Subgroups — C₂×C₃²×D₇₁⋊C₇ #31822063

Order 17892.

120 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	C₃²×D₇₁⋊C₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×D₇₁⋊C₇	C₃	1	4	yes	maximal, direct product factor
4	C₃²×C₇₁⋊C₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃×D₇₁⋊C₇	C₂×C₃	1	8	yes	direct product factor
6	C₂×C₃×C₇₁⋊C₇	C₂×C₃	1	4	yes	complemented
7	C₂×C₃²×D₇₁	C₇	1	1	yes	maximal, characteristic, complemented, Hall
8	C₂×D₇₁⋊C₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
9	C₃×C₇₁⋊C₇	C₂²×C₃	1	4	yes	complemented
10	C₂×C₃²×C₇₁	C₂×C₇	1	1	yes	characteristic, complemented
11	C₃²×D₇₁	C₂×C₇	1	2	yes	complemented
12	C₂×C₇₁⋊C₇	C₂×C₃²	1	1	yes	characteristic, complemented
13	D₇₁⋊C₇	C₂×C₃²	1	2	yes	direct product factor
14	C₂×C₃×D₇₁	C₃×C₇	1	4	yes	complemented
15	C₃²×C₇₁	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
16	C₇₁⋊C₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₇₁	C₂×C₃×C₇	1	4	yes	complemented
18	C₃×D₇₁	C₂×C₃×C₇	1	8	yes	complemented
19	C₂×D₇₁	C₃²×C₇	1	1	yes	characteristic, complemented, Hall
20	C₂²×C₃²×C₇	—	71	1	no	maximal, Hall
21	C₃×C₇₁	C₂²×C₃×C₇	1	4	yes	complemented
22	C₂×C₇₁	C₂×C₃²×C₇	1	1	yes	characteristic, complemented
23	D₇₁	C₂×C₃²×C₇	1	2	yes	complemented
24	C₂×C₃²×C₇	—	71	2	no	—
25	C₂×C₃²×C₇	—	71	1	no	—
26	C₂²×C₃×C₇	—	71	4	no	—
27	C₇₁	C₂²×C₃²×C₇	1	1	yes	characteristic, complemented, Sylow
28	C₃²×C₇	—	71	1	no	Hall
29	C₂×C₃×C₇	—	71	8	no	—
30	C₂×C₃×C₇	—	71	4	no	—
31	C₂²×C₃²	—	71	1	no	Hall
32	C₂²×C₇	—	71	1	no	Hall
33	C₃×C₇	—	71	4	no	—
34	C₂×C₃²	—	71	2	no	—
35	C₂×C₃²	D₇₁⋊C₇	1	1	yes	characteristic, direct product factor
36	C₂×C₇	—	71	2	no	—
37	C₂×C₇	—	71	1	no	—
38	C₂²×C₃	—	71	4	no	—
39	C₃²	C₂×D₇₁⋊C₇	1	1	yes	characteristic, direct product factor, Sylow
40	C₇	—	71	1	no	Sylow
41	C₂×C₃	—	71	8	no	—
42	C₂×C₃	C₃×D₇₁⋊C₇	1	4	yes	direct product factor
43	C₂²	—	71	1	no	Sylow
44	C₃	C₂×C₃×D₇₁⋊C₇	1	4	yes	direct product factor
45	C₂	—	71	2	no	—
46	C₂	C₃²×D₇₁⋊C₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃²	D₇₁⋊C₇
commutator, cosocle, hypocenter	C₇₁	C₂²×C₃²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇₁	C₂×C₇
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₇₁⋊C₇	C₂	semidirect
2. C₃²×D₇₁⋊C₇	C₂	direct (N × H)
3. C₂×C₃×D₇₁⋊C₇	C₃	direct (N × H)
4. C₃²×C₇₁⋊C₇	C₂²	semidirect
5. C₃×D₇₁⋊C₇	C₂×C₃	direct (N × H)
6. C₂×C₃×C₇₁⋊C₇	C₂×C₃	semidirect
7. C₂×C₃²×D₇₁	C₇	semidirect
8. C₂×D₇₁⋊C₇	C₃²	direct (N × H)
9. C₃×C₇₁⋊C₇	C₂²×C₃	semidirect
10. C₂×C₃²×C₇₁	C₂×C₇	semidirect
11. C₃²×D₇₁	C₂×C₇	semidirect
12. C₂×C₇₁⋊C₇	C₂×C₃²	semidirect
13. D₇₁⋊C₇	C₂×C₃²	direct (N × H)
14. C₂×C₃×D₇₁	C₃×C₇	semidirect
15. C₃²×C₇₁	C₂²×C₇	semidirect
16. C₇₁⋊C₇	C₂²×C₃²	semidirect
17. C₂×C₃×C₇₁	C₂×C₃×C₇	semidirect
18. C₃×D₇₁	C₂×C₃×C₇	semidirect
19. C₂×D₇₁	C₃²×C₇	semidirect
21. C₃×C₇₁	C₂²×C₃×C₇	semidirect
22. C₂×C₇₁	C₂×C₃²×C₇	semidirect
23. D₇₁	C₂×C₃²×C₇	semidirect
27. C₇₁	C₂²×C₃²×C₇	semidirect
35. C₂×C₃²	D₇₁⋊C₇	direct (N × H)
39. C₃²	C₂×D₇₁⋊C₇	direct (N × H)
42. C₂×C₃	C₃×D₇₁⋊C₇	direct (N × H)
44. C₃	C₂×C₃×D₇₁⋊C₇	direct (N × H)
46. C₂	C₃²×D₇₁⋊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
C₃²×D₇₁⋊C₇	2
C₂×C₃×D₇₁⋊C₇	3
C₂×C₃²×D₇₁	7
C₂²×C₃²×C₇	71

Subgroups — C2×C32×D71⋊C7 #31822063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×D₇₁⋊C₇ #31822063