Subgroups — C₂×ΓL(1,𝔽₈)×D₂₂₁ #32915554

Order 18564.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₂₂₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₁₃×C₁₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×D₂₂₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₃×C₁₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×D₂₂₁	C₂×C₃	1	2	yes	complemented
6	C₂×C₇×C₁₃×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₃×D₂₂₁	—	7	1	no	maximal, Hall
8	C₇×C₁₃×C₁₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×ΓL(1,𝔽₈)×D₁₇	—	13	1	no	maximal, Hall
10	C₂×C₃×C₁₃×C₁₇	—	7	1	no	—
11	C₃×D₂₂₁	—	7	2	no	—
12	C₂×ΓL(1,𝔽₈)×D₁₃	—	17	1	no	maximal, Hall
13	C₂×D₂₂₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₁₇×ΓL(1,𝔽₈)	D₁₃	1	1	yes	characteristic, complemented
15	ΓL(1,𝔽₈)×D₁₇	—	13	2	no	—
16	C₃×C₁₃×C₁₇	—	7	1	no	Hall
17	C₂×C₁₃×ΓL(1,𝔽₈)	D₁₇	1	1	yes	characteristic, complemented
18	ΓL(1,𝔽₈)×D₁₃	—	17	2	no	—
19	C₂×C₇×D₁₇	—	13	1	no	Hall
20	C₂×C₁₃×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	D₂₂₁	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
22	C₂×C₇×D₁₃	—	17	1	no	Hall
23	C₁₇×ΓL(1,𝔽₈)	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
24	C₁₃×ΓL(1,𝔽₈)	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
25	C₂×C₇×C₁₇	C₃×D₁₃	1	1	yes	characteristic, complemented
26	C₇×D₁₇	—	13	2	no	—
27	C₁₃×C₁₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
28	C₂×C₃×D₁₇	—	91	1	no	Hall
29	C₂×C₇×C₁₃	C₃×D₁₇	1	1	yes	characteristic, complemented
30	C₇×D₁₃	—	17	2	no	—
31	C₂×C₃×D₁₃	—	119	1	no	Hall
32	C₇×C₁₇	C₂×C₃×D₁₃	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₁₇	—	7	1	no	—
34	C₃×D₁₇	—	91	2	no	—
35	C₇×C₁₃	C₂×C₃×D₁₇	1	1	yes	characteristic, complemented, Hall
36	C₂²×ΓL(1,𝔽₈)	—	221	1	no	Hall
37	C₂×C₃×C₁₃	—	7	1	no	—
38	C₃×D₁₃	—	119	2	no	—
39	C₂×D₁₇	—	13	1	no	Hall
40	C₂×D₁₃	—	17	1	no	Hall
41	C₃×C₁₇	—	7	1	no	Hall
42	C₂×ΓL(1,𝔽₈)	—	221	2	no	—
43	C₂×ΓL(1,𝔽₈)	D₂₂₁	1	1	yes	characteristic, direct product factor
44	C₃×C₁₃	—	7	1	no	Hall
45	C₂×C₁₇	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
46	D₁₇	—	13	2	no	—
47	C₂²×C₇	—	221	1	no	Hall
48	C₂×C₁₃	ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, complemented
49	D₁₃	—	17	2	no	—
50	ΓL(1,𝔽₈)	C₂×D₂₂₁	1	1	yes	characteristic, direct product factor, Hall
51	C₁₇	C₂×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₇	—	221	2	no	—
53	C₂×C₇	C₃×D₂₂₁	1	1	yes	characteristic, complemented
54	C₁₃	C₂×ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₃	—	1547	1	no	Hall
56	C₇	C₂×C₃×D₂₂₁	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃	—	7	1	no	—
58	C₂×C₃	—	1547	2	no	—
59	C₂²	—	221	1	no	Sylow
60	C₃	—	7	1	no	Sylow
61	C₂	—	221	2	no	—
62	C₂	ΓL(1,𝔽₈)×D₂₂₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)×D₂₂₁
commutator, cosocle, hypocenter	C₇×C₁₃×C₁₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₃×C₁₇	C₂×C₃
LCS	G ▷ C₇×C₁₃×C₁₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₂₂₁	C₂	direct (N × H)
2. C₂×C₁₃×C₁₇×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂×C₇×D₂₂₁	C₃	semidirect
4. C₁₃×C₁₇×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₇×D₂₂₁	C₂×C₃	semidirect
6. C₂×C₇×C₁₃×C₁₇	C₂×C₃	semidirect
8. C₇×C₁₃×C₁₇	C₂²×C₃	semidirect
13. C₂×D₂₂₁	ΓL(1,𝔽₈)	direct (N × H)
14. C₂×C₁₇×ΓL(1,𝔽₈)	D₁₃	semidirect
17. C₂×C₁₃×ΓL(1,𝔽₈)	D₁₇	semidirect
20. C₂×C₁₃×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
21. D₂₂₁	C₂×ΓL(1,𝔽₈)	direct (N × H)
23. C₁₇×ΓL(1,𝔽₈)	C₂×D₁₃	semidirect
24. C₁₃×ΓL(1,𝔽₈)	C₂×D₁₇	semidirect
25. C₂×C₇×C₁₇	C₃×D₁₃	semidirect
27. C₁₃×C₁₇	C₂²×ΓL(1,𝔽₈)	semidirect
29. C₂×C₇×C₁₃	C₃×D₁₇	semidirect
32. C₇×C₁₇	C₂×C₃×D₁₃	semidirect
35. C₇×C₁₃	C₂×C₃×D₁₇	semidirect
43. C₂×ΓL(1,𝔽₈)	D₂₂₁	direct (N × H)
45. C₂×C₁₇	ΓL(1,𝔽₈)×D₁₃	semidirect
48. C₂×C₁₃	ΓL(1,𝔽₈)×D₁₇	semidirect
50. ΓL(1,𝔽₈)	C₂×D₂₂₁	direct (N × H)
51. C₁₇	C₂×ΓL(1,𝔽₈)×D₁₃	semidirect
53. C₂×C₇	C₃×D₂₂₁	semidirect
54. C₁₃	C₂×ΓL(1,𝔽₈)×D₁₇	semidirect
56. C₇	C₂×C₃×D₂₂₁	semidirect
62. C₂	ΓL(1,𝔽₈)×D₂₂₁	direct (N × H)

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×D₂₂₁	2
C₂×C₁₃×C₁₇×ΓL(1,𝔽₈)	2
C₂×C₇×D₂₂₁	3
C₂×C₃×D₂₂₁	7
C₂×ΓL(1,𝔽₈)×D₁₇	13
C₂×ΓL(1,𝔽₈)×D₁₃	17

Subgroups — C2×ΓL(1,𝔽8)×D221 #32915554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×D₂₂₁ #32915554