Subgroups — C₃×C₇²×D₁₅ #17984407

Order 4410.

180 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	C₇²×D₁₅	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₇²×D₅	—	3	1	no	maximal
4	C₃×D₃×C₇²	—	5	1	no	maximal, Hall
5	C₃×C₅×C₇²	—	2	1	no	—
6	C₃×C₅×C₇²	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₇×D₁₅	C₇	1	8	yes	maximal, direct product factor
9	C₇²×D₅	—	3	1	no	Hall
10	C₃²×C₇²	D₅	1	1	yes	characteristic, complemented, Hall
11	C₃²×C₅×C₇	C₂×C₇	1	8	yes	complemented
12	C₂×C₃×C₇²	—	15	1	no	—
13	D₃×C₇²	—	5	1	no	—
14	C₅×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₇×D₁₅	C₃×C₇	1	8	yes	direct product factor
16	C₃×C₇×D₅	—	3	8	no	—
17	C₃×C₇²	—	2	1	no	—
18	C₃×C₇²	C₃×D₅	1	1	yes	characteristic, complemented
19	C₃×C₇²	D₁₅	1	1	yes	characteristic, direct product factor
20	C₃×D₃×C₇	—	5	8	no	—
21	C₃×C₅×C₇	—	2	8	no	—
22	C₃×C₅×C₇	D₃×C₇	1	8	yes	complemented
23	C₃×C₅×C₇	C₂×C₃×C₇	1	8	yes	complemented
24	C₂×C₇²	—	15	1	no	Hall
25	C₃×D₁₅	C₇²	1	1	yes	characteristic, direct product factor, Hall
26	C₇×D₅	—	3	8	no	—
27	C₃²×C₇	C₇×D₅	1	8	yes	complemented
28	C₇²	C₃×D₁₅	1	1	yes	characteristic, direct product factor, Sylow
29	C₃²×C₅	C₂×C₇²	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₇	—	15	8	no	—
31	D₃×C₇	—	5	8	no	—
32	C₅×C₇	C₃×D₃×C₇	1	8	yes	complemented
33	D₁₅	C₃×C₇²	1	1	yes	characteristic, direct product factor
34	C₃×D₅	—	3	1	no	—
35	C₃×C₇	—	2	8	no	—
36	C₃×C₇	C₃×C₇×D₅	1	8	yes	complemented
37	C₃×C₇	C₇×D₁₅	1	8	yes	direct product factor
38	C₃×D₃	—	5	1	no	Hall
39	C₃×C₅	—	2	1	no	—
40	C₃×C₅	D₃×C₇²	1	1	yes	characteristic, complemented
41	C₃×C₅	C₂×C₃×C₇²	1	1	yes	characteristic, complemented
42	C₂×C₇	—	15	8	no	—
43	D₅	—	3	1	no	Hall
44	C₃²	C₇²×D₅	1	1	yes	characteristic, complemented, Sylow
45	C₇	C₃×C₇×D₁₅	1	8	yes	direct product factor
46	C₂×C₃	—	15	1	no	—
47	D₃	—	5	1	no	—
48	C₅	C₃×D₃×C₇²	1	1	yes	characteristic, complemented, Sylow
49	C₃	—	2	1	no	—
50	C₃	C₃×C₇²×D₅	1	1	yes	characteristic, complemented
51	C₃	C₇²×D₁₅	1	1	yes	characteristic, direct product factor
52	C₂	—	15	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₇²	D₁₅
commutator, cosocle, hypocenter	C₃×C₅	C₂×C₃×C₇²
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇²	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃×C₇²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₇²	C₂	semidirect
2. C₇²×D₁₅	C₃	direct (N × H)
6. C₃×C₅×C₇²	D₃	semidirect
7. C₃×C₅×C₇²	C₂×C₃	semidirect
8. C₃×C₇×D₁₅	C₇	direct (N × H)
10. C₃²×C₇²	D₅	semidirect
11. C₃²×C₅×C₇	C₂×C₇	semidirect
14. C₅×C₇²	C₃×D₃	semidirect
15. C₇×D₁₅	C₃×C₇	direct (N × H)
18. C₃×C₇²	C₃×D₅	semidirect
19. C₃×C₇²	D₁₅	direct (N × H)
22. C₃×C₅×C₇	D₃×C₇	semidirect
23. C₃×C₅×C₇	C₂×C₃×C₇	semidirect
25. C₃×D₁₅	C₇²	direct (N × H)
27. C₃²×C₇	C₇×D₅	semidirect
28. C₇²	C₃×D₁₅	direct (N × H)
29. C₃²×C₅	C₂×C₇²	semidirect
32. C₅×C₇	C₃×D₃×C₇	semidirect
33. D₁₅	C₃×C₇²	direct (N × H)
36. C₃×C₇	C₃×C₇×D₅	semidirect
37. C₃×C₇	C₇×D₁₅	direct (N × H)
40. C₃×C₅	D₃×C₇²	semidirect
41. C₃×C₅	C₂×C₃×C₇²	semidirect
44. C₃²	C₇²×D₅	semidirect
45. C₇	C₃×C₇×D₁₅	direct (N × H)
48. C₅	C₃×D₃×C₇²	semidirect
50. C₃	C₃×C₇²×D₅	semidirect
51. C₃	C₇²×D₁₅	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
C₇²×D₁₅	3
C₃×C₇²×D₅	3
C₃×D₃×C₇²	5
C₃×C₇×D₁₅	7

Subgroups — C3×C72×D15 #17984407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇²×D₁₅ #17984407