Subgroups — C₅²×C₃⋊(C₃⋊(C₇⋊D₇)) #36348079

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	C₅²×C₃⋊(C₇⋊D₇)	—	3	4	no	maximal
3	C₅×C₃⋊(C₃⋊(C₇⋊D₇))	C₅	1	6	yes	maximal, direct product factor
4	C₃×C₅²×C₇²	D₃	1	4	yes	complemented
5	C₅²×C₃⋊D₂₁	—	7	8	no	maximal
6	C₅²×C₇⋊D₇	—	9	1	no	Hall
7	C₃²×C₅×C₇²	C₂×C₅	1	6	yes	complemented
8	C₃²×C₅²×C₇	D₇	1	8	yes	complemented
9	C₅×C₃⋊(C₇⋊D₇)	—	3	24	no	—
10	C₅²×C₇²	C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
11	C₅²×D₂₁	—	21	32	no	—
12	C₃⋊(C₃⋊(C₇⋊D₇))	C₅²	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₅×C₇²	C₅×D₃	1	24	yes	complemented
14	C₅×C₃⋊D₂₁	—	7	48	no	—
15	C₃×C₅²×C₇	D₂₁	1	32	yes	complemented
16	C₅×C₇⋊D₇	—	9	6	no	—
17	C₅²×C₃⋊D₃	—	49	1	no	Hall
18	C₃²×C₇²	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
19	C₅²×D₇	—	63	8	no	—
20	C₃²×C₅×C₇	C₅×D₇	1	48	yes	complemented
21	C₃⋊(C₇⋊D₇)	—	3	4	no	—
22	C₅×C₇²	C₅×C₃⋊D₃	1	6	yes	complemented
23	C₃²×C₅²	C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
24	C₅×D₂₁	—	21	192	no	—
25	C₅²×C₇	C₃⋊D₂₁	1	8	yes	complemented
26	C₅²×D₃	—	147	4	no	—
27	C₃×C₇²	C₅²×D₃	1	4	yes	complemented
28	C₃⋊D₂₁	—	7	8	no	—
29	C₃×C₅×C₇	C₅×D₂₁	1	192	yes	complemented
30	C₇⋊D₇	—	9	1	no	Hall
31	C₅×C₃⋊D₃	—	49	6	no	—
32	C₃×C₅²	C₃⋊(C₇⋊D₇)	1	4	yes	complemented
33	C₅×D₇	—	63	48	no	—
34	C₃²×C₇	C₅²×D₇	1	8	yes	complemented
35	C₂×C₅²	—	441	1	no	Hall
36	C₇²	C₅²×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
37	C₃²×C₅	C₅×C₇⋊D₇	1	6	yes	complemented
38	D₂₁	—	21	32	no	—
39	C₅×C₇	C₅×C₃⋊D₂₁	1	48	yes	complemented
40	C₅×D₃	—	147	24	no	—
41	C₅²	C₃⋊(C₃⋊(C₇⋊D₇))	1	1	yes	characteristic, direct product factor, Sylow
42	C₃×C₇	C₅²×D₂₁	1	32	yes	complemented
43	C₃⋊D₃	—	49	1	no	Hall
44	C₃×C₅	C₅×C₃⋊(C₇⋊D₇)	1	24	yes	complemented
45	D₇	—	63	8	no	—
46	C₂×C₅	—	441	6	no	—
47	C₃²	C₅²×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
48	C₇	C₅²×C₃⋊D₂₁	1	8	yes	complemented
49	D₃	—	147	4	no	—
50	C₅	C₅×C₃⋊(C₃⋊(C₇⋊D₇))	1	6	yes	direct product factor
51	C₃	C₅²×C₃⋊(C₇⋊D₇)	1	4	yes	complemented
52	C₂	—	441	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅²	C₃⋊(C₃⋊(C₇⋊D₇))
commutator, cosocle, hypocenter	C₃²×C₇²	C₂×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇²	C₂
LCS	G ▷ C₃²×C₇²
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₇²	C₂	semidirect
3. C₅×C₃⋊(C₃⋊(C₇⋊D₇))	C₅	direct (N × H)
4. C₃×C₅²×C₇²	D₃	semidirect
7. C₃²×C₅×C₇²	C₂×C₅	semidirect
8. C₃²×C₅²×C₇	D₇	semidirect
10. C₅²×C₇²	C₃⋊D₃	semidirect
12. C₃⋊(C₃⋊(C₇⋊D₇))	C₅²	direct (N × H)
13. C₃×C₅×C₇²	C₅×D₃	semidirect
15. C₃×C₅²×C₇	D₂₁	semidirect
18. C₃²×C₇²	C₂×C₅²	semidirect
20. C₃²×C₅×C₇	C₅×D₇	semidirect
22. C₅×C₇²	C₅×C₃⋊D₃	semidirect
23. C₃²×C₅²	C₇⋊D₇	semidirect
25. C₅²×C₇	C₃⋊D₂₁	semidirect
27. C₃×C₇²	C₅²×D₃	semidirect
29. C₃×C₅×C₇	C₅×D₂₁	semidirect
32. C₃×C₅²	C₃⋊(C₇⋊D₇)	semidirect
34. C₃²×C₇	C₅²×D₇	semidirect
36. C₇²	C₅²×C₃⋊D₃	semidirect
37. C₃²×C₅	C₅×C₇⋊D₇	semidirect
39. C₅×C₇	C₅×C₃⋊D₂₁	semidirect
41. C₅²	C₃⋊(C₃⋊(C₇⋊D₇))	direct (N × H)
42. C₃×C₇	C₅²×D₂₁	semidirect
44. C₃×C₅	C₅×C₃⋊(C₇⋊D₇)	semidirect
47. C₃²	C₅²×C₇⋊D₇	semidirect
48. C₇	C₅²×C₃⋊D₂₁	semidirect
50. C₅	C₅×C₃⋊(C₃⋊(C₇⋊D₇))	direct (N × H)
51. C₃	C₅²×C₃⋊(C₇⋊D₇)	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
C₅²×C₃⋊(C₇⋊D₇)	3
C₅×C₃⋊(C₃⋊(C₇⋊D₇))	5
C₅²×C₃⋊D₂₁	7

Subgroups — C52×C3⋊(C3⋊(C7⋊D7)) #36348079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₃⋊(C₃⋊(C₇⋊D₇)) #36348079