Subgroups — C₇²×C₃⋊(C₃⋊(C₅⋊D₅)) #36348083

Order 22050.

960 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₇²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×C₃⋊(C₅⋊D₅)	—	3	4	no	maximal
3	C₇²×C₃⋊D₁₅	—	5	6	no	maximal
4	C₃×C₅²×C₇²	D₃	1	4	yes	complemented
5	C₇×C₃⋊(C₃⋊(C₅⋊D₅))	C₇	1	8	yes	maximal, direct product factor
6	C₇²×C₅⋊D₅	—	9	1	no	Hall
7	C₃²×C₅×C₇²	D₅	1	6	yes	complemented
8	C₃²×C₅²×C₇	C₂×C₇	1	8	yes	complemented
9	C₇²×D₁₅	—	15	24	no	—
10	C₅²×C₇²	C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
11	C₇×C₃⋊(C₅⋊D₅)	—	3	32	no	—
12	C₇²×C₃⋊D₃	—	25	1	no	Hall
13	C₃×C₅×C₇²	D₁₅	1	24	yes	complemented
14	C₇×C₃⋊D₁₅	—	5	48	no	—
15	C₃×C₅²×C₇	D₃×C₇	1	32	yes	complemented
16	C₇²×D₅	—	45	6	no	—
17	C₃⋊(C₃⋊(C₅⋊D₅))	C₇²	1	1	yes	characteristic, direct product factor, Hall
18	C₃²×C₇²	C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
19	C₇×C₅⋊D₅	—	9	8	no	—
20	C₃²×C₅×C₇	C₇×D₅	1	48	yes	complemented
21	D₃×C₇²	—	75	4	no	—
22	C₅×C₇²	C₃⋊D₁₅	1	6	yes	complemented
23	C₃²×C₅²	C₂×C₇²	1	1	yes	characteristic, complemented, Hall
24	C₇×D₁₅	—	15	192	no	—
25	C₅²×C₇	C₇×C₃⋊D₃	1	8	yes	complemented
26	C₃⋊(C₅⋊D₅)	—	3	4	no	—
27	C₃×C₇²	C₃⋊(C₅⋊D₅)	1	4	yes	complemented
28	C₇×C₃⋊D₃	—	25	8	no	—
29	C₃×C₅×C₇	C₇×D₁₅	1	192	yes	complemented
30	C₂×C₇²	—	225	1	no	Hall
31	C₃⋊D₁₅	—	5	6	no	—
32	C₃×C₅²	D₃×C₇²	1	4	yes	complemented
33	C₇×D₅	—	45	48	no	—
34	C₃²×C₇	C₇×C₅⋊D₅	1	8	yes	complemented
35	C₅⋊D₅	—	9	1	no	Hall
36	C₇²	C₃⋊(C₃⋊(C₅⋊D₅))	1	1	yes	characteristic, direct product factor, Sylow
37	C₃²×C₅	C₇²×D₅	1	6	yes	complemented
38	D₃×C₇	—	75	32	no	—
39	C₅×C₇	C₇×C₃⋊D₁₅	1	48	yes	complemented
40	D₁₅	—	15	24	no	—
41	C₅²	C₇²×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
42	C₃×C₇	C₇×C₃⋊(C₅⋊D₅)	1	32	yes	complemented
43	C₃⋊D₃	—	25	1	no	Hall
44	C₃×C₅	C₇²×D₁₅	1	24	yes	complemented
45	C₂×C₇	—	225	8	no	—
46	D₅	—	45	6	no	—
47	C₃²	C₇²×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
48	C₇	C₇×C₃⋊(C₃⋊(C₅⋊D₅))	1	8	yes	direct product factor
49	D₃	—	75	4	no	—
50	C₅	C₇²×C₃⋊D₁₅	1	6	yes	complemented
51	C₃	C₇²×C₃⋊(C₅⋊D₅)	1	4	yes	complemented
52	C₂	—	225	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇²	C₃⋊(C₃⋊(C₅⋊D₅))
commutator, cosocle, hypocenter	C₃²×C₅²	C₂×C₇²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇²	C₂
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₇²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₇²	C₂	semidirect
4. C₃×C₅²×C₇²	D₃	semidirect
5. C₇×C₃⋊(C₃⋊(C₅⋊D₅))	C₇	direct (N × H)
7. C₃²×C₅×C₇²	D₅	semidirect
8. C₃²×C₅²×C₇	C₂×C₇	semidirect
10. C₅²×C₇²	C₃⋊D₃	semidirect
13. C₃×C₅×C₇²	D₁₅	semidirect
15. C₃×C₅²×C₇	D₃×C₇	semidirect
17. C₃⋊(C₃⋊(C₅⋊D₅))	C₇²	direct (N × H)
18. C₃²×C₇²	C₅⋊D₅	semidirect
20. C₃²×C₅×C₇	C₇×D₅	semidirect
22. C₅×C₇²	C₃⋊D₁₅	semidirect
23. C₃²×C₅²	C₂×C₇²	semidirect
25. C₅²×C₇	C₇×C₃⋊D₃	semidirect
27. C₃×C₇²	C₃⋊(C₅⋊D₅)	semidirect
29. C₃×C₅×C₇	C₇×D₁₅	semidirect
32. C₃×C₅²	D₃×C₇²	semidirect
34. C₃²×C₇	C₇×C₅⋊D₅	semidirect
36. C₇²	C₃⋊(C₃⋊(C₅⋊D₅))	direct (N × H)
37. C₃²×C₅	C₇²×D₅	semidirect
39. C₅×C₇	C₇×C₃⋊D₁₅	semidirect
41. C₅²	C₇²×C₃⋊D₃	semidirect
42. C₃×C₇	C₇×C₃⋊(C₅⋊D₅)	semidirect
44. C₃×C₅	C₇²×D₁₅	semidirect
47. C₃²	C₇²×C₅⋊D₅	semidirect
48. C₇	C₇×C₃⋊(C₃⋊(C₅⋊D₅))	direct (N × H)
50. C₅	C₇²×C₃⋊D₁₅	semidirect
51. C₃	C₇²×C₃⋊(C₅⋊D₅)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅²×C₇²	2
C₇²×C₃⋊(C₅⋊D₅)	3
C₇²×C₃⋊D₁₅	5
C₇×C₃⋊(C₃⋊(C₅⋊D₅))	7

Subgroups — C72×C3⋊(C3⋊(C5⋊D5)) #36348083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×C₃⋊(C₃⋊(C₅⋊D₅)) #36348083