Subgroups — C₂×C₅×D₇×C₃₁⋊C₃ #27189201

Order 13020.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₇×C₃₁⋊C₃	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₅×C₇×C₃₁⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₇×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅×C₇×C₃₁⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×D₇×C₃₁⋊C₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₅×D₇×C₃₁	C₂×C₃	1	2	yes	complemented
7	C₂×C₅×C₇×C₃₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂²×C₅×C₃₁⋊C₃	—	7	1	no	maximal, Hall
9	C₂×C₇×C₃₁⋊C₃	C₂×C₅	1	1	yes	characteristic, complemented
10	D₇×C₃₁⋊C₃	C₂×C₅	1	2	yes	direct product factor
11	C₅×C₇×C₃₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×C₅×C₃₁⋊C₃	—	7	2	no	—
13	C₂×C₅×C₃₁⋊C₃	D₇	1	1	yes	characteristic, direct product factor
14	C₂×D₇×C₃₁	C₃×C₅	1	1	yes	characteristic, complemented, Hall
15	C₇×C₃₁⋊C₃	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
16	C₂²×C₅×C₃₁	—	7	1	no	Hall
17	C₅×C₃₁⋊C₃	C₂×D₇	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₇×C₃₁	C₂×C₃×C₅	1	1	yes	characteristic, complemented
19	D₇×C₃₁	C₂×C₃×C₅	1	2	yes	complemented
20	C₂×C₃×C₅×D₇	—	31	1	no	maximal, Hall
21	C₂²×C₃₁⋊C₃	—	7	1	no	Hall
22	C₂×C₅×C₃₁	—	7	2	no	—
23	C₂×C₅×C₃₁	C₃×D₇	1	1	yes	characteristic, complemented
24	C₇×C₃₁	C₂²×C₃×C₅	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₅×C₇	—	31	1	no	—
26	C₃×C₅×D₇	—	31	2	no	—
27	C₂×C₃₁⋊C₃	—	7	2	no	—
28	C₂×C₃₁⋊C₃	C₅×D₇	1	1	yes	characteristic, direct product factor
29	C₅×C₃₁	C₂×C₃×D₇	1	1	yes	characteristic, complemented, Hall
30	C₂×C₅×D₇	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor, Hall
31	C₂²×C₃₁	—	7	1	no	Hall
32	C₃×C₅×C₇	—	31	1	no	Hall
33	C₃₁⋊C₃	C₂×C₅×D₇	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₃×D₇	—	31	1	no	Hall
35	C₂×C₅×C₇	C₂×C₃₁⋊C₃	1	1	yes	characteristic, complemented
36	C₅×D₇	C₂×C₃₁⋊C₃	1	2	yes	direct product factor
37	C₂×C₃₁	—	7	2	no	—
38	C₂×C₃₁	C₃×C₅×D₇	1	1	yes	characteristic, complemented
39	C₂²×C₃×C₅	—	217	1	no	Hall
40	C₂×C₃×C₇	—	31	1	no	—
41	C₃×D₇	—	31	2	no	—
42	C₅×C₇	C₂²×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
43	C₃₁	C₂×C₃×C₅×D₇	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃×C₅	—	31	1	no	—
45	C₂×C₃×C₅	—	217	2	no	—
46	C₂×D₇	C₅×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor, Hall
47	C₃×C₇	—	31	1	no	Hall
48	C₂²×C₅	—	7	1	no	Hall
49	C₃×C₅	—	31	1	no	Hall
50	C₂×C₇	C₂×C₅×C₃₁⋊C₃	1	1	yes	characteristic, complemented
51	D₇	C₂×C₅×C₃₁⋊C₃	1	2	yes	direct product factor
52	C₂²×C₃	—	217	1	no	Hall
53	C₂×C₅	—	7	2	no	—
54	C₂×C₅	D₇×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
55	C₇	C₂²×C₅×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃	—	31	1	no	—
57	C₂×C₃	—	217	2	no	—
58	C₅	C₂×D₇×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²	—	7	1	no	Sylow
60	C₃	—	31	1	no	Sylow
61	C₂	—	7	2	no	—
62	C₂	C₅×D₇×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	D₇×C₃₁⋊C₃
commutator, cosocle, hypocenter	C₇×C₃₁	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇×C₃₁	C₂×C₃
LCS	G ▷ C₇×C₃₁
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₅×D₇×C₃₁⋊C₃	C₂	direct (N × H)
2. C₂×C₅×C₇×C₃₁⋊C₃	C₂	semidirect
3. C₂×C₅×D₇×C₃₁	C₃	semidirect
4. C₅×C₇×C₃₁⋊C₃	C₂²	semidirect
5. C₂×D₇×C₃₁⋊C₃	C₅	direct (N × H)
6. C₅×D₇×C₃₁	C₂×C₃	semidirect
7. C₂×C₅×C₇×C₃₁	C₂×C₃	semidirect
9. C₂×C₇×C₃₁⋊C₃	C₂×C₅	semidirect
10. D₇×C₃₁⋊C₃	C₂×C₅	direct (N × H)
11. C₅×C₇×C₃₁	C₂²×C₃	semidirect
13. C₂×C₅×C₃₁⋊C₃	D₇	direct (N × H)
14. C₂×D₇×C₃₁	C₃×C₅	semidirect
15. C₇×C₃₁⋊C₃	C₂²×C₅	semidirect
17. C₅×C₃₁⋊C₃	C₂×D₇	direct (N × H)
18. C₂×C₇×C₃₁	C₂×C₃×C₅	semidirect
19. D₇×C₃₁	C₂×C₃×C₅	semidirect
23. C₂×C₅×C₃₁	C₃×D₇	semidirect
24. C₇×C₃₁	C₂²×C₃×C₅	semidirect
28. C₂×C₃₁⋊C₃	C₅×D₇	direct (N × H)
29. C₅×C₃₁	C₂×C₃×D₇	semidirect
30. C₂×C₅×D₇	C₃₁⋊C₃	direct (N × H)
33. C₃₁⋊C₃	C₂×C₅×D₇	direct (N × H)
35. C₂×C₅×C₇	C₂×C₃₁⋊C₃	semidirect
36. C₅×D₇	C₂×C₃₁⋊C₃	direct (N × H)
38. C₂×C₃₁	C₃×C₅×D₇	semidirect
42. C₅×C₇	C₂²×C₃₁⋊C₃	semidirect
43. C₃₁	C₂×C₃×C₅×D₇	semidirect
46. C₂×D₇	C₅×C₃₁⋊C₃	direct (N × H)
50. C₂×C₇	C₂×C₅×C₃₁⋊C₃	semidirect
51. D₇	C₂×C₅×C₃₁⋊C₃	direct (N × H)
54. C₂×C₅	D₇×C₃₁⋊C₃	direct (N × H)
55. C₇	C₂²×C₅×C₃₁⋊C₃	semidirect
58. C₅	C₂×D₇×C₃₁⋊C₃	direct (N × H)
62. C₂	C₅×D₇×C₃₁⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×D₇×C₃₁⋊C₃	2
C₂×C₅×C₇×C₃₁⋊C₃	2
C₂×C₅×D₇×C₃₁	3
C₂×D₇×C₃₁⋊C₃	5
C₂²×C₅×C₃₁⋊C₃	7
C₂×C₃×C₅×D₇	31

Subgroups — C2×C5×D7×C31⋊C3 #27189201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₇×C₃₁⋊C₃ #27189201