Subgroups — C₃²×C₇⋊#66772 #36339446

Order 21420.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₇×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇⋊#66772	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×C₇×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×C₇⋊#595	—	5	1	no	maximal, Hall
5	C₃×C₅×C₇×D₁₇	C₂×C₃	1	4	yes	—
6	C₃²×#66772	—	7	1	no	maximal, Hall
7	C₇⋊#66772	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₃²×C₇×D₁₇	D₅	1	1	yes	characteristic
9	C₃×C₅×C₇×C₁₇	C₃×C₄	1	4	yes	complemented
10	C₃²×C₅×D₁₇	D₇	1	1	yes	characteristic
11	C₃×C₇⋊#595	—	5	4	no	—
12	C₃²×Dic₃₅	—	17	1	no	maximal, Hall
13	C₅×C₇×D₁₇	C₂×C₃²	1	1	yes	characteristic
14	C₃²×C₇×C₁₇	Dic₅	1	1	yes	characteristic, complemented, Hall
15	C₃×#66772	—	7	4	no	—
16	C₃²×C₅×C₁₇	Dic₇	1	1	yes	characteristic, complemented, Hall
17	C₃×C₇×D₁₇	C₃×D₅	1	4	yes	—
18	C₂×C₃²×C₅×C₇	—	17	1	no	—
19	C₃²×#595	—	35	1	no	Hall
20	C₅×C₇×C₁₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
21	C₃×C₅×D₁₇	C₃×D₇	1	4	yes	—
22	C₇⋊#595	—	5	1	no	Hall
23	C₃×Dic₃₅	—	17	4	no	—
24	C₃×C₇×C₁₇	C₃×Dic₅	1	4	yes	complemented
25	#66772	—	7	1	no	Hall
26	C₃²×C₅×C₇	#595	1	1	yes	characteristic, complemented, Hall
27	C₃²×D₁₇	D₃₅	1	1	yes	characteristic
28	C₃×C₅×C₁₇	C₃×Dic₇	1	4	yes	complemented
29	C₃²×Dic₇	—	85	1	no	Hall
30	C₇×D₁₇	C₃²×D₅	1	1	yes	characteristic
31	C₂×C₃×C₅×C₇	—	17	4	no	—
32	C₃×#595	—	35	4	no	—
33	C₃²×Dic₅	—	119	1	no	Hall
34	C₅×D₁₇	C₃²×D₇	1	1	yes	characteristic
35	C₃²×C₁₇	Dic₃₅	1	1	yes	characteristic, complemented, Hall
36	Dic₃₅	—	17	1	no	Hall
37	C₂×C₃²×C₇	—	17	1	no	—
38	C₇×C₁₇	C₃²×Dic₅	1	1	yes	characteristic, complemented, Hall
39	C₃×C₅×C₇	C₃×#595	1	4	yes	complemented
40	C₃×D₁₇	C₃×D₃₅	1	4	yes	—
41	C₂×C₃²×C₅	—	17	1	no	—
42	C₅×C₁₇	C₃²×Dic₇	1	1	yes	characteristic, complemented, Hall
43	C₃×Dic₇	—	85	4	no	—
44	C₂×C₅×C₇	—	17	1	no	—
45	#595	—	35	1	no	Hall
46	C₃²×C₇	#66772	1	1	yes	characteristic, complemented, Hall
47	C₃×Dic₅	—	119	4	no	—
48	C₃×C₁₇	C₃×Dic₃₅	1	4	yes	complemented
49	C₃²×C₅	C₇⋊#595	1	1	yes	characteristic, complemented, Hall
50	C₂×C₃×C₇	—	17	4	no	—
51	C₃²×C₄	—	595	1	no	Hall
52	C₅×C₇	C₃²×#595	1	1	yes	characteristic, complemented, Hall
53	D₁₇	C₃²×D₃₅	1	1	yes	characteristic
54	C₂×C₃×C₅	—	17	4	no	—
55	Dic₇	—	85	1	no	Hall
56	C₃×C₇	C₃×#66772	1	4	yes	complemented
57	Dic₅	—	119	1	no	Hall
58	C₂×C₃²	—	17	1	no	—
59	C₁₇	C₃²×Dic₃₅	1	1	yes	characteristic, complemented, Sylow
60	C₃×C₅	C₃×C₇⋊#595	1	4	yes	complemented
61	C₂×C₇	—	17	1	no	—
62	C₃×C₄	—	595	4	no	—
63	C₂×C₅	—	17	1	no	—
64	C₃²	C₇⋊#66772	1	1	yes	characteristic, direct product factor, Sylow
65	C₇	C₃²×#66772	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	17	4	no	—
67	C₅	C₃²×C₇⋊#595	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	595	1	no	Sylow
69	C₃	C₃×C₇⋊#66772	1	4	yes	direct product factor
70	C₂	—	17	1	no	—

	subgroup	G/H
center, hypercenter	C₃²	C₇⋊#66772
commutator, hypocenter	C₅×C₇×C₁₇	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇×C₁₇	C₄
prime radical	C₃²×C₅×C₇×D₁₇	C₂
cosocle	C₅×C₇×D₁₇	C₂×C₃²
LCS	G ▷ C₅×C₇×C₁₇
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
2. C₃×C₇⋊#66772	C₃	direct (N × H)
3. C₃²×C₅×C₇×C₁₇	C₄	semidirect
7. C₇⋊#66772	C₃²	direct (N × H)
9. C₃×C₅×C₇×C₁₇	C₃×C₄	semidirect
14. C₃²×C₇×C₁₇	Dic₅	semidirect
16. C₃²×C₅×C₁₇	Dic₇	semidirect
20. C₅×C₇×C₁₇	C₃²×C₄	semidirect
24. C₃×C₇×C₁₇	C₃×Dic₅	semidirect
26. C₃²×C₅×C₇	#595	semidirect
28. C₃×C₅×C₁₇	C₃×Dic₇	semidirect
35. C₃²×C₁₇	Dic₃₅	semidirect
38. C₇×C₁₇	C₃²×Dic₅	semidirect
39. C₃×C₅×C₇	C₃×#595	semidirect
42. C₅×C₁₇	C₃²×Dic₇	semidirect
46. C₃²×C₇	#66772	semidirect
48. C₃×C₁₇	C₃×Dic₃₅	semidirect
49. C₃²×C₅	C₇⋊#595	semidirect
52. C₅×C₇	C₃²×#595	semidirect
56. C₃×C₇	C₃×#66772	semidirect
59. C₁₇	C₃²×Dic₃₅	semidirect
60. C₃×C₅	C₃×C₇⋊#595	semidirect
64. C₃²	C₇⋊#66772	direct (N × H)
65. C₇	C₃²×#66772	semidirect
67. C₅	C₃²×C₇⋊#595	semidirect
69. C₃	C₃×C₇⋊#66772	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅×C₇×D₁₇	2
C₃×C₇⋊#66772	3
C₃²×C₇⋊#595	5
C₃²×#66772	7
C₃²×Dic₃₅	17

Subgroups — C32×C7⋊#66772 #36339446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₇⋊#66772 #36339446