Subgroups — C₂×C₃×D₃×C₇₁⋊C₇ #31822065

Order 17892.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₇₁⋊C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×C₇₁⋊C₇	C₂	1	2	yes	maximal, direct product factor
3	C₂²×C₃×C₇₁⋊C₇	—	3	1	no	maximal
4	C₂×D₃×C₇₁⋊C₇	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃²×C₇₁⋊C₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₃×C₇₁⋊C₇	C₂×C₃	1	2	yes	direct product factor
7	C₂×C₃×C₇₁⋊C₇	—	2	1	no	—
8	C₂×C₃×C₇₁⋊C₇	—	3	2	no	—
9	C₂×C₃×C₇₁⋊C₇	D₃	1	1	yes	characteristic, direct product factor
10	C₂×C₃×C₇₁⋊C₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃×D₃×C₇₁	C₇	1	1	yes	maximal, characteristic, complemented, Hall
12	C₂²×C₇₁⋊C₇	—	3	1	no	Hall
13	C₃×C₇₁⋊C₇	—	2	1	no	—
14	C₃×C₇₁⋊C₇	C₂×D₃	1	1	yes	characteristic, direct product factor
15	C₃×C₇₁⋊C₇	C₂²×C₃	1	1	yes	characteristic, complemented
16	C₂×C₃²×C₇₁	C₂×C₇	1	1	yes	characteristic, complemented
17	C₃×D₃×C₇₁	C₂×C₇	1	2	yes	complemented
18	C₂×C₇₁⋊C₇	—	3	2	no	—
19	C₂×C₇₁⋊C₇	C₃×D₃	1	1	yes	characteristic, direct product factor
20	C₂²×C₃×C₇₁	—	3	1	no	—
21	C₂×D₃×C₇₁	C₃×C₇	1	1	yes	characteristic, complemented
22	C₃²×C₇₁	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
23	C₇₁⋊C₇	C₂×C₃×D₃	1	1	yes	characteristic, direct product factor, Hall
24	C₂×C₃×C₇₁	—	2	1	no	—
25	C₂×C₃×C₇₁	—	3	2	no	—
26	C₂×C₃×C₇₁	D₃×C₇	1	1	yes	characteristic, complemented
27	C₂×C₃×C₇₁	C₂×C₃×C₇	1	1	yes	characteristic, complemented
28	D₃×C₇₁	C₂×C₃×C₇	1	2	yes	complemented
29	C₂²×C₇₁	—	3	1	no	Hall
30	C₂×C₃×D₃×C₇	—	71	1	no	maximal, Hall
31	C₃×C₇₁	—	2	1	no	—
32	C₃×C₇₁	C₂×D₃×C₇	1	1	yes	characteristic, complemented
33	C₃×C₇₁	C₂²×C₃×C₇	1	1	yes	characteristic, complemented
34	C₂×C₇₁	—	3	2	no	—
35	C₂×C₇₁	C₃×D₃×C₇	1	1	yes	characteristic, complemented
36	C₂×C₃²×C₇	—	71	1	no	—
37	C₃×D₃×C₇	—	71	2	no	—
38	C₂²×C₃×C₇	—	213	1	no	—
39	C₂×D₃×C₇	—	71	1	no	—
40	C₇₁	C₂×C₃×D₃×C₇	1	1	yes	characteristic, complemented, Sylow
41	C₃²×C₇	—	71	1	no	Hall
42	C₂×C₃×C₇	—	71	1	no	—
43	C₂×C₃×C₇	—	71	1	no	—
44	C₂×C₃×C₇	—	142	1	no	—
45	C₂×C₃×C₇	—	213	2	no	—
46	D₃×C₇	—	71	2	no	—
47	C₂×C₃×D₃	C₇₁⋊C₇	1	1	yes	characteristic, direct product factor, Hall
48	C₂²×C₇	—	213	1	no	Hall
49	C₃×C₇	—	71	1	no	—
50	C₃×C₇	—	71	1	no	—
51	C₃×C₇	—	142	1	no	—
52	C₂×C₃²	C₂×C₇₁⋊C₇	1	1	yes	characteristic, complemented
53	C₃×D₃	C₂×C₇₁⋊C₇	1	2	yes	direct product factor
54	C₂×C₇	—	71	1	no	—
55	C₂×C₇	—	213	2	no	—
56	C₂²×C₃	—	3	1	no	—
57	C₂×D₃	C₃×C₇₁⋊C₇	1	1	yes	characteristic, direct product factor
58	C₃²	C₂²×C₇₁⋊C₇	1	1	yes	characteristic, complemented, Sylow
59	C₇	—	71	1	no	Sylow
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	3	2	no	—
62	C₂×C₃	C₂×C₃×C₇₁⋊C₇	1	1	yes	characteristic, complemented
63	C₂×C₃	D₃×C₇₁⋊C₇	1	1	yes	characteristic, direct product factor
64	D₃	C₂×C₃×C₇₁⋊C₇	1	2	yes	direct product factor
65	C₂²	—	3	1	no	Sylow
66	C₃	—	2	1	no	—
67	C₃	C₂×D₃×C₇₁⋊C₇	1	1	yes	characteristic, direct product factor
68	C₃	C₂²×C₃×C₇₁⋊C₇	1	1	yes	characteristic, complemented
69	C₂	—	3	2	no	—
70	C₂	C₃×D₃×C₇₁⋊C₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×C₇₁⋊C₇
commutator, cosocle, hypocenter	C₃×C₇₁	C₂²×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇₁	C₂×C₇
LCS	G ▷ C₃×C₇₁
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₇₁⋊C₇	C₂	semidirect
2. C₃×D₃×C₇₁⋊C₇	C₂	direct (N × H)
4. C₂×D₃×C₇₁⋊C₇	C₃	direct (N × H)
5. C₃²×C₇₁⋊C₇	C₂²	semidirect
6. D₃×C₇₁⋊C₇	C₂×C₃	direct (N × H)
9. C₂×C₃×C₇₁⋊C₇	D₃	direct (N × H)
10. C₂×C₃×C₇₁⋊C₇	C₂×C₃	semidirect
11. C₂×C₃×D₃×C₇₁	C₇	semidirect
14. C₃×C₇₁⋊C₇	C₂×D₃	direct (N × H)
15. C₃×C₇₁⋊C₇	C₂²×C₃	semidirect
16. C₂×C₃²×C₇₁	C₂×C₇	semidirect
17. C₃×D₃×C₇₁	C₂×C₇	semidirect
19. C₂×C₇₁⋊C₇	C₃×D₃	direct (N × H)
21. C₂×D₃×C₇₁	C₃×C₇	semidirect
22. C₃²×C₇₁	C₂²×C₇	semidirect
23. C₇₁⋊C₇	C₂×C₃×D₃	direct (N × H)
26. C₂×C₃×C₇₁	D₃×C₇	semidirect
27. C₂×C₃×C₇₁	C₂×C₃×C₇	semidirect
28. D₃×C₇₁	C₂×C₃×C₇	semidirect
32. C₃×C₇₁	C₂×D₃×C₇	semidirect
33. C₃×C₇₁	C₂²×C₃×C₇	semidirect
35. C₂×C₇₁	C₃×D₃×C₇	semidirect
40. C₇₁	C₂×C₃×D₃×C₇	semidirect
47. C₂×C₃×D₃	C₇₁⋊C₇	direct (N × H)
52. C₂×C₃²	C₂×C₇₁⋊C₇	semidirect
53. C₃×D₃	C₂×C₇₁⋊C₇	direct (N × H)
57. C₂×D₃	C₃×C₇₁⋊C₇	direct (N × H)
58. C₃²	C₂²×C₇₁⋊C₇	semidirect
62. C₂×C₃	C₂×C₃×C₇₁⋊C₇	semidirect
63. C₂×C₃	D₃×C₇₁⋊C₇	direct (N × H)
64. D₃	C₂×C₃×C₇₁⋊C₇	direct (N × H)
67. C₃	C₂×D₃×C₇₁⋊C₇	direct (N × H)
68. C₃	C₂²×C₃×C₇₁⋊C₇	semidirect
70. C₂	C₃×D₃×C₇₁⋊C₇	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×C₇₁⋊C₇	2
C₃×D₃×C₇₁⋊C₇	2
C₂²×C₃×C₇₁⋊C₇	3
C₂×D₃×C₇₁⋊C₇	3
C₂×C₃×D₃×C₇₁	7
C₂×C₃×D₃×C₇	71

Subgroups — C2×C3×D3×C71⋊C7 #31822065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₃×C₇₁⋊C₇ #31822065