Subgroups — Q₈×C₇⋊SL(2,𝔽₃) #13060784

Order 1344.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇⋊SL(2,𝔽₃)	C₂	1	3	yes	maximal
2	C₇×Q₈²	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×C₇⋊SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
4	C₂×Q₈×ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₇×C₄⋊Q₈	—	3	3	no	—
6	C₄×C₇×Q₈	—	3	1	no	—
7	C₄×C₇×Q₈	C₂×C₃	1	3	yes	—
8	Q₈×SL(2,𝔽₃)	—	7	1	no	maximal, Hall
9	C₇⋊SL(2,𝔽₃)	Q₈	1	1	yes	characteristic, direct product factor
10	Q₈×ΓL(1,𝔽₈)	—	4	4	no	—
11	C₂×C₄×ΓL(1,𝔽₈)	—	4	3	no	—
12	C₂×C₇×Q₈	—	3	2	no	—
13	C₂×C₇×Q₈	A₄	1	1	yes	characteristic
14	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
15	C₇×C₄⋊C₄	—	3	3	no	—
16	C₇×C₄⋊C₄	—	3	3	no	—
17	C₄²×C₇	—	3	3	no	—
18	C₄×SL(2,𝔽₃)	—	7	3	no	—
19	C₂²×ΓL(1,𝔽₈)	—	4	1	no	—
20	C₄×ΓL(1,𝔽₈)	—	4	6	no	—
21	Q₈²	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
22	C₇×Q₈	—	3	1	no	—
23	C₇×Q₈	—	12	2	no	—
24	C₇×Q₈	SL(2,𝔽₃)	1	4	yes	complemented
25	C₇×Q₈	C₃×Q₈	1	1	yes	characteristic, complemented
26	C₂×C₄×C₇	—	3	1	no	—
27	C₂×C₄×C₇	—	3	3	no	—
28	C₂×C₄×C₇	C₂×A₄	1	3	yes	—
29	C₂×C₃×Q₈	—	28	1	no	—
30	C₂×SL(2,𝔽₃)	—	7	1	no	—
31	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
32	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
33	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
34	C₄⋊Q₈	—	3	3	no	—
35	C₄×Q₈	—	3	1	no	—
36	C₄×Q₈	C₂×ΓL(1,𝔽₈)	1	3	yes	—
37	C₂²×C₇	C₂²×A₄	1	1	yes	characteristic
38	C₄×C₇	—	3	1	no	—
39	C₄×C₇	—	3	1	no	—
40	C₄×C₇	—	6	3	no	—
41	C₄×C₇	C₂×SL(2,𝔽₃)	1	6	yes	—
42	C₃×Q₈	—	28	4	no	—
43	C₂×C₃×C₄	—	28	3	no	—
44	SL(2,𝔽₃)	—	7	1	no	—
45	ΓL(1,𝔽₈)	—	4	1	no	Hall
46	C₂×Q₈	—	3	2	no	—
47	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
48	C₂×Q₈	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
49	C₄⋊C₄	—	3	3	no	—
50	C₄⋊C₄	—	3	3	no	—
51	C₄²	—	3	3	no	—
52	C₂×C₇	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic
53	C₂×C₇	Q₈×A₄	1	1	yes	characteristic
54	C₂×C₇	#994	1	1	yes	characteristic
55	C₂²×C₃	—	28	1	no	—
56	C₃×C₄	—	28	6	no	—
57	Q₈	—	3	1	no	—
58	Q₈	—	12	2	no	—
59	Q₈	Q₈×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
60	Q₈	C₇⋊SL(2,𝔽₃)	1	4	yes	direct product factor
61	C₂×C₄	—	3	1	no	—
62	C₂×C₄	—	3	3	no	—
63	C₂×C₄	C₂×C₂²⋊ΓL(1,𝔽₈)	1	3	yes	—
64	C₇	Q₈×SL(2,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	28	1	no	—
66	C₂×C₃	—	28	1	no	—
67	C₂×C₃	—	28	1	no	—
68	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
69	C₄	—	3	1	no	—
70	C₄	—	3	1	no	—
71	C₄	—	6	3	no	—
72	C₄	C₂×C₇⋊SL(2,𝔽₃)	1	6	yes	—
73	C₃	—	28	1	no	Sylow
74	C₂	C₂²×C₇⋊SL(2,𝔽₃)	1	1	yes	characteristic
75	C₂	Q₈×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
76	C₂	#10621615	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle	C₂×C₇×Q₈	C₂²×C₃
Fitting	C₇×Q₈²	C₃
socle	C₂²×C₇	C₂²×A₄
prime radical	C₂×C₇⋊SL(2,𝔽₃)	C₂²
hypercenter	C₂×Q₈	C₂²⋊ΓL(1,𝔽₈)
hypocenter	C₇×Q₈	C₃×Q₈
LCS	G ▷ C₂×C₇×Q₈ ▷ C₇×Q₈
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. C₇×Q₈²	C₃	semidirect
9. C₇⋊SL(2,𝔽₃)	Q₈	direct (N × H)
21. Q₈²	ΓL(1,𝔽₈)	semidirect
24. C₇×Q₈	SL(2,𝔽₃)	semidirect
25. C₇×Q₈	C₃×Q₈	semidirect
59. Q₈	Q₈×ΓL(1,𝔽₈)	semidirect
60. Q₈	C₇⋊SL(2,𝔽₃)	direct (N × H)
64. C₇	Q₈×SL(2,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇⋊SL(2,𝔽₃)	2
C₇×Q₈²	3
C₂×Q₈×ΓL(1,𝔽₈)	4
Q₈×SL(2,𝔽₃)	7

Subgroups — Q8×C7⋊SL(2,𝔽3) #13060784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₇⋊SL(2,𝔽₃) #13060784