Subgroups — C₂³×C₇×SL(2,𝔽₃) #13062303

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×C₃×C₇	—	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)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₃×C₇	—	4	8	no	—
10	C₂³×C₃×C₇	—	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)	C₂×C₇	1	7	yes	direct product factor
19	C₂²×C₃×C₇	—	4	28	no	—
20	C₂²×C₃×C₇	—	4	7	no	—
21	C₂³×Q₈	C₃×C₇	1	1	yes	characteristic, complemented, Sylow
22	C₂³×C₇	SL(2,𝔽₃)	1	8	yes	direct product factor
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	—	4	1	no	—
30	C₂×SL(2,𝔽₃)	C₂²×C₇	1	7	yes	direct product factor
31	C₂×C₃×C₇	—	4	14	no	—
32	C₂×C₃×C₇	—	4	1	no	—
33	C₂²×Q₈	—	3	7	no	—
34	C₂²×Q₈	C₂×C₃×C₇	1	7	yes	complemented
35	C₂³×C₄	—	3	1	no	—
36	C₂²×C₇	C₂×SL(2,𝔽₃)	1	28	yes	direct product factor
37	C₂²×C₇	C₂²×A₄	1	7	yes	—
38	C₄×C₇	—	3	7	no	—
39	C₄×C₇	—	3	1	no	—
40	C₂³×C₃	—	4	8	no	—
41	C₂³×C₃	—	4	7	no	—
42	SL(2,𝔽₃)	C₂³×C₇	1	1	yes	characteristic, direct product factor
43	C₃×C₇	—	4	1	no	Hall
44	C₂⁴	C₇×A₄	1	1	yes	characteristic
45	C₂×Q₈	—	3	14	no	—
46	C₂×Q₈	—	3	21	no	—
47	C₂×Q₈	C₂²×C₃×C₇	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	direct product factor
51	C₂×C₇	C₂³×A₄	1	1	yes	characteristic
52	C₂²×C₃	—	4	28	no	—
53	C₂²×C₃	—	4	7	no	—
54	C₂³	C₇×SL(2,𝔽₃)	1	8	yes	direct product factor
55	C₂³	C₂×C₇×A₄	1	7	yes	—
56	Q₈	—	3	7	no	—
57	Q₈	—	3	14	no	—
58	Q₈	C₂³×C₃×C₇	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, direct product factor, Sylow
62	C₂×C₃	—	4	14	no	—
63	C₂×C₃	—	4	1	no	—
64	C₂²	C₂×C₇×SL(2,𝔽₃)	1	28	yes	direct product factor
65	C₂²	C₂²×C₇×A₄	1	7	yes	—
66	C₄	—	3	1	no	—
67	C₄	—	3	7	no	—
68	C₃	—	4	1	no	Sylow
69	C₂	C₂²×C₇×SL(2,𝔽₃)	1	14	yes	direct product factor
70	C₂	C₂³×C₇×A₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂⁴×C₇	A₄
commutator, cosocle, hypocenter	Q₈	C₂³×C₃×C₇
Frattini	C₂	C₂³×C₇×A₄
Fitting	C₂³×C₇×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂⁴×C₇

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
8. C₂³×SL(2,𝔽₃)	C₇	direct (N × H)
11. C₇×SL(2,𝔽₃)	C₂³	direct (N × H)
15. C₂×C₇×Q₈	C₂²×C₃	semidirect
18. C₂²×SL(2,𝔽₃)	C₂×C₇	direct (N × H)
21. C₂³×Q₈	C₃×C₇	semidirect
22. C₂³×C₇	SL(2,𝔽₃)	direct (N × H)
26. C₇×Q₈	C₂³×C₃	semidirect
30. C₂×SL(2,𝔽₃)	C₂²×C₇	direct (N × H)
34. C₂²×Q₈	C₂×C₃×C₇	semidirect
36. C₂²×C₇	C₂×SL(2,𝔽₃)	direct (N × H)
42. SL(2,𝔽₃)	C₂³×C₇	direct (N × H)
47. C₂×Q₈	C₂²×C₃×C₇	semidirect
50. C₂×C₇	C₂²×SL(2,𝔽₃)	direct (N × H)
54. C₂³	C₇×SL(2,𝔽₃)	direct (N × H)
58. Q₈	C₂³×C₃×C₇	semidirect
61. C₇	C₂³×SL(2,𝔽₃)	direct (N × H)
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×C₃×C₇	4
C₂³×SL(2,𝔽₃)	7

Subgroups — C23×C7×SL(2,𝔽3) #13062303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×SL(2,𝔽₃) #13062303