Subgroups — C₃×A₄×C₇₁⋊C₇ #31822062

Order 17892.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇₁⋊C₇	C₃	1	1	yes	maximal, characteristic, complemented
2	A₄×C₇₁⋊C₇	C₃	1	3	yes	maximal, direct product factor
3	C₃²×C₇₁⋊C₇	—	4	1	no	maximal, Hall
4	C₂×C₃×C₇₁⋊C₇	—	3	1	no	—
5	C₃×A₄×C₇₁	C₇	1	1	yes	maximal, characteristic, complemented, Hall
6	C₂²×C₇₁⋊C₇	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃×C₇₁⋊C₇	—	4	3	no	—
8	C₃×C₇₁⋊C₇	A₄	1	1	yes	characteristic, direct product factor
9	C₂×C₇₁⋊C₇	—	3	1	no	—
10	C₂²×C₃×C₇₁	C₃×C₇	1	1	yes	characteristic, complemented
11	A₄×C₇₁	C₃×C₇	1	3	yes	complemented
12	C₃²×C₇₁	—	4	1	no	Hall
13	C₇₁⋊C₇	C₃×A₄	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃×C₇₁	—	3	1	no	—
15	C₂²×C₇₁	C₃²×C₇	1	1	yes	characteristic, complemented, Hall
16	C₃×C₇×A₄	—	71	1	no	maximal, Hall
17	C₃×C₇₁	—	4	3	no	—
18	C₃×C₇₁	C₇×A₄	1	1	yes	characteristic, complemented
19	C₂×C₇₁	—	3	1	no	—
20	C₂²×C₃×C₇	—	71	1	no	—
21	C₇×A₄	—	71	3	no	—
22	C₇₁	C₃×C₇×A₄	1	1	yes	characteristic, complemented, Sylow
23	C₃²×C₇	—	284	1	no	Hall
24	C₂×C₃×C₇	—	213	1	no	—
25	C₃×A₄	C₇₁⋊C₇	1	1	yes	characteristic, direct product factor, Hall
26	C₂²×C₇	—	71	1	no	Hall
27	C₃×C₇	—	284	3	no	—
28	C₃×C₇	—	71	1	no	—
29	C₂×C₇	—	213	1	no	—
30	C₂²×C₃	C₃×C₇₁⋊C₇	1	1	yes	characteristic, complemented
31	A₄	C₃×C₇₁⋊C₇	1	3	yes	direct product factor
32	C₃²	—	4	1	no	Sylow
33	C₇	—	71	1	no	Sylow
34	C₂×C₃	—	3	1	no	—
35	C₂²	C₃²×C₇₁⋊C₇	1	1	yes	characteristic, complemented, Sylow
36	C₃	—	4	3	no	—
37	C₃	A₄×C₇₁⋊C₇	1	1	yes	characteristic, direct product factor
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃	A₄×C₇₁⋊C₇
commutator, cosocle, hypocenter	C₂²×C₇₁	C₃²×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇₁	C₃×C₇
LCS	G ▷ C₂²×C₇₁
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇₁⋊C₇	C₃	semidirect
2. A₄×C₇₁⋊C₇	C₃	direct (N × H)
5. C₃×A₄×C₇₁	C₇	semidirect
6. C₂²×C₇₁⋊C₇	C₃²	semidirect
8. C₃×C₇₁⋊C₇	A₄	direct (N × H)
10. C₂²×C₃×C₇₁	C₃×C₇	semidirect
11. A₄×C₇₁	C₃×C₇	semidirect
13. C₇₁⋊C₇	C₃×A₄	direct (N × H)
15. C₂²×C₇₁	C₃²×C₇	semidirect
18. C₃×C₇₁	C₇×A₄	semidirect
22. C₇₁	C₃×C₇×A₄	semidirect
25. C₃×A₄	C₇₁⋊C₇	direct (N × H)
30. C₂²×C₃	C₃×C₇₁⋊C₇	semidirect
31. A₄	C₃×C₇₁⋊C₇	direct (N × H)
35. C₂²	C₃²×C₇₁⋊C₇	semidirect
37. C₃	A₄×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
C₂²×C₃×C₇₁⋊C₇	3
A₄×C₇₁⋊C₇	3
C₃²×C₇₁⋊C₇	4
C₃×A₄×C₇₁	7
C₃×C₇×A₄	71

Subgroups — C3×A4×C71⋊C7 #31822062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×A₄×C₇₁⋊C₇ #31822062