Subgroups — C₃²×C₁₇×Dic₃₅ #36339331

Order 21420.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×C₇×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₇×Dic₃₅	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×C₇×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×C₁₇×Dic₇	—	5	1	no	maximal, Hall
5	C₂×C₃×C₅×C₇×C₁₇	C₂×C₃	1	4	yes	—
6	C₃²×C₁₇×Dic₅	—	7	1	no	maximal, Hall
7	C₁₇×Dic₃₅	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₂×C₃²×C₇×C₁₇	D₅	1	1	yes	characteristic
9	C₃×C₅×C₇×C₁₇	C₃×C₄	1	4	yes	complemented
10	C₂×C₃²×C₅×C₁₇	D₇	1	1	yes	characteristic
11	C₃×C₁₇×Dic₇	—	5	4	no	—
12	C₃²×Dic₃₅	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₅×C₇×C₁₇	C₂×C₃²	1	1	yes	characteristic
14	C₃²×C₇×C₁₇	Dic₅	1	1	yes	characteristic, complemented, Hall
15	C₃×C₁₇×Dic₅	—	7	4	no	—
16	C₃²×C₅×C₁₇	Dic₇	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₇×C₁₇	C₃×D₅	1	4	yes	—
18	C₂×C₃²×C₅×C₇	C₂×C₁₇	1	1	yes	characteristic
19	C₃²×C₄×C₁₇	—	35	1	no	Hall
20	C₅×C₇×C₁₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃×C₅×C₁₇	C₃×D₇	1	4	yes	—
22	C₁₇×Dic₇	—	5	1	no	Hall
23	C₃×Dic₃₅	C₃×C₁₇	1	4	yes	direct product factor
24	C₃×C₇×C₁₇	C₃×Dic₅	1	4	yes	complemented
25	C₁₇×Dic₅	—	7	1	no	Hall
26	C₃²×C₅×C₇	C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
27	C₂×C₃²×C₁₇	D₃₅	1	1	yes	characteristic
28	C₃×C₅×C₁₇	C₃×Dic₇	1	4	yes	complemented
29	C₃²×Dic₇	—	5	1	no	Hall
30	C₂×C₇×C₁₇	C₃²×D₅	1	1	yes	characteristic
31	C₂×C₃×C₅×C₇	C₂×C₃×C₁₇	1	4	yes	—
32	C₃×C₄×C₁₇	—	35	4	no	—
33	C₃²×Dic₅	—	7	1	no	Hall
34	C₂×C₅×C₁₇	C₃²×D₇	1	1	yes	characteristic
35	C₃²×C₁₇	Dic₃₅	1	1	yes	characteristic, direct product factor, Hall
36	Dic₃₅	C₃²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
37	C₂×C₃²×C₇	D₅×C₁₇	1	1	yes	characteristic
38	C₇×C₁₇	C₃²×Dic₅	1	1	yes	characteristic, complemented, Hall
39	C₃×C₅×C₇	C₃×C₄×C₁₇	1	4	yes	complemented
40	C₂×C₃×C₁₇	C₃×D₃₅	1	4	yes	—
41	C₂×C₃²×C₅	D₇×C₁₇	1	1	yes	characteristic
42	C₅×C₁₇	C₃²×Dic₇	1	1	yes	characteristic, complemented, Hall
43	C₃×Dic₇	—	5	4	no	—
44	C₂×C₅×C₇	C₂×C₃²×C₁₇	1	1	yes	characteristic
45	C₄×C₁₇	—	35	1	no	Hall
46	C₃²×C₇	C₁₇×Dic₅	1	1	yes	characteristic, complemented, Hall
47	C₃×Dic₅	—	7	4	no	—
48	C₃×C₁₇	C₃×Dic₃₅	1	4	yes	direct product factor
49	C₃²×C₅	C₁₇×Dic₇	1	1	yes	characteristic, complemented, Hall
50	C₂×C₃×C₇	C₃×D₅×C₁₇	1	4	yes	—
51	C₃²×C₄	—	35	1	no	Hall
52	C₅×C₇	C₃²×C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
53	C₂×C₁₇	C₃²×D₃₅	1	1	yes	characteristic
54	C₂×C₃×C₅	C₃×D₇×C₁₇	1	4	yes	—
55	Dic₇	—	5	1	no	Hall
56	C₃×C₇	C₃×C₁₇×Dic₅	1	4	yes	complemented
57	Dic₅	—	7	1	no	Hall
58	C₂×C₃²	C₁₇×D₃₅	1	1	yes	characteristic
59	C₁₇	C₃²×Dic₃₅	1	1	yes	characteristic, direct product factor, Sylow
60	C₃×C₅	C₃×C₁₇×Dic₇	1	4	yes	complemented
61	C₂×C₇	C₃²×D₅×C₁₇	1	1	yes	characteristic
62	C₃×C₄	—	35	4	no	—
63	C₂×C₅	C₃²×D₇×C₁₇	1	1	yes	characteristic
64	C₃²	C₁₇×Dic₃₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₇	C₃²×C₁₇×Dic₅	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	C₃×C₁₇×D₃₅	1	4	yes	—
67	C₅	C₃²×C₁₇×Dic₇	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	35	1	no	Sylow
69	C₃	C₃×C₁₇×Dic₃₅	1	4	yes	direct product factor
70	C₂	C₃²×C₁₇×D₃₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₁₇	D₃₅
commutator, hypocenter	C₅×C₇	C₃²×C₄×C₁₇
Frattini	C₂	C₃²×C₁₇×D₃₅
Fitting, socle, prime radical	C₂×C₃²×C₅×C₇×C₁₇	C₂
cosocle	C₂×C₅×C₇	C₂×C₃²×C₁₇
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂×C₃²×C₁₇

Complements

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

N	H ≅ G/N	type
2. C₃×C₁₇×Dic₃₅	C₃	direct (N × H)
3. C₃²×C₅×C₇×C₁₇	C₄	semidirect
7. C₁₇×Dic₃₅	C₃²	direct (N × H)
9. C₃×C₅×C₇×C₁₇	C₃×C₄	semidirect
12. C₃²×Dic₃₅	C₁₇	direct (N × H)
14. C₃²×C₇×C₁₇	Dic₅	semidirect
16. C₃²×C₅×C₁₇	Dic₇	semidirect
20. C₅×C₇×C₁₇	C₃²×C₄	semidirect
23. C₃×Dic₃₅	C₃×C₁₇	direct (N × H)
24. C₃×C₇×C₁₇	C₃×Dic₅	semidirect
26. C₃²×C₅×C₇	C₄×C₁₇	semidirect
28. C₃×C₅×C₁₇	C₃×Dic₇	semidirect
35. C₃²×C₁₇	Dic₃₅	direct (N × H)
36. Dic₃₅	C₃²×C₁₇	direct (N × H)
38. C₇×C₁₇	C₃²×Dic₅	semidirect
39. C₃×C₅×C₇	C₃×C₄×C₁₇	semidirect
42. C₅×C₁₇	C₃²×Dic₇	semidirect
46. C₃²×C₇	C₁₇×Dic₅	semidirect
48. C₃×C₁₇	C₃×Dic₃₅	direct (N × H)
49. C₃²×C₅	C₁₇×Dic₇	semidirect
52. C₅×C₇	C₃²×C₄×C₁₇	semidirect
56. C₃×C₇	C₃×C₁₇×Dic₅	semidirect
59. C₁₇	C₃²×Dic₃₅	direct (N × H)
60. C₃×C₅	C₃×C₁₇×Dic₇	semidirect
64. C₃²	C₁₇×Dic₃₅	direct (N × H)
65. C₇	C₃²×C₁₇×Dic₅	semidirect
67. C₅	C₃²×C₁₇×Dic₇	semidirect
69. C₃	C₃×C₁₇×Dic₃₅	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×C₃²×C₅×C₇×C₁₇	2
C₃×C₁₇×Dic₃₅	3
C₃²×C₁₇×Dic₇	5
C₃²×C₁₇×Dic₅	7
C₃²×Dic₃₅	17

Subgroups — C32×C17×Dic35 #36339331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₁₇×Dic₃₅ #36339331