Subgroups — C₂×C₅×C₇×D₂₁₇ #44477197

Order 30380.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×D₂₁₇	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₅×C₇²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₇²×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×D₂₁₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₅×C₇×D₃₁	—	7	1	no	maximal
6	C₂×C₅×D₂₁₇	C₇	1	1	yes	maximal, characteristic, direct product factor
7	C₇×D₂₁₇	C₂×C₅	1	2	yes	direct product factor
8	C₂×C₇²×C₃₁	C₂×C₅	1	1	yes	characteristic, complemented
9	C₅×D₂₁₇	C₂×C₇	1	2	yes	direct product factor
10	C₅×C₇×D₃₁	—	7	2	no	—
11	C₂×C₅×C₇×C₃₁	—	2	3	no	—
12	C₂×C₅×C₇×C₃₁	D₇	1	1	yes	characteristic, complemented
13	C₂×C₅×C₇×C₃₁	C₂×C₇	1	1	yes	characteristic, complemented
14	C₇²×C₃₁	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
15	C₅×C₇×C₃₁	—	2	3	no	—
16	C₅×C₇×C₃₁	C₂×D₇	1	1	yes	characteristic, complemented
17	C₅×C₇×C₃₁	C₂²×C₇	1	1	yes	characteristic, complemented
18	C₂×C₅×C₇×D₇	—	31	1	no	maximal, Hall
19	C₂×D₂₁₇	C₅×C₇	1	1	yes	characteristic, direct product factor
20	C₂×C₇×D₃₁	—	7	1	no	—
21	C₂×C₅×D₃₁	—	7	1	no	Hall
22	C₂×C₅×C₇²	D₃₁	1	1	yes	characteristic, complemented
23	C₅×C₇×D₇	—	31	2	no	—
24	C₂×C₇×C₃₁	—	2	3	no	—
25	C₂×C₇×C₃₁	C₅×D₇	1	1	yes	characteristic, complemented
26	C₂×C₇×C₃₁	C₂×C₅×C₇	1	1	yes	characteristic, complemented
27	D₂₁₇	C₂×C₅×C₇	1	2	yes	direct product factor
28	C₇×D₃₁	—	7	2	no	—
29	C₂×C₅×C₃₁	C₇×D₇	1	1	yes	characteristic, complemented
30	C₅×D₃₁	—	7	2	no	—
31	C₅×C₇²	C₂×D₃₁	1	1	yes	characteristic, complemented, Hall
32	C₇×C₃₁	—	2	3	no	—
33	C₇×C₃₁	C₂×C₅×D₇	1	1	yes	characteristic, complemented
34	C₇×C₃₁	C₂²×C₅×C₇	1	1	yes	characteristic, complemented
35	C₂×C₇×D₇	—	31	1	no	Hall
36	C₅×C₃₁	C₂×C₇×D₇	1	1	yes	characteristic, complemented, Hall
37	C₂²×C₅×C₇	—	217	1	no	—
38	C₂×C₅×D₇	—	31	1	no	—
39	C₂×D₃₁	—	7	1	no	Hall
40	C₂×C₇²	C₅×D₃₁	1	1	yes	characteristic, complemented
41	C₇×D₇	—	31	2	no	—
42	C₂×C₅×C₇	—	2	3	no	—
43	C₂×C₅×C₇	—	217	2	no	—
44	C₂×C₅×C₇	C₇×D₃₁	1	1	yes	characteristic, complemented
45	C₂×C₅×C₇	D₂₁₇	1	1	yes	characteristic, direct product factor
46	C₅×D₇	—	31	2	no	—
47	C₂×C₃₁	C₅×C₇×D₇	1	1	yes	characteristic, complemented
48	D₃₁	—	7	2	no	—
49	C₇²	C₂×C₅×D₃₁	1	1	yes	characteristic, complemented, Sylow
50	C₅×C₇	—	2	3	no	—
51	C₅×C₇	C₂×C₇×D₃₁	1	1	yes	characteristic, complemented
52	C₅×C₇	C₂×D₂₁₇	1	1	yes	characteristic, direct product factor
53	C₃₁	C₂×C₅×C₇×D₇	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₇	—	217	1	no	—
55	C₂×D₇	—	31	1	no	—
56	C₂²×C₅	—	217	1	no	Hall
57	C₂×C₇	—	2	3	no	—
58	C₂×C₇	—	217	2	no	—
59	C₂×C₇	C₅×C₇×D₃₁	1	1	yes	characteristic, complemented
60	C₂×C₇	C₅×D₂₁₇	1	1	yes	characteristic, direct product factor
61	D₇	—	31	2	no	—
62	C₂×C₅	—	217	2	no	—
63	C₂×C₅	C₇×D₂₁₇	1	1	yes	characteristic, direct product factor
64	C₇	—	2	3	no	—
65	C₇	C₂×C₅×D₂₁₇	1	1	yes	characteristic, direct product factor
66	C₇	C₂×C₅×C₇×D₃₁	1	1	yes	characteristic, complemented
67	C₅	C₂×C₇×D₂₁₇	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²	—	217	1	no	Sylow
69	C₂	—	217	2	no	—
70	C₂	C₅×C₇×D₂₁₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅×C₇	D₂₁₇
commutator, cosocle, hypocenter	C₇×C₃₁	C₂²×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇²×C₃₁	C₂
LCS	G ▷ C₇×C₃₁
UCS	C₁ ◁ C₂×C₅×C₇

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×D₂₁₇	C₂	direct (N × H)
2. C₂×C₅×C₇²×C₃₁	C₂	semidirect
3. C₅×C₇²×C₃₁	C₂²	semidirect
4. C₂×C₇×D₂₁₇	C₅	direct (N × H)
6. C₂×C₅×D₂₁₇	C₇	direct (N × H)
7. C₇×D₂₁₇	C₂×C₅	direct (N × H)
8. C₂×C₇²×C₃₁	C₂×C₅	semidirect
9. C₅×D₂₁₇	C₂×C₇	direct (N × H)
12. C₂×C₅×C₇×C₃₁	D₇	semidirect
13. C₂×C₅×C₇×C₃₁	C₂×C₇	semidirect
14. C₇²×C₃₁	C₂²×C₅	semidirect
16. C₅×C₇×C₃₁	C₂×D₇	semidirect
17. C₅×C₇×C₃₁	C₂²×C₇	semidirect
19. C₂×D₂₁₇	C₅×C₇	direct (N × H)
22. C₂×C₅×C₇²	D₃₁	semidirect
25. C₂×C₇×C₃₁	C₅×D₇	semidirect
26. C₂×C₇×C₃₁	C₂×C₅×C₇	semidirect
27. D₂₁₇	C₂×C₅×C₇	direct (N × H)
29. C₂×C₅×C₃₁	C₇×D₇	semidirect
31. C₅×C₇²	C₂×D₃₁	semidirect
33. C₇×C₃₁	C₂×C₅×D₇	semidirect
34. C₇×C₃₁	C₂²×C₅×C₇	semidirect
36. C₅×C₃₁	C₂×C₇×D₇	semidirect
40. C₂×C₇²	C₅×D₃₁	semidirect
44. C₂×C₅×C₇	C₇×D₃₁	semidirect
45. C₂×C₅×C₇	D₂₁₇	direct (N × H)
47. C₂×C₃₁	C₅×C₇×D₇	semidirect
49. C₇²	C₂×C₅×D₃₁	semidirect
51. C₅×C₇	C₂×C₇×D₃₁	semidirect
52. C₅×C₇	C₂×D₂₁₇	direct (N × H)
53. C₃₁	C₂×C₅×C₇×D₇	semidirect
59. C₂×C₇	C₅×C₇×D₃₁	semidirect
60. C₂×C₇	C₅×D₂₁₇	direct (N × H)
63. C₂×C₅	C₇×D₂₁₇	direct (N × H)
65. C₇	C₂×C₅×D₂₁₇	direct (N × H)
66. C₇	C₂×C₅×C₇×D₃₁	semidirect
67. C₅	C₂×C₇×D₂₁₇	direct (N × H)
70. C₂	C₅×C₇×D₂₁₇	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×C₇×D₂₁₇	2
C₂×C₅×C₇²×C₃₁	2
C₂×C₇×D₂₁₇	5
C₂×C₅×C₇×D₃₁	7
C₂×C₅×D₂₁₇	7
C₂×C₅×C₇×D₇	31

Subgroups — C2×C5×C7×D217 #44477197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₇×D₂₁₇ #44477197