Subgroups — C₈×C₇⋊SL(2,𝔽₃) #13055074

Order 1344.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇⋊SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₇×C₈×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×C₇⋊SL(2,𝔽₃)	C₄	1	1	yes	characteristic
4	C₂×C₈×ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
6	C₇×C₄⋊C₈	—	3	1	no	—
7	C₄×C₇×C₈	—	3	1	no	—
8	C₈×SL(2,𝔽₃)	—	7	1	no	maximal, Hall
9	C₇⋊SL(2,𝔽₃)	C₈	1	1	yes	characteristic, direct product factor
10	C₂×C₄×ΓL(1,𝔽₈)	—	4	1	no	—
11	C₈×ΓL(1,𝔽₈)	—	4	2	no	—
12	C₂×C₇×Q₈	C₃×C₄	1	1	yes	characteristic
13	C₂×C₇×C₈	—	3	1	no	—
14	C₂×C₇×C₈	A₄	1	1	yes	characteristic
15	C₇×C₄⋊C₄	—	3	1	no	—
16	C₄²×C₇	—	3	1	no	—
17	C₄×SL(2,𝔽₃)	—	7	1	no	—
18	C₂²×ΓL(1,𝔽₈)	—	4	1	no	—
19	C₄×ΓL(1,𝔽₈)	—	4	1	no	—
20	C₄×ΓL(1,𝔽₈)	—	4	1	no	—
21	C₈×Q₈	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
22	C₇×Q₈	—	3	1	no	—
23	C₇×Q₈	C₃×C₈	1	1	yes	characteristic, complemented
24	C₂×C₄×C₇	—	3	1	no	—
25	C₂×C₄×C₇	—	3	1	no	—
26	C₂×C₄×C₇	C₂×A₄	1	1	yes	characteristic
27	C₇×C₈	—	6	1	no	—
28	C₇×C₈	SL(2,𝔽₃)	1	2	yes	complemented
29	C₂×SL(2,𝔽₃)	—	7	1	no	—
30	C₂×C₃×C₈	—	28	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	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
35	C₄⋊C₈	—	3	1	no	—
36	C₄×C₈	—	3	1	no	—
37	C₂²×C₇	C₄×A₄	1	1	yes	characteristic
38	C₄×C₇	—	3	1	no	—
39	C₄×C₇	—	3	1	no	—
40	C₄×C₇	—	6	1	no	—
41	C₄×C₇	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
42	C₄×C₇	#41⋊C₃	1	1	yes	characteristic
43	C₂×C₃×C₄	—	28	1	no	—
44	SL(2,𝔽₃)	—	7	1	no	—
45	C₃×C₈	—	28	2	no	—
46	ΓL(1,𝔽₈)	—	4	1	no	Hall
47	C₂×Q₈	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
48	C₂×C₈	—	3	1	no	—
49	C₂×C₈	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
50	C₄⋊C₄	—	3	1	no	—
51	C₄²	—	3	1	no	—
52	C₂×C₇	C₄×SL(2,𝔽₃)	1	1	yes	characteristic
53	C₂×C₇	C₈×A₄	1	1	yes	characteristic
54	C₂×C₇	#131⋊C₃	1	1	yes	characteristic
55	C₂²×C₃	—	28	1	no	—
56	C₃×C₄	—	28	1	no	—
57	C₃×C₄	—	28	1	no	—
58	Q₈	—	3	1	no	—
59	Q₈	C₈×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
60	C₂×C₄	—	3	1	no	—
61	C₂×C₄	—	3	1	no	—
62	C₂×C₄	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
63	C₈	—	6	1	no	—
64	C₈	C₇⋊SL(2,𝔽₃)	1	2	yes	direct product factor
65	C₇	C₈×SL(2,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	28	1	no	—
67	C₂×C₃	—	28	1	no	—
68	C₂×C₃	—	28	1	no	—
69	C₂²	C₄×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
70	C₄	—	3	1	no	—
71	C₄	—	3	1	no	—
72	C₄	—	6	1	no	—
73	C₄	C₂×C₇⋊SL(2,𝔽₃)	1	1	yes	characteristic
74	C₄	#41⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
75	C₃	—	28	1	no	Sylow
76	C₂	C₄×C₇⋊SL(2,𝔽₃)	1	1	yes	characteristic
77	C₂	C₈×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
78	C₂	#131⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₈	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₇×Q₈	C₃×C₈
Frattini	C₂×C₄	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×C₈×Q₈	C₃
socle	C₂²×C₇	C₄×A₄
prime radical	C₂×C₇⋊SL(2,𝔽₃)	C₄
cosocle	C₄×C₇×Q₈	C₂×C₃
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₇×C₈×Q₈	C₃	semidirect
9. C₇⋊SL(2,𝔽₃)	C₈	direct (N × H)
21. C₈×Q₈	ΓL(1,𝔽₈)	semidirect
23. C₇×Q₈	C₃×C₈	semidirect
28. C₇×C₈	SL(2,𝔽₃)	semidirect
59. Q₈	C₈×ΓL(1,𝔽₈)	semidirect
64. C₈	C₇⋊SL(2,𝔽₃)	direct (N × H)
65. C₇	C₈×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×C₈×Q₈	3
C₂×C₈×ΓL(1,𝔽₈)	4
C₈×SL(2,𝔽₃)	7

Subgroups — C8×C7⋊SL(2,𝔽3) #13055074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₇⋊SL(2,𝔽₃) #13055074