Subgroups — C₃×C₅×C₁₃×#5997 #50304336

Order 38220.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₁₃×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₃×#5997	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₇²×C₁₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₁₃×#5997	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×C₁₃×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic
6	C₃×C₁₃×C₇⋊D₇	C₂×C₅	1	1	yes	characteristic
7	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₃×C₅×#5997	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×C₁₃×D₇	—	14	4	no	—
10	C₁₃×#5997	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
11	C₃×C₇²×C₁₃	C₄×C₅	1	1	yes	characteristic, complemented, Hall
12	C₃×C₅×C₇⋊D₇	C₂×C₁₃	1	1	yes	characteristic
13	C₃×C₅×C₇×C₁₃	—	2	4	no	—
14	C₁₃×C₇⋊D₇	C₂×C₃×C₅	1	1	yes	characteristic
15	C₅×#5997	C₃×C₁₃	1	1	yes	characteristic, direct product factor, Hall
16	C₅×C₁₃×D₇	—	14	4	no	—
17	C₃×C₄×C₅×C₁₃	—	49	1	no	maximal, Hall
18	C₃×C₅×C₇²	C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
19	C₇²×C₁₃	C₃×C₄×C₅	1	1	yes	characteristic, complemented, Hall
20	C₃×#5997	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₁₃×D₇	—	14	4	no	—
22	C₅×C₇⋊D₇	C₂×C₃×C₁₃	1	1	yes	characteristic
23	C₅×C₇×C₁₃	—	2	4	no	—
24	C₂×C₃×C₅×C₁₃	—	49	1	no	—
25	C₃×C₇⋊D₇	C₂×C₅×C₁₃	1	1	yes	characteristic
26	C₃×C₇×C₁₃	—	2	4	no	—
27	C₄×C₅×C₁₃	—	49	1	no	Hall
28	C₅×C₇²	C₃×C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
29	C₃×C₅×D₇	—	14	4	no	—
30	#5997	C₃×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
31	C₃×C₅×C₁₃	#5997	1	1	yes	characteristic, direct product factor, Hall
32	C₁₃×D₇	—	14	4	no	—
33	C₃×C₄×C₁₃	—	49	1	no	Hall
34	C₃×C₇²	C₄×C₅×C₁₃	1	1	yes	characteristic, complemented, Hall
35	C₂×C₅×C₁₃	—	49	1	no	—
36	C₃×C₅×C₇	—	2	4	no	—
37	C₇⋊D₇	C₂×C₃×C₅×C₁₃	1	1	yes	characteristic
38	C₇×C₁₃	—	2	4	no	—
39	C₂×C₃×C₁₃	—	49	1	no	—
40	C₅×D₇	—	14	4	no	—
41	C₅×C₁₃	C₃×#5997	1	1	yes	characteristic, direct product factor, Hall
42	C₃×C₄×C₅	—	49	1	no	Hall
43	C₄×C₁₃	—	49	1	no	Hall
44	C₇²	C₃×C₄×C₅×C₁₃	1	1	yes	characteristic, complemented, Sylow
45	C₃×D₇	—	14	4	no	—
46	C₃×C₁₃	C₅×#5997	1	1	yes	characteristic, direct product factor, Hall
47	C₅×C₇	—	2	4	no	—
48	C₂×C₃×C₅	—	49	1	no	—
49	C₂×C₁₃	—	49	1	no	—
50	C₃×C₇	—	2	4	no	—
51	C₄×C₅	—	49	1	no	Hall
52	C₃×C₅	C₁₃×#5997	1	1	yes	characteristic, direct product factor, Hall
53	D₇	—	14	4	no	—
54	C₁₃	C₃×C₅×#5997	1	1	yes	characteristic, direct product factor, Sylow
55	C₃×C₄	—	49	1	no	Hall
56	C₂×C₅	—	49	1	no	—
57	C₇	—	2	4	no	—
58	C₂×C₃	—	49	1	no	—
59	C₅	C₃×C₁₃×#5997	1	1	yes	characteristic, direct product factor, Sylow
60	C₄	—	49	1	no	Sylow
61	C₃	C₅×C₁₃×#5997	1	1	yes	characteristic, direct product factor, Sylow
62	C₂	—	49	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅×C₁₃	#5997
commutator, hypocenter	C₇²	C₃×C₄×C₅×C₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇²×C₁₃	C₄
prime radical	C₃×C₅×C₁₃×C₇⋊D₇	C₂
cosocle	C₇⋊D₇	C₂×C₃×C₅×C₁₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₃×C₅×C₁₃

Complements

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

N	H ≅ G/N	type
2. C₅×C₁₃×#5997	C₃	direct (N × H)
3. C₃×C₅×C₇²×C₁₃	C₄	semidirect
4. C₃×C₁₃×#5997	C₅	direct (N × H)
7. C₅×C₇²×C₁₃	C₃×C₄	semidirect
8. C₃×C₅×#5997	C₁₃	direct (N × H)
10. C₁₃×#5997	C₃×C₅	direct (N × H)
11. C₃×C₇²×C₁₃	C₄×C₅	semidirect
15. C₅×#5997	C₃×C₁₃	direct (N × H)
18. C₃×C₅×C₇²	C₄×C₁₃	semidirect
19. C₇²×C₁₃	C₃×C₄×C₅	semidirect
20. C₃×#5997	C₅×C₁₃	direct (N × H)
28. C₅×C₇²	C₃×C₄×C₁₃	semidirect
30. #5997	C₃×C₅×C₁₃	direct (N × H)
31. C₃×C₅×C₁₃	#5997	direct (N × H)
34. C₃×C₇²	C₄×C₅×C₁₃	semidirect
41. C₅×C₁₃	C₃×#5997	direct (N × H)
44. C₇²	C₃×C₄×C₅×C₁₃	semidirect
46. C₃×C₁₃	C₅×#5997	direct (N × H)
52. C₃×C₅	C₁₃×#5997	direct (N × H)
54. C₁₃	C₃×C₅×#5997	direct (N × H)
59. C₅	C₃×C₁₃×#5997	direct (N × H)
61. C₃	C₅×C₁₃×#5997	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×C₁₃×C₇⋊D₇	2
C₅×C₁₃×#5997	3
C₃×C₁₃×#5997	5
C₃×C₅×#5997	13
C₃×C₄×C₅×C₁₃	49

Subgroups — C3×C5×C13×#5997 #50304336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₁₃×#5997 #50304336