Subgroups — C₃²×C₄×C₇₁⋊C₇ #31822034

Order 17892.

72 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
2	C₃×C₄×C₇₁⋊C₇	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₇₁⋊C₇	C₄	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₃×C₇₁⋊C₇	C₂×C₃	1	4	yes	—
5	C₃²×C₄×C₇₁	C₇	1	1	yes	maximal, characteristic, complemented, Hall
6	C₄×C₇₁⋊C₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃×C₇₁⋊C₇	C₃×C₄	1	4	yes	direct product factor
8	C₂×C₃²×C₇₁	C₂×C₇	1	1	yes	characteristic
9	C₂×C₇₁⋊C₇	C₂×C₃²	1	1	yes	characteristic
10	C₃×C₄×C₇₁	C₃×C₇	1	4	yes	complemented
11	C₃²×C₇₁	C₄×C₇	1	1	yes	characteristic, complemented, Hall
12	C₇₁⋊C₇	C₃²×C₄	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₃×C₇₁	C₂×C₃×C₇	1	4	yes	—
14	C₄×C₇₁	C₃²×C₇	1	1	yes	characteristic, complemented, Hall
15	C₃²×C₄×C₇	—	71	1	no	maximal, Hall
16	C₃×C₇₁	C₃×C₄×C₇	1	4	yes	complemented
17	C₂×C₇₁	C₂×C₃²×C₇	1	1	yes	characteristic
18	C₂×C₃²×C₇	—	71	1	no	—
19	C₃×C₄×C₇	—	71	4	no	—
20	C₇₁	C₃²×C₄×C₇	1	1	yes	characteristic, complemented, Sylow
21	C₃²×C₇	—	71	1	no	Hall
22	C₂×C₃×C₇	—	71	4	no	—
23	C₃²×C₄	C₇₁⋊C₇	1	1	yes	characteristic, direct product factor, Hall
24	C₄×C₇	—	71	1	no	Hall
25	C₃×C₇	—	71	4	no	—
26	C₂×C₃²	C₂×C₇₁⋊C₇	1	1	yes	characteristic
27	C₂×C₇	—	71	1	no	—
28	C₃×C₄	C₃×C₇₁⋊C₇	1	4	yes	direct product factor
29	C₃²	C₄×C₇₁⋊C₇	1	1	yes	characteristic, direct product factor, Sylow
30	C₇	—	71	1	no	Sylow
31	C₂×C₃	C₂×C₃×C₇₁⋊C₇	1	4	yes	—
32	C₄	C₃²×C₇₁⋊C₇	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	C₃×C₄×C₇₁⋊C₇	1	4	yes	direct product factor
34	C₂	C₂×C₃²×C₇₁⋊C₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃²×C₄	C₇₁⋊C₇
commutator, hypocenter	C₇₁	C₃²×C₄×C₇
Frattini	C₂	C₂×C₃²×C₇₁⋊C₇
Fitting	C₃²×C₄×C₇₁	C₇
socle	C₂×C₃²×C₇₁	C₂×C₇
prime radical	C₂×C₃²×C₇₁⋊C₇	C₂
cosocle	C₂×C₇₁	C₂×C₃²×C₇
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₃²×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇₁⋊C₇	C₃	direct (N × H)
3. C₃²×C₇₁⋊C₇	C₄	direct (N × H)
5. C₃²×C₄×C₇₁	C₇	semidirect
6. C₄×C₇₁⋊C₇	C₃²	direct (N × H)
7. C₃×C₇₁⋊C₇	C₃×C₄	direct (N × H)
10. C₃×C₄×C₇₁	C₃×C₇	semidirect
11. C₃²×C₇₁	C₄×C₇	semidirect
12. C₇₁⋊C₇	C₃²×C₄	direct (N × H)
14. C₄×C₇₁	C₃²×C₇	semidirect
16. C₃×C₇₁	C₃×C₄×C₇	semidirect
20. C₇₁	C₃²×C₄×C₇	semidirect
23. C₃²×C₄	C₇₁⋊C₇	direct (N × H)
28. C₃×C₄	C₃×C₇₁⋊C₇	direct (N × H)
29. C₃²	C₄×C₇₁⋊C₇	direct (N × H)
32. C₄	C₃²×C₇₁⋊C₇	direct (N × H)
33. C₃	C₃×C₄×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×C₄×C₇₁⋊C₇	3
C₃²×C₄×C₇₁	7
C₃²×C₄×C₇	71

Subgroups — C32×C4×C71⋊C7 #31822034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₇₁⋊C₇ #31822034