Subgroups — C₃²×C₅×C₇×D₇ #17984399

Order 4410.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₇²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅×C₇×D₇	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₇×D₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₅×C₇²	C₂×C₃	1	4	yes	complemented
5	C₂×C₃²×C₅×C₇	—	7	1	no	maximal
6	C₃²×C₅×D₇	C₇	1	1	yes	maximal, characteristic, direct product factor
7	C₅×C₇×D₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₃²×C₇²	C₂×C₅	1	1	yes	characteristic, complemented, Hall
9	C₃²×C₅×C₇	—	2	3	no	—
10	C₃²×C₅×C₇	D₇	1	1	yes	characteristic, direct product factor
11	C₃²×C₅×C₇	C₂×C₇	1	1	yes	characteristic, complemented
12	C₃×C₇×D₇	C₃×C₅	1	4	yes	direct product factor
13	C₅×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₅×C₇	—	7	4	no	—
15	C₃×C₅×D₇	C₃×C₇	1	4	yes	direct product factor
16	C₃×C₇²	C₂×C₃×C₅	1	4	yes	complemented
17	C₂×C₃²×C₇	—	7	1	no	—
18	C₃²×D₇	C₅×C₇	1	1	yes	characteristic, direct product factor
19	C₃×C₅×C₇	—	2	12	no	—
20	C₃×C₅×C₇	C₃×D₇	1	4	yes	direct product factor
21	C₃×C₅×C₇	C₂×C₃×C₇	1	4	yes	complemented
22	C₇×D₇	C₃²×C₅	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₃²×C₅	—	7	1	no	Hall
24	C₂×C₅×C₇	—	7	1	no	—
25	C₅×D₇	C₃²×C₇	1	1	yes	characteristic, direct product factor
26	C₃²×C₇	—	2	3	no	—
27	C₃²×C₇	C₅×D₇	1	1	yes	characteristic, direct product factor
28	C₃²×C₇	C₂×C₅×C₇	1	1	yes	characteristic, complemented
29	C₇²	C₂×C₃²×C₅	1	1	yes	characteristic, complemented, Sylow
30	C₃²×C₅	C₇×D₇	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₇	—	7	4	no	—
32	C₃×D₇	C₃×C₅×C₇	1	4	yes	direct product factor
33	C₅×C₇	—	2	3	no	—
34	C₅×C₇	C₃²×D₇	1	1	yes	characteristic, direct product factor
35	C₅×C₇	C₂×C₃²×C₇	1	1	yes	characteristic, complemented
36	C₂×C₃×C₅	—	7	4	no	—
37	C₃×C₇	—	2	12	no	—
38	C₃×C₇	C₃×C₅×D₇	1	4	yes	direct product factor
39	C₃×C₇	C₂×C₃×C₅×C₇	1	4	yes	complemented
40	C₂×C₃²	—	7	1	no	Hall
41	C₃×C₅	C₃×C₇×D₇	1	4	yes	direct product factor
42	C₂×C₇	—	7	1	no	—
43	D₇	C₃²×C₅×C₇	1	1	yes	characteristic, direct product factor
44	C₂×C₅	—	7	1	no	Hall
45	C₃²	C₅×C₇×D₇	1	1	yes	characteristic, direct product factor, Sylow
46	C₇	—	2	3	no	—
47	C₇	C₃²×C₅×D₇	1	1	yes	characteristic, direct product factor
48	C₇	C₂×C₃²×C₅×C₇	1	1	yes	characteristic, complemented
49	C₂×C₃	—	7	4	no	—
50	C₅	C₃²×C₇×D₇	1	1	yes	characteristic, direct product factor, Sylow
51	C₃	C₃×C₅×C₇×D₇	1	4	yes	direct product factor
52	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃²×C₅×C₇	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₃²×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇²	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₃²×C₅×C₇

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₇²	C₂	semidirect
2. C₃×C₅×C₇×D₇	C₃	direct (N × H)
3. C₃²×C₇×D₇	C₅	direct (N × H)
4. C₃×C₅×C₇²	C₂×C₃	semidirect
6. C₃²×C₅×D₇	C₇	direct (N × H)
7. C₅×C₇×D₇	C₃²	direct (N × H)
8. C₃²×C₇²	C₂×C₅	semidirect
10. C₃²×C₅×C₇	D₇	direct (N × H)
11. C₃²×C₅×C₇	C₂×C₇	semidirect
12. C₃×C₇×D₇	C₃×C₅	direct (N × H)
13. C₅×C₇²	C₂×C₃²	semidirect
15. C₃×C₅×D₇	C₃×C₇	direct (N × H)
16. C₃×C₇²	C₂×C₃×C₅	semidirect
18. C₃²×D₇	C₅×C₇	direct (N × H)
20. C₃×C₅×C₇	C₃×D₇	direct (N × H)
21. C₃×C₅×C₇	C₂×C₃×C₇	semidirect
22. C₇×D₇	C₃²×C₅	direct (N × H)
25. C₅×D₇	C₃²×C₇	direct (N × H)
27. C₃²×C₇	C₅×D₇	direct (N × H)
28. C₃²×C₇	C₂×C₅×C₇	semidirect
29. C₇²	C₂×C₃²×C₅	semidirect
30. C₃²×C₅	C₇×D₇	direct (N × H)
32. C₃×D₇	C₃×C₅×C₇	direct (N × H)
34. C₅×C₇	C₃²×D₇	direct (N × H)
35. C₅×C₇	C₂×C₃²×C₇	semidirect
38. C₃×C₇	C₃×C₅×D₇	direct (N × H)
39. C₃×C₇	C₂×C₃×C₅×C₇	semidirect
41. C₃×C₅	C₃×C₇×D₇	direct (N × H)
43. D₇	C₃²×C₅×C₇	direct (N × H)
45. C₃²	C₅×C₇×D₇	direct (N × H)
47. C₇	C₃²×C₅×D₇	direct (N × H)
48. C₇	C₂×C₃²×C₅×C₇	semidirect
50. C₅	C₃²×C₇×D₇	direct (N × H)
51. C₃	C₃×C₅×C₇×D₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅×C₇²	2
C₃×C₅×C₇×D₇	3
C₃²×C₇×D₇	5
C₂×C₃²×C₅×C₇	7
C₃²×C₅×D₇	7

Subgroups — C32×C5×C7×D7 #17984399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×C₇×D₇ #17984399