Subgroups — C₂²×C₃₁×#63271 #46799628

Order 33852.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#63271	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₇×C₁₃×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₃₁×#63271	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₇×C₁₃×C₃₁	C₂×C₃	1	3	yes	complemented
5	C₂²×C₃₁×C₁₃⋊C₃	—	7	1	no	maximal, Hall
6	C₇×C₁₃×C₃₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
7	C₂²×ΓL(1,𝔽₈)×C₃₁	—	13	1	no	maximal, Hall
8	C₂×C₃₁×C₁₃⋊C₃	—	7	3	no	—
9	C₂²×C₁₃×C₃₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
10	C₂×ΓL(1,𝔽₈)×C₃₁	—	13	3	no	—
11	C₃₁×C₁₃⋊C₃	—	7	1	no	Hall
12	C₂²×#63271	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₇×C₃₁	C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
14	C₂×C₁₃×C₃₁	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
15	ΓL(1,𝔽₈)×C₃₁	—	13	1	no	Hall
16	C₂×#63271	C₂×C₃₁	1	3	yes	direct product factor
17	C₂×C₇×C₃₁	C₂×C₁₃⋊C₃	1	3	yes	complemented
18	C₁₃×C₃₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
19	C₂²×C₃×C₃₁	—	91	1	no	Hall
20	C₂²×C₇×C₁₃	C₃×C₃₁	1	1	yes	characteristic, complemented, Hall
21	#63271	C₂²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
22	C₇×C₃₁	C₂²×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
23	C₂×C₃×C₃₁	—	91	3	no	—
24	C₂×C₇×C₁₃	C₂×C₃×C₃₁	1	3	yes	complemented
25	C₂²×C₁₃⋊C₃	—	7	1	no	Hall
26	C₂²×C₃₁	#63271	1	1	yes	characteristic, direct product factor, Hall
27	C₃×C₃₁	—	91	1	no	Hall
28	C₇×C₁₃	C₂²×C₃×C₃₁	1	1	yes	characteristic, complemented, Hall
29	C₂²×ΓL(1,𝔽₈)	—	13	1	no	Hall
30	C₂×C₁₃⋊C₃	—	7	3	no	—
31	C₂×C₃₁	C₂×#63271	1	3	yes	direct product factor
32	C₂²×C₁₃	ΓL(1,𝔽₈)×C₃₁	1	1	yes	characteristic, complemented, Hall
33	C₂×ΓL(1,𝔽₈)	—	13	3	no	—
34	C₁₃⋊C₃	—	7	1	no	Hall
35	C₃₁	C₂²×#63271	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×C₇	C₃₁×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
37	C₂×C₁₃	C₂×ΓL(1,𝔽₈)×C₃₁	1	3	yes	complemented
38	ΓL(1,𝔽₈)	—	13	1	no	Hall
39	C₂×C₇	C₂×C₃₁×C₁₃⋊C₃	1	3	yes	complemented
40	C₁₃	C₂²×ΓL(1,𝔽₈)×C₃₁	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₃	—	91	1	no	Hall
42	C₇	C₂²×C₃₁×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	91	3	no	—
44	C₂²	C₃₁×#63271	1	1	yes	characteristic, direct product factor, Sylow
45	C₃	—	91	1	no	Sylow
46	C₂	C₂×C₃₁×#63271	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃₁	#63271
commutator, cosocle, hypocenter	C₇×C₁₃	C₂²×C₃×C₃₁
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₁₃×C₃₁	C₃
LCS	G ▷ C₇×C₁₃
UCS	C₁ ◁ C₂²×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#63271	C₂	direct (N × H)
2. C₂²×C₇×C₁₃×C₃₁	C₃	semidirect
3. C₃₁×#63271	C₂²	direct (N × H)
4. C₂×C₇×C₁₃×C₃₁	C₂×C₃	semidirect
6. C₇×C₁₃×C₃₁	C₂²×C₃	semidirect
9. C₂²×C₁₃×C₃₁	ΓL(1,𝔽₈)	semidirect
12. C₂²×#63271	C₃₁	direct (N × H)
13. C₂²×C₇×C₃₁	C₁₃⋊C₃	semidirect
14. C₂×C₁₃×C₃₁	C₂×ΓL(1,𝔽₈)	semidirect
16. C₂×#63271	C₂×C₃₁	direct (N × H)
17. C₂×C₇×C₃₁	C₂×C₁₃⋊C₃	semidirect
18. C₁₃×C₃₁	C₂²×ΓL(1,𝔽₈)	semidirect
20. C₂²×C₇×C₁₃	C₃×C₃₁	semidirect
21. #63271	C₂²×C₃₁	direct (N × H)
22. C₇×C₃₁	C₂²×C₁₃⋊C₃	semidirect
24. C₂×C₇×C₁₃	C₂×C₃×C₃₁	semidirect
26. C₂²×C₃₁	#63271	direct (N × H)
28. C₇×C₁₃	C₂²×C₃×C₃₁	semidirect
31. C₂×C₃₁	C₂×#63271	direct (N × H)
32. C₂²×C₁₃	ΓL(1,𝔽₈)×C₃₁	semidirect
35. C₃₁	C₂²×#63271	direct (N × H)
36. C₂²×C₇	C₃₁×C₁₃⋊C₃	semidirect
37. C₂×C₁₃	C₂×ΓL(1,𝔽₈)×C₃₁	semidirect
39. C₂×C₇	C₂×C₃₁×C₁₃⋊C₃	semidirect
40. C₁₃	C₂²×ΓL(1,𝔽₈)×C₃₁	semidirect
42. C₇	C₂²×C₃₁×C₁₃⋊C₃	semidirect
44. C₂²	C₃₁×#63271	direct (N × H)
46. C₂	C₂×C₃₁×#63271	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#63271	2
C₂²×C₇×C₁₃×C₃₁	3
C₂²×C₃₁×C₁₃⋊C₃	7
C₂²×ΓL(1,𝔽₈)×C₃₁	13
C₂²×#63271	31

Subgroups — C22×C31×#63271 #46799628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#63271 #46799628