
23	C₂×SL(2,𝔽₃)	C₂×C₁₃	1	3	yes	direct product factor
24	C₃×C₁₃	—	4	1	no	Hall
25	C₂²×Q₈	C₃×C₁₃	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₁₃	C₂×SL(2,𝔽₃)	1	6	yes	direct product factor
27	C₂×C₁₃	C₂²×A₄	1	1	yes	characteristic
28	C₂³×C₃	—	4	1	no	—
29	SL(2,𝔽₃)	C₂²×C₁₃	1	1	yes	characteristic, direct product factor
30	C₂×Q₈	—	3	3	no	—
31	C₂×Q₈	C₂×C₃×C₁₃	1	3	yes	complemented
32	C₂²×C₄	—	3	1	no	—
33	C₁₃	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
34	C₂²×C₃	—	4	4	no	—
35	C₂²×C₃	—	4	3	no	—
36	C₂³	A₄×C₁₃	1	1	yes	characteristic
37	Q₈	—	3	3	no	—
38	Q₈	—	3	2	no	—
39	Q₈	C₂²×C₃×C₁₃	1	1	yes	characteristic, complemented
40	C₂×C₄	—	3	3	no	—
41	C₂×C₄	—	3	3	no	—
42	C₂×C₃	—	4	6	no	—
43	C₂×C₃	—	4	1	no	—
44	C₂²	C₁₃×SL(2,𝔽₃)	1	4	yes	direct product factor
45	C₂²	C₂×A₄×C₁₃	1	3	yes	—
46	C₄	—	3	3	no	—
47	C₄	—	3	1	no	—
48	C₃	—	4	1	no	Sylow
49	C₂	C₂×C₁₃×SL(2,𝔽₃)	1	6	yes	direct product factor
50	C₂	C₂²×A₄×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂³×C₁₃	A₄
commutator, cosocle, hypocenter	Q₈	C₂²×C₃×C₁₃
Frattini	C₂	C₂²×A₄×C₁₃
Fitting	C₂²×Q₈×C₁₃	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂³×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×SL(2,𝔽₃)	C₂	direct (N × H)
2. C₂²×Q₈×C₁₃	C₃	semidirect
4. C₁₃×SL(2,𝔽₃)	C₂²	direct (N × H)
6. C₂×Q₈×C₁₃	C₂×C₃	semidirect
13. Q₈×C₁₃	C₂²×C₃	semidirect
16. C₂²×SL(2,𝔽₃)	C₁₃	direct (N × H)
19. C₂²×C₁₃	SL(2,𝔽₃)	direct (N × H)
23. C₂×SL(2,𝔽₃)	C₂×C₁₃	direct (N × H)
25. C₂²×Q₈	C₃×C₁₃	semidirect
26. C₂×C₁₃	C₂×SL(2,𝔽₃)	direct (N × H)
29. SL(2,𝔽₃)	C₂²×C₁₃	direct (N × H)
31. C₂×Q₈	C₂×C₃×C₁₃	semidirect
33. C₁₃	C₂²×SL(2,𝔽₃)	direct (N × H)
39. Q₈	C₂²×C₃×C₁₃	semidirect
44. C₂²	C₁₃×SL(2,𝔽₃)	direct (N × H)
49. C₂	C₂×C₁₃×SL(2,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×SL(2,𝔽₃)	2
C₂²×Q₈×C₁₃	3
C₂³×C₃×C₁₃	4
C₂²×SL(2,𝔽₃)	13

Subgroups — C22×C13×SL(2,𝔽3) #11929255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×SL(2,𝔽₃) #11929255