Subgroups — C₅×C₇²×C₃⋊D₁₅ #36348080

Order 22050.

600 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	C₅×C₇²×D₁₅	—	3	4	no	maximal
3	C₇²×C₃⋊D₁₅	C₅	1	1	yes	maximal, characteristic, direct product factor
4	C₅×C₇²×C₃⋊D₃	—	5	1	no	maximal
5	C₃×C₅²×C₇²	D₃	1	4	yes	complemented
6	C₅×C₇×C₃⋊D₁₅	C₇	1	8	yes	maximal, direct product factor
7	C₅×C₇²×D₅	—	9	1	no	Hall
8	C₃²×C₅×C₇²	—	2	2	no	—
9	C₃²×C₅×C₇²	D₅	1	1	yes	characteristic, complemented
10	C₃²×C₅×C₇²	C₂×C₅	1	1	yes	characteristic, complemented
11	C₃²×C₅²×C₇	C₂×C₇	1	8	yes	complemented
12	C₇²×D₁₅	—	3	4	no	—
13	C₅×D₃×C₇²	—	15	4	no	—
14	C₅²×C₇²	C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
15	C₅×C₇×D₁₅	—	3	32	no	—
16	C₇²×C₃⋊D₃	—	5	1	no	Hall
17	C₃×C₅×C₇²	—	2	8	no	—
18	C₃×C₅×C₇²	C₅×D₃	1	4	yes	complemented
19	C₃×C₅×C₇²	D₁₅	1	4	yes	complemented
20	C₇×C₃⋊D₁₅	C₅×C₇	1	8	yes	direct product factor
21	C₅×C₇×C₃⋊D₃	—	5	8	no	—
22	C₃×C₅²×C₇	D₃×C₇	1	32	yes	complemented
23	C₂×C₅×C₇²	—	45	1	no	—
24	C₇²×D₅	—	9	1	no	—
25	C₅×C₃⋊D₁₅	C₇²	1	1	yes	characteristic, direct product factor, Hall
26	C₃²×C₇²	C₅×D₅	1	1	yes	characteristic, complemented, Hall
27	C₅×C₇×D₅	—	9	8	no	—
28	C₃²×C₅×C₇	—	2	16	no	—
29	C₃²×C₅×C₇	C₇×D₅	1	8	yes	complemented
30	C₃²×C₅×C₇	C₂×C₅×C₇	1	8	yes	complemented
31	D₃×C₇²	—	15	4	no	—
32	C₅×C₇²	—	2	2	no	—
33	C₅×C₇²	C₅×C₃⋊D₃	1	1	yes	characteristic, complemented
34	C₅×C₇²	C₃⋊D₁₅	1	1	yes	characteristic, direct product factor
35	C₃²×C₅²	C₂×C₇²	1	1	yes	characteristic, complemented, Hall
36	C₇×D₁₅	—	3	32	no	—
37	C₅×D₃×C₇	—	15	32	no	—
38	C₅²×C₇	C₇×C₃⋊D₃	1	8	yes	complemented
39	C₅×D₁₅	—	3	4	no	—
40	C₃×C₇²	C₅×D₁₅	1	4	yes	complemented
41	C₇×C₃⋊D₃	—	5	8	no	—
42	C₃×C₅×C₇	—	2	64	no	—
43	C₃×C₅×C₇	C₅×D₃×C₇	1	32	yes	complemented
44	C₃×C₅×C₇	C₇×D₁₅	1	32	yes	complemented
45	C₂×C₇²	—	45	1	no	Hall
46	C₃⋊D₁₅	C₅×C₇²	1	1	yes	characteristic, direct product factor
47	C₅×C₃⋊D₃	—	5	1	no	—
48	C₃×C₅²	D₃×C₇²	1	4	yes	complemented
49	C₂×C₅×C₇	—	45	8	no	—
50	C₇×D₅	—	9	8	no	—
51	C₃²×C₇	C₅×C₇×D₅	1	8	yes	complemented
52	C₅×D₅	—	9	1	no	Hall
53	C₇²	C₅×C₃⋊D₁₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₃²×C₅	—	2	2	no	—
55	C₃²×C₅	C₇²×D₅	1	1	yes	characteristic, complemented
56	C₃²×C₅	C₂×C₅×C₇²	1	1	yes	characteristic, complemented
57	D₃×C₇	—	15	32	no	—
58	C₅×C₇	—	2	16	no	—
59	C₅×C₇	C₅×C₇×C₃⋊D₃	1	8	yes	complemented
60	C₅×C₇	C₇×C₃⋊D₁₅	1	8	yes	direct product factor
61	D₁₅	—	3	4	no	—
62	C₅×D₃	—	15	4	no	—
63	C₅²	C₇²×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
64	C₃×C₇	C₅×C₇×D₁₅	1	32	yes	complemented
65	C₃⋊D₃	—	5	1	no	Hall
66	C₃×C₅	—	2	8	no	—
67	C₃×C₅	C₅×D₃×C₇²	1	4	yes	complemented
68	C₃×C₅	C₇²×D₁₅	1	4	yes	complemented
69	C₂×C₇	—	45	8	no	—
70	C₂×C₅	—	45	1	no	—
71	D₅	—	9	1	no	—
72	C₃²	C₅×C₇²×D₅	1	1	yes	characteristic, complemented, Sylow
73	C₇	C₅×C₇×C₃⋊D₁₅	1	8	yes	direct product factor
74	D₃	—	15	4	no	—
75	C₅	—	2	2	no	—
76	C₅	C₅×C₇²×C₃⋊D₃	1	1	yes	characteristic, complemented
77	C₅	C₇²×C₃⋊D₁₅	1	1	yes	characteristic, direct product factor
78	C₃	C₅×C₇²×D₁₅	1	4	yes	complemented
79	C₂	—	45	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₇²	C₃⋊D₁₅
commutator, cosocle, hypocenter	C₃²×C₅	C₂×C₅×C₇²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇²	C₂
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₅×C₇²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₇²	C₂	semidirect
3. C₇²×C₃⋊D₁₅	C₅	direct (N × H)
5. C₃×C₅²×C₇²	D₃	semidirect
6. C₅×C₇×C₃⋊D₁₅	C₇	direct (N × H)
9. C₃²×C₅×C₇²	D₅	semidirect
10. C₃²×C₅×C₇²	C₂×C₅	semidirect
11. C₃²×C₅²×C₇	C₂×C₇	semidirect
14. C₅²×C₇²	C₃⋊D₃	semidirect
18. C₃×C₅×C₇²	C₅×D₃	semidirect
19. C₃×C₅×C₇²	D₁₅	semidirect
20. C₇×C₃⋊D₁₅	C₅×C₇	direct (N × H)
22. C₃×C₅²×C₇	D₃×C₇	semidirect
25. C₅×C₃⋊D₁₅	C₇²	direct (N × H)
26. C₃²×C₇²	C₅×D₅	semidirect
29. C₃²×C₅×C₇	C₇×D₅	semidirect
30. C₃²×C₅×C₇	C₂×C₅×C₇	semidirect
33. C₅×C₇²	C₅×C₃⋊D₃	semidirect
34. C₅×C₇²	C₃⋊D₁₅	direct (N × H)
35. C₃²×C₅²	C₂×C₇²	semidirect
38. C₅²×C₇	C₇×C₃⋊D₃	semidirect
40. C₃×C₇²	C₅×D₁₅	semidirect
43. C₃×C₅×C₇	C₅×D₃×C₇	semidirect
44. C₃×C₅×C₇	C₇×D₁₅	semidirect
46. C₃⋊D₁₅	C₅×C₇²	direct (N × H)
48. C₃×C₅²	D₃×C₇²	semidirect
51. C₃²×C₇	C₅×C₇×D₅	semidirect
53. C₇²	C₅×C₃⋊D₁₅	direct (N × H)
55. C₃²×C₅	C₇²×D₅	semidirect
56. C₃²×C₅	C₂×C₅×C₇²	semidirect
59. C₅×C₇	C₅×C₇×C₃⋊D₃	semidirect
60. C₅×C₇	C₇×C₃⋊D₁₅	direct (N × H)
63. C₅²	C₇²×C₃⋊D₃	semidirect
64. C₃×C₇	C₅×C₇×D₁₅	semidirect
67. C₃×C₅	C₅×D₃×C₇²	semidirect
68. C₃×C₅	C₇²×D₁₅	semidirect
72. C₃²	C₅×C₇²×D₅	semidirect
73. C₇	C₅×C₇×C₃⋊D₁₅	direct (N × H)
76. C₅	C₅×C₇²×C₃⋊D₃	semidirect
77. C₅	C₇²×C₃⋊D₁₅	direct (N × H)
78. C₃	C₅×C₇²×D₁₅	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
C₅×C₇²×D₁₅	3
C₇²×C₃⋊D₁₅	5
C₅×C₇²×C₃⋊D₃	5
C₅×C₇×C₃⋊D₁₅	7

Subgroups — C5×C72×C3⋊D15 #36348080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇²×C₃⋊D₁₅ #36348080