Subgroups — C₂³×C₇⋊SL(2,𝔽₃) #13062312

Order 1344.

560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇⋊SL(2,𝔽₃)	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₇×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂⁴×ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₂×C₇⋊SL(2,𝔽₃)	C₂²	1	7	yes	direct product factor
5	C₂²×C₇×Q₈	—	3	7	no	—
6	C₂²×C₇×Q₈	C₂×C₃	1	7	yes	complemented
7	C₂³×C₄×C₇	—	3	1	no	—
8	C₂³×SL(2,𝔽₃)	—	7	1	no	maximal, Hall
9	C₂³×ΓL(1,𝔽₈)	—	4	8	no	—
10	C₂³×ΓL(1,𝔽₈)	—	4	7	no	—
11	C₇⋊SL(2,𝔽₃)	C₂³	1	1	yes	characteristic, direct product factor
12	C₂⁴×C₇	A₄	1	1	yes	characteristic
13	C₂×C₇×Q₈	—	3	14	no	—
14	C₂×C₇×Q₈	—	3	21	no	—
15	C₂×C₇×Q₈	C₂²×C₃	1	7	yes	complemented
16	C₂²×C₄×C₇	—	3	7	no	—
17	C₂²×C₄×C₇	—	3	7	no	—
18	C₂²×SL(2,𝔽₃)	—	7	7	no	—
19	C₂²×ΓL(1,𝔽₈)	—	4	28	no	—
20	C₂²×ΓL(1,𝔽₈)	—	4	7	no	—
21	C₂³×Q₈	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
22	C₂³×C₇	SL(2,𝔽₃)	1	8	yes	complemented
23	C₂³×C₇	C₂×A₄	1	7	yes	—
24	C₇×Q₈	—	3	7	no	—
25	C₇×Q₈	—	3	14	no	—
26	C₇×Q₈	C₂³×C₃	1	1	yes	characteristic, complemented
27	C₂×C₄×C₇	—	3	21	no	—
28	C₂×C₄×C₇	—	3	7	no	—
29	C₂⁴×C₃	—	28	1	no	—
30	C₂×SL(2,𝔽₃)	—	7	7	no	—
31	C₂×ΓL(1,𝔽₈)	—	4	14	no	—
32	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
33	C₂²×Q₈	—	3	7	no	—
34	C₂²×Q₈	C₂×ΓL(1,𝔽₈)	1	7	yes	complemented
35	C₂³×C₄	—	3	1	no	—
36	C₂²×C₇	C₂×SL(2,𝔽₃)	1	28	yes	complemented
37	C₂²×C₇	C₂²×A₄	1	7	yes	—
38	C₄×C₇	—	3	7	no	—
39	C₄×C₇	—	3	1	no	—
40	C₂³×C₃	—	28	8	no	—
41	C₂³×C₃	—	28	7	no	—
42	SL(2,𝔽₃)	—	7	1	no	—
43	ΓL(1,𝔽₈)	—	4	1	no	Hall
44	C₂⁴	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
45	C₂×Q₈	—	3	14	no	—
46	C₂×Q₈	—	3	21	no	—
47	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	7	yes	complemented
48	C₂²×C₄	—	3	7	no	—
49	C₂²×C₄	—	3	7	no	—
50	C₂×C₇	C₂²×SL(2,𝔽₃)	1	14	yes	complemented
51	C₂×C₇	C₂³×A₄	1	1	yes	characteristic
52	C₂²×C₃	—	28	28	no	—
53	C₂²×C₃	—	28	7	no	—
54	C₂³	C₇⋊SL(2,𝔽₃)	1	8	yes	direct product factor
55	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	7	yes	—
56	Q₈	—	3	7	no	—
57	Q₈	—	3	14	no	—
58	Q₈	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
59	C₂×C₄	—	3	21	no	—
60	C₂×C₄	—	3	7	no	—
61	C₇	C₂³×SL(2,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₃	—	28	14	no	—
63	C₂×C₃	—	28	1	no	—
64	C₂²	C₂×C₇⋊SL(2,𝔽₃)	1	28	yes	direct product factor
65	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	7	yes	—
66	C₄	—	3	1	no	—
67	C₄	—	3	7	no	—
68	C₃	—	28	1	no	Sylow
69	C₂	C₂²×C₇⋊SL(2,𝔽₃)	1	14	yes	direct product factor
70	C₂	C₂³×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇×Q₈	C₂³×C₃
Frattini	C₂	C₂³×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₂³×C₇×Q₈	C₃
socle	C₂⁴×C₇	A₄
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇⋊SL(2,𝔽₃)	C₂	direct (N × H)
2. C₂³×C₇×Q₈	C₃	semidirect
4. C₂×C₇⋊SL(2,𝔽₃)	C₂²	direct (N × H)
6. C₂²×C₇×Q₈	C₂×C₃	semidirect
11. C₇⋊SL(2,𝔽₃)	C₂³	direct (N × H)
15. C₂×C₇×Q₈	C₂²×C₃	semidirect
21. C₂³×Q₈	ΓL(1,𝔽₈)	semidirect
22. C₂³×C₇	SL(2,𝔽₃)	semidirect
26. C₇×Q₈	C₂³×C₃	semidirect
34. C₂²×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
36. C₂²×C₇	C₂×SL(2,𝔽₃)	semidirect
47. C₂×Q₈	C₂²×ΓL(1,𝔽₈)	semidirect
50. C₂×C₇	C₂²×SL(2,𝔽₃)	semidirect
54. C₂³	C₇⋊SL(2,𝔽₃)	direct (N × H)
58. Q₈	C₂³×ΓL(1,𝔽₈)	semidirect
61. C₇	C₂³×SL(2,𝔽₃)	semidirect
64. C₂²	C₂×C₇⋊SL(2,𝔽₃)	direct (N × H)
69. C₂	C₂²×C₇⋊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
C₂²×C₇⋊SL(2,𝔽₃)	2
C₂³×C₇×Q₈	3
C₂⁴×ΓL(1,𝔽₈)	4
C₂³×SL(2,𝔽₃)	7

Subgroups — C23×C7⋊SL(2,𝔽3) #13062312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇⋊SL(2,𝔽₃) #13062312