Subgroups — C₁₃⋊C₃×AGL(1,𝔽₃₁) #49136556

Order 36270.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃⋊C₃×(C₃₁⋊C₃)⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	#14559023⋊C₅	C₃	1	1	yes	maximal, characteristic, complemented
3	#14559022⋊C₅	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃⋊C₃×D₃₁⋊C₅	C₃	1	1	yes	maximal, characteristic, complemented
5	C₁₃×AGL(1,𝔽₃₁)	C₃	1	1	yes	maximal, characteristic, complemented
6	C₁₃⋊C₃×D₃₁⋊C₃	C₅	1	1	yes	maximal, characteristic, complemented, Hall
7	#11925881⋊C₅	C₂×C₃	1	1	yes	characteristic, complemented
8	#11925882⋊C₅	C₂×C₃	1	1	yes	characteristic, complemented
9	C₁₃⋊C₃×ΓL(1,𝔽₃₂)	C₂×C₃	1	1	yes	characteristic, complemented
10	C₁₃×(C₃₁⋊C₃)⋊C₅	C₂×C₃	1	1	yes	characteristic, complemented
11	C₁₃×D₃₁⋊C₅	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₁₃⋊C₃×C₃₁⋊C₃	C₂×C₅	1	1	yes	characteristic, complemented, Hall
13	C₃×AGL(1,𝔽₃₁)	—	13	1	no	maximal, Hall
14	#14559023	C₃×C₅	1	1	yes	characteristic, complemented
15	#14559022	C₃×C₅	1	1	yes	characteristic, complemented
16	C₁₃⋊C₃×D₃₁	C₃×C₅	1	1	yes	characteristic, complemented
17	C₁₃×D₃₁⋊C₃	C₃×C₅	1	1	yes	characteristic, complemented
18	C₁₃×ΓL(1,𝔽₃₂)	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
19	C₃×(C₃₁⋊C₃)⋊C₅	—	13	1	no	Hall
20	#11925882	C₂×C₃×C₅	1	1	yes	characteristic, complemented
21	#11925881	C₂×C₃×C₅	1	1	yes	characteristic, complemented
22	C₁₃×C₃₁⋊C₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
23	C₃₁×C₁₃⋊C₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
24	C₂×C₃×C₅×C₁₃⋊C₃	—	31	1	no	maximal, Hall
25	C₃×D₃₁⋊C₅	—	13	1	no	—
26	AGL(1,𝔽₃₁)	—	13	1	no	—
27	AGL(1,𝔽₃₁)	—	13	1	no	—
28	AGL(1,𝔽₃₁)	C₁₃⋊C₃	1	1	yes	characteristic, direct product factor
29	C₁₃×D₃₁	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
30	C₃×C₅×C₁₃⋊C₃	—	31	1	no	Hall
31	C₃×D₃₁⋊C₃	—	13	1	no	Hall
32	C₃×ΓL(1,𝔽₃₂)	—	13	1	no	—
33	(C₃₁⋊C₃)⋊C₅	—	13	1	no	—
34	(C₃₁⋊C₃)⋊C₅	—	13	1	no	—
35	(C₃₁⋊C₃)⋊C₅	C₂×C₁₃⋊C₃	1	1	yes	characteristic, complemented
36	C₁₃×C₃₁	C₂×C₃²×C₅	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₅×C₁₃	—	31	1	no	—
38	C₂×C₅×C₁₃⋊C₃	—	31	1	no	—
39	C₂×C₅×C₁₃⋊C₃	—	31	1	no	—
40	C₂×C₅×C₁₃⋊C₃	—	31	1	no	—
41	D₃₁⋊C₅	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
42	C₃×C₃₁⋊C₃	—	13	1	no	Hall
43	C₂×C₃×C₁₃⋊C₃	—	31	1	no	Hall
44	C₃×C₅×C₁₃	—	31	1	no	—
45	C₅×C₁₃⋊C₃	—	31	1	no	—
46	C₅×C₁₃⋊C₃	—	31	1	no	—
47	C₅×C₁₃⋊C₃	—	31	1	no	—
48	C₃×D₃₁	—	13	1	no	—
49	D₃₁⋊C₃	—	13	1	no	—
50	D₃₁⋊C₃	—	13	1	no	—
51	D₃₁⋊C₃	C₅×C₁₃⋊C₃	1	1	yes	characteristic, complemented
52	ΓL(1,𝔽₃₂)	C₂×C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
53	C₂×C₅×C₁₃	—	31	1	no	Hall
54	C₃×C₁₃⋊C₃	—	31	1	no	Hall
55	C₃×C₃₁	—	13	1	no	—
56	C₃₁⋊C₃	—	13	1	no	—
57	C₃₁⋊C₃	—	13	1	no	—
58	C₃₁⋊C₃	C₂×C₅×C₁₃⋊C₃	1	1	yes	characteristic, complemented
59	C₂×C₃²×C₅	—	403	1	no	Hall
60	C₂×C₃×C₁₃	—	31	1	no	—
61	C₂×C₁₃⋊C₃	—	31	1	no	—
62	C₂×C₁₃⋊C₃	—	31	1	no	—
63	C₂×C₁₃⋊C₃	—	31	1	no	—
64	C₅×C₁₃	—	31	1	no	Hall
65	D₃₁	C₃×C₅×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
66	C₃²×C₅	—	403	1	no	Hall
67	C₃×C₁₃	—	31	1	no	—
68	C₁₃⋊C₃	—	31	1	no	—
69	C₁₃⋊C₃	—	31	1	no	—
70	C₁₃⋊C₃	AGL(1,𝔽₃₁)	1	1	yes	characteristic, direct product factor
71	C₃₁	C₂×C₃×C₅×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₃×C₅	—	31	1	no	—
73	C₂×C₃×C₅	—	403	1	no	—
74	C₂×C₃×C₅	—	403	1	no	—
75	C₂×C₃×C₅	—	403	1	no	—
76	C₂×C₁₃	—	31	1	no	Hall
77	C₂×C₃²	—	403	1	no	Hall
78	C₃×C₅	—	31	1	no	—
79	C₃×C₅	—	403	1	no	—
80	C₃×C₅	—	403	1	no	—
81	C₃×C₅	—	403	1	no	—
82	C₁₃	C₃×AGL(1,𝔽₃₁)	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₅	—	31	1	no	Hall
84	C₃²	—	403	1	no	Sylow
85	C₂×C₃	—	31	1	no	—
86	C₂×C₃	—	403	1	no	—
87	C₂×C₃	—	403	1	no	—
88	C₂×C₃	—	403	1	no	—
89	C₅	—	31	1	no	Sylow
90	C₃	—	13	1	no	—
91	C₃	—	31	1	no	—
92	C₃	—	403	1	no	—
93	C₃	—	403	1	no	—
94	C₂	—	31	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃×C₃₁	C₂×C₃²×C₅
LCS	G ▷ C₁₃×C₃₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₁₃⋊C₃×(C₃₁⋊C₃)⋊C₅	C₂	semidirect
2. #14559023⋊C₅	C₃	semidirect
3. #14559022⋊C₅	C₃	semidirect
4. C₁₃⋊C₃×D₃₁⋊C₅	C₃	semidirect
5. C₁₃×AGL(1,𝔽₃₁)	C₃	semidirect
6. C₁₃⋊C₃×D₃₁⋊C₃	C₅	semidirect
7. #11925881⋊C₅	C₂×C₃	semidirect
8. #11925882⋊C₅	C₂×C₃	semidirect
9. C₁₃⋊C₃×ΓL(1,𝔽₃₂)	C₂×C₃	semidirect
10. C₁₃×(C₃₁⋊C₃)⋊C₅	C₂×C₃	semidirect
11. C₁₃×D₃₁⋊C₅	C₃²	semidirect
12. C₁₃⋊C₃×C₃₁⋊C₃	C₂×C₅	semidirect
14. #14559023	C₃×C₅	semidirect
15. #14559022	C₃×C₅	semidirect
16. C₁₃⋊C₃×D₃₁	C₃×C₅	semidirect
17. C₁₃×D₃₁⋊C₃	C₃×C₅	semidirect
18. C₁₃×ΓL(1,𝔽₃₂)	C₂×C₃²	semidirect
20. #11925882	C₂×C₃×C₅	semidirect
21. #11925881	C₂×C₃×C₅	semidirect
22. C₁₃×C₃₁⋊C₃	C₂×C₃×C₅	semidirect
23. C₃₁×C₁₃⋊C₃	C₂×C₃×C₅	semidirect
28. AGL(1,𝔽₃₁)	C₁₃⋊C₃	direct (N × H)
29. C₁₃×D₃₁	C₃²×C₅	semidirect
35. (C₃₁⋊C₃)⋊C₅	C₂×C₁₃⋊C₃	semidirect
36. C₁₃×C₃₁	C₂×C₃²×C₅	semidirect
41. D₃₁⋊C₅	C₃×C₁₃⋊C₃	semidirect
51. D₃₁⋊C₃	C₅×C₁₃⋊C₃	semidirect
52. ΓL(1,𝔽₃₂)	C₂×C₃×C₁₃⋊C₃	semidirect
58. C₃₁⋊C₃	C₂×C₅×C₁₃⋊C₃	semidirect
65. D₃₁	C₃×C₅×C₁₃⋊C₃	semidirect
70. C₁₃⋊C₃	AGL(1,𝔽₃₁)	direct (N × H)
71. C₃₁	C₂×C₃×C₅×C₁₃⋊C₃	semidirect
82. C₁₃	C₃×AGL(1,𝔽₃₁)	semidirect

Maximal subgroups

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

subgroup	index
C₁₃⋊C₃×(C₃₁⋊C₃)⋊C₅	2
#14559023⋊C₅	3
#14559022⋊C₅	3
C₁₃⋊C₃×D₃₁⋊C₅	3
C₁₃×AGL(1,𝔽₃₁)	3
C₁₃⋊C₃×D₃₁⋊C₃	5
C₃×AGL(1,𝔽₃₁)	13
C₂×C₃×C₅×C₁₃⋊C₃	31

Subgroups — C13⋊C3×AGL(1,𝔽31) #49136556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃⋊C₃×AGL(1,𝔽₃₁) #49136556