Subgroups — C₂×C₃⋊(C₃⋊(D₇₁⋊C₇)) #31822066

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	C₃⋊(C₃⋊(D₇₁⋊C₇))	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⋊(D₇₁⋊C₇)	—	3	4	no	maximal
4	C₃²×C₇₁⋊C₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃⋊(D₇₁⋊C₇)	—	3	8	no	—
6	C₂×C₃×C₇₁⋊C₇	D₃	1	4	yes	complemented
7	C₂×C₃⋊D₂₁₃	C₇	1	1	yes	maximal, characteristic, complemented, Hall
8	C₂×D₇₁⋊C₇	—	9	1	no	Hall
9	C₃×C₇₁⋊C₇	C₂×D₃	1	4	yes	complemented
10	C₂×C₃²×C₇₁	C₂×C₇	1	1	yes	characteristic, complemented
11	C₃⋊D₂₁₃	C₂×C₇	1	2	yes	complemented
12	C₂×C₇₁⋊C₇	C₃⋊D₃	1	1	yes	characteristic, complemented
13	D₇₁⋊C₇	—	9	2	no	—
14	C₂×D₂₁₃	—	3	4	no	—
15	C₃²×C₇₁	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
16	C₇₁⋊C₇	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₇₁	D₃×C₇	1	4	yes	complemented
18	D₂₁₃	—	3	8	no	—
19	C₂×D₇₁	—	9	1	no	Hall
20	C₂×C₇×C₃⋊D₃	—	71	1	no	maximal, Hall
21	C₃×C₇₁	C₂×D₃×C₇	1	4	yes	complemented
22	C₂×C₇₁	C₇×C₃⋊D₃	1	1	yes	characteristic, complemented
23	D₇₁	—	9	2	no	—
24	C₂×C₃²×C₇	—	71	1	no	—
25	C₇×C₃⋊D₃	—	71	2	no	—
26	C₂×D₃×C₇	—	213	4	no	—
27	C₇₁	C₂×C₇×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
28	C₃²×C₇	—	71	1	no	Hall
29	C₂×C₃×C₇	—	71	4	no	—
30	D₃×C₇	—	213	8	no	—
31	C₂×C₃⋊D₃	—	71	1	no	Hall
32	C₂²×C₇	—	639	1	no	Hall
33	C₃×C₇	—	71	4	no	—
34	C₂×C₃²	D₇₁⋊C₇	1	1	yes	characteristic, complemented
35	C₃⋊D₃	—	71	2	no	—
36	C₂×C₇	—	71	1	no	—
37	C₂×C₇	—	639	2	no	—
38	C₂×D₃	—	213	4	no	—
39	C₃²	C₂×D₇₁⋊C₇	1	1	yes	characteristic, complemented, Sylow
40	C₇	—	71	1	no	Sylow
41	C₂×C₃	C₃⋊(D₇₁⋊C₇)	1	4	yes	complemented
42	D₃	—	213	8	no	—
43	C₂²	—	639	1	no	Sylow
44	C₃	C₂×C₃⋊(D₇₁⋊C₇)	1	4	yes	complemented
45	C₂	—	639	2	no	—
46	C₂	C₃⋊(C₃⋊(D₇₁⋊C₇))	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₃⋊(C₃⋊(D₇₁⋊C₇))
commutator, cosocle, hypocenter	C₃²×C₇₁	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇₁	C₂×C₇
LCS	G ▷ C₃²×C₇₁
UCS	C₁ ◁ C₂

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⋊(C₃⋊(D₇₁⋊C₇))	C₂	direct (N × H)
4. C₃²×C₇₁⋊C₇	C₂²	semidirect
6. C₂×C₃×C₇₁⋊C₇	D₃	semidirect
7. C₂×C₃⋊D₂₁₃	C₇	semidirect
9. C₃×C₇₁⋊C₇	C₂×D₃	semidirect
10. C₂×C₃²×C₇₁	C₂×C₇	semidirect
11. C₃⋊D₂₁₃	C₂×C₇	semidirect
12. C₂×C₇₁⋊C₇	C₃⋊D₃	semidirect
15. C₃²×C₇₁	C₂²×C₇	semidirect
16. C₇₁⋊C₇	C₂×C₃⋊D₃	semidirect
17. C₂×C₃×C₇₁	D₃×C₇	semidirect
21. C₃×C₇₁	C₂×D₃×C₇	semidirect
22. C₂×C₇₁	C₇×C₃⋊D₃	semidirect
27. C₇₁	C₂×C₇×C₃⋊D₃	semidirect
34. C₂×C₃²	D₇₁⋊C₇	semidirect
39. C₃²	C₂×D₇₁⋊C₇	semidirect
41. C₂×C₃	C₃⋊(D₇₁⋊C₇)	semidirect
44. C₃	C₂×C₃⋊(D₇₁⋊C₇)	semidirect
46. C₂	C₃⋊(C₃⋊(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
C₃⋊(C₃⋊(D₇₁⋊C₇))	2
C₂×C₃⋊(D₇₁⋊C₇)	3
C₂×C₃⋊D₂₁₃	7
C₂×C₇×C₃⋊D₃	71

Subgroups — C2×C3⋊(C3⋊(D71⋊C7)) #31822066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊(C₃⋊(D₇₁⋊C₇)) #31822066