Subgroups — C₇×C₃⋊D₃×D₇₁ #31822091

Order 17892.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₃⋊D₂₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₃⋊D₃×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₇×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₇×D₇₁	—	3	4	no	maximal
5	C₃²×C₇×C₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×D₂₁₃	—	3	4	no	—
7	D₃×C₇×C₇₁	—	3	4	no	—
8	C₃×C₇×D₇₁	D₃	1	4	yes	complemented
9	C₃⋊D₃×D₇₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₇×D₇₁	—	9	1	no	Hall
11	C₃×C₇×C₇₁	C₂×D₃	1	4	yes	complemented
12	C₃⋊D₂₁₃	C₂×C₇	1	1	yes	characteristic, complemented
13	C₃⋊D₃×C₇₁	C₂×C₇	1	1	yes	characteristic, complemented
14	C₃²×D₇₁	C₂×C₇	1	1	yes	characteristic, complemented
15	C₂×C₇×C₇₁	—	9	1	no	—
16	C₇×D₇₁	—	9	1	no	—
17	C₇×D₇₁	C₃⋊D₃	1	1	yes	characteristic, direct product factor
18	D₃×D₇₁	—	3	4	no	—
19	C₃²×C₇₁	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
20	C₇×C₇₁	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
21	D₂₁₃	—	3	4	no	—
22	C₃×D₇₁	D₃×C₇	1	4	yes	complemented
23	D₃×C₇₁	—	3	4	no	—
24	C₂×D₇₁	—	9	1	no	Hall
25	C₂×C₇×C₃⋊D₃	—	71	1	no	maximal, Hall
26	C₃×C₇₁	C₂×D₃×C₇	1	4	yes	complemented
27	C₂×C₇₁	—	9	1	no	—
28	D₇₁	—	9	1	no	—
29	D₇₁	C₇×C₃⋊D₃	1	1	yes	characteristic, direct product factor
30	C₂×C₃²×C₇	—	71	1	no	—
31	C₇×C₃⋊D₃	—	71	1	no	—
32	C₇×C₃⋊D₃	D₇₁	1	1	yes	characteristic, direct product factor
33	C₂×D₃×C₇	—	213	4	no	—
34	C₇₁	C₂×C₇×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
35	C₃²×C₇	C₂×D₇₁	1	1	yes	characteristic, complemented, Hall
36	C₂×C₃×C₇	—	71	4	no	—
37	D₃×C₇	—	3	4	no	—
38	D₃×C₇	—	213	4	no	—
39	C₂×C₃⋊D₃	—	71	1	no	Hall
40	C₂²×C₇	—	639	1	no	Hall
41	C₃×C₇	D₃×D₇₁	1	4	yes	complemented
42	C₂×C₃²	—	71	1	no	—
43	C₃⋊D₃	—	71	1	no	—
44	C₃⋊D₃	C₇×D₇₁	1	1	yes	characteristic, direct product factor
45	C₂×C₇	—	9	1	no	—
46	C₂×C₇	—	71	1	no	—
47	C₂×C₇	—	639	1	no	—
48	C₂×D₃	—	213	4	no	—
49	C₃²	C₂×C₇×D₇₁	1	1	yes	characteristic, complemented, Sylow
50	C₇	C₃⋊D₃×D₇₁	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃	—	71	4	no	—
52	D₃	—	3	4	no	—
53	D₃	—	213	4	no	—
54	C₂²	—	639	1	no	Sylow
55	C₃	D₃×C₇×D₇₁	1	4	yes	complemented
56	C₂	—	9	1	no	—
57	C₂	—	71	1	no	—
58	C₂	—	639	1	no	—

	subgroup	G/H
center, hypercenter	C₇	C₃⋊D₃×D₇₁
commutator, cosocle, hypocenter	C₃²×C₇₁	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₇₁	C₂²
LCS	G ▷ C₃²×C₇₁
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₇×C₃⋊D₂₁₃	C₂	semidirect
2. C₇×C₃⋊D₃×C₇₁	C₂	semidirect
3. C₃²×C₇×D₇₁	C₂	semidirect
5. C₃²×C₇×C₇₁	C₂²	semidirect
8. C₃×C₇×D₇₁	D₃	semidirect
9. C₃⋊D₃×D₇₁	C₇	direct (N × H)
11. C₃×C₇×C₇₁	C₂×D₃	semidirect
12. C₃⋊D₂₁₃	C₂×C₇	semidirect
13. C₃⋊D₃×C₇₁	C₂×C₇	semidirect
14. C₃²×D₇₁	C₂×C₇	semidirect
17. C₇×D₇₁	C₃⋊D₃	direct (N × H)
19. C₃²×C₇₁	C₂²×C₇	semidirect
20. C₇×C₇₁	C₂×C₃⋊D₃	semidirect
22. C₃×D₇₁	D₃×C₇	semidirect
26. C₃×C₇₁	C₂×D₃×C₇	semidirect
29. D₇₁	C₇×C₃⋊D₃	direct (N × H)
32. C₇×C₃⋊D₃	D₇₁	direct (N × H)
34. C₇₁	C₂×C₇×C₃⋊D₃	semidirect
35. C₃²×C₇	C₂×D₇₁	semidirect
41. C₃×C₇	D₃×D₇₁	semidirect
44. C₃⋊D₃	C₇×D₇₁	direct (N × H)
49. C₃²	C₂×C₇×D₇₁	semidirect
50. C₇	C₃⋊D₃×D₇₁	direct (N × H)
55. C₃	D₃×C₇×D₇₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×C₃⋊D₂₁₃	2
C₇×C₃⋊D₃×C₇₁	2
C₃²×C₇×D₇₁	2
D₃×C₇×D₇₁	3
C₃⋊D₃×D₇₁	7
C₂×C₇×C₃⋊D₃	71

Subgroups — C7×C3⋊D3×D71 #31822091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₃⋊D₃×D₇₁ #31822091