Subgroups — C₂×C₁₃⋊GL(2,𝔽₃) #11929123

Order 1248.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃⋊GL(2,𝔽₃)	C₂	1	2	yes	maximal, direct product factor
3	C₂×#6106	—	3	1	no	maximal, Hall
4	C₂²×D₃₉	—	4	1	no	maximal
5	C₁₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
6	C₂×Q₈×C₁₃	D₃	1	1	yes	characteristic, complemented
7	C₂×D₅₂	—	3	1	no	—
8	#6106	—	3	2	no	—
9	#6106	—	3	2	no	—
10	C₂×#1055	—	3	1	no	—
11	C₂²×C₃×C₁₃	—	4	1	no	—
12	C₂×D₃₉	—	4	2	no	—
13	C₂×D₃₉	—	4	4	no	—
14	C₂²×D₁₃	—	6	1	no	—
15	Q₈×C₁₃	—	3	1	no	—
16	Q₈×C₁₃	C₂×D₃	1	1	yes	characteristic, complemented
17	C₂×C₄×C₁₃	—	3	1	no	—
18	D₅₂	—	3	2	no	—
19	D₅₂	—	6	1	no	—
20	#1055	—	3	2	no	—
21	C₂×GL(2,𝔽₃)	—	13	1	no	maximal, Hall
22	C₂×C₃×C₁₃	—	4	2	no	—
23	C₂×C₃×C₁₃	—	4	1	no	—
24	D₃₉	—	4	4	no	—
25	C₂²×C₁₃	S₄	1	1	yes	characteristic
26	C₂×D₁₃	—	6	2	no	—
27	C₂×D₁₃	—	12	2	no	—
28	C₄×C₁₃	—	3	1	no	—
29	C₄×C₁₃	—	3	1	no	—
30	C₂×SL(2,𝔽₃)	D₁₃	1	1	yes	characteristic, complemented
31	GL(2,𝔽₃)	—	13	2	no	—
32	C₃×C₁₃	—	4	1	no	Hall
33	C₂×SD₁₆	—	39	1	no	Sylow
34	C₂×C₁₃	GL(2,𝔽₃)	1	2	yes	complemented
35	C₂×C₁₃	C₂×S₄	1	1	yes	characteristic
36	D₁₃	—	12	2	no	—
37	C₂²×D₃	—	52	1	no	—
38	SL(2,𝔽₃)	C₂×D₁₃	1	1	yes	characteristic, complemented
39	C₂×Q₈	D₃₉	1	1	yes	characteristic, complemented
40	C₂×D₄	—	39	1	no	—
41	SD₁₆	—	39	2	no	—
42	SD₁₆	—	39	2	no	—
43	C₂×C₈	—	39	1	no	—
44	C₁₃	C₂×GL(2,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₃	—	4	1	no	—
46	C₂×D₃	—	52	2	no	—
47	C₂×D₃	—	52	4	no	—
48	C₂³	—	78	1	no	—
49	Q₈	—	3	1	no	—
50	Q₈	C₂×D₃₉	1	1	yes	characteristic, complemented
51	D₄	—	39	2	no	—
52	D₄	—	78	1	no	—
53	C₂×C₄	—	3	1	no	—
54	C₈	—	39	2	no	—
55	C₂×C₃	—	4	2	no	—
56	C₂×C₃	—	4	1	no	—
57	D₃	—	52	4	no	—
58	C₂²	—	78	2	no	—
59	C₂²	—	156	2	no	—
60	C₂²	A₄⋊D₁₃	1	1	yes	characteristic
61	C₄	—	3	1	no	—
62	C₄	—	3	1	no	—
63	C₃	—	4	1	no	Sylow
64	C₂	—	156	2	no	—
65	C₂	C₁₃⋊GL(2,𝔽₃)	1	2	yes	direct product factor
66	C₂	C₂×A₄⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	A₄⋊D₁₃
commutator, cosocle, hypocenter	C₁₃×SL(2,𝔽₃)	C₂²
Frattini	C₂	C₂×A₄⋊D₁₃
Fitting	C₂×Q₈×C₁₃	D₃
socle	C₂²×C₁₃	S₄
LCS	G ▷ C₁₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×SL(2,𝔽₃)	C₂	semidirect
2. C₁₃⋊GL(2,𝔽₃)	C₂	direct (N × H)
5. C₁₃×SL(2,𝔽₃)	C₂²	semidirect
6. C₂×Q₈×C₁₃	D₃	semidirect
16. Q₈×C₁₃	C₂×D₃	semidirect
30. C₂×SL(2,𝔽₃)	D₁₃	semidirect
34. C₂×C₁₃	GL(2,𝔽₃)	semidirect
38. SL(2,𝔽₃)	C₂×D₁₃	semidirect
39. C₂×Q₈	D₃₉	semidirect
44. C₁₃	C₂×GL(2,𝔽₃)	semidirect
50. Q₈	C₂×D₃₉	semidirect
65. C₂	C₁₃⋊GL(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
C₁₃⋊GL(2,𝔽₃)	2
C₂×#6106	3
C₂²×D₃₉	4
C₂×GL(2,𝔽₃)	13

Subgroups — C2×C13⋊GL(2,𝔽3) #11929123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃⋊GL(2,𝔽₃) #11929123