Subgroups — C₃×C₇²×D₇₅ #36347775

Order 22050.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇²×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×D₇₅	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₇²×D₂₅	—	3	1	no	maximal
4	C₃×C₇²×D₁₅	—	5	1	no	maximal
5	C₃×C₇²×C₂₅	—	2	1	no	—
6	C₃×C₇²×C₂₅	D₃	1	1	yes	characteristic, complemented
7	C₃×C₇²×C₂₅	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₇×D₇₅	C₇	1	8	yes	maximal, direct product factor
9	C₇²×D₂₅	—	3	1	no	Hall
10	C₃²×C₅×C₇²	D₅	1	1	yes	characteristic
11	C₃²×C₇×C₂₅	C₂×C₇	1	8	yes	complemented
12	C₇²×D₁₅	—	5	1	no	—
13	C₃×C₇²×D₅	—	15	1	no	—
14	C₇²×C₂₅	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₇×D₇₅	C₃×C₇	1	8	yes	direct product factor
16	C₃×C₇×D₂₅	—	3	8	no	—
17	C₃×D₃×C₇²	—	25	1	no	Hall
18	C₃×C₅×C₇²	—	2	1	no	—
19	C₃×C₅×C₇²	C₃×D₅	1	1	yes	characteristic
20	C₃×C₅×C₇²	D₁₅	1	1	yes	characteristic
21	C₃×C₇×D₁₅	—	5	8	no	—
22	C₃×C₇×C₂₅	—	2	8	no	—
23	C₃×C₇×C₂₅	D₃×C₇	1	8	yes	complemented
24	C₃×C₇×C₂₅	C₂×C₃×C₇	1	8	yes	complemented
25	C₇²×D₅	—	15	1	no	—
26	C₃×D₇₅	C₇²	1	1	yes	characteristic, direct product factor, Hall
27	C₃²×C₇²	D₂₅	1	1	yes	characteristic, complemented, Hall
28	C₇×D₂₅	—	3	8	no	—
29	C₃²×C₅×C₇	C₇×D₅	1	8	yes	—
30	C₂×C₃×C₇²	—	75	1	no	—
31	D₃×C₇²	—	25	1	no	—
32	C₅×C₇²	C₃×D₁₅	1	1	yes	characteristic
33	C₃²×C₂₅	C₂×C₇²	1	1	yes	characteristic, complemented, Hall
34	C₇×D₁₅	—	5	8	no	—
35	C₃×C₇×D₅	—	15	8	no	—
36	C₇×C₂₅	C₃×D₃×C₇	1	8	yes	complemented
37	D₇₅	C₃×C₇²	1	1	yes	characteristic, direct product factor
38	C₃×D₂₅	—	3	1	no	—
39	C₃×C₇²	—	2	1	no	—
40	C₃×C₇²	C₃×D₂₅	1	1	yes	characteristic, complemented
41	C₃×C₇²	D₇₅	1	1	yes	characteristic, direct product factor
42	C₃×D₃×C₇	—	25	8	no	—
43	C₃×C₅×C₇	—	2	8	no	—
44	C₃×C₅×C₇	C₃×C₇×D₅	1	8	yes	—
45	C₃×C₅×C₇	C₇×D₁₅	1	8	yes	—
46	C₂×C₇²	—	75	1	no	Hall
47	C₃×D₁₅	—	5	1	no	—
48	C₃×C₂₅	—	2	1	no	—
49	C₃×C₂₅	D₃×C₇²	1	1	yes	characteristic, complemented
50	C₃×C₂₅	C₂×C₃×C₇²	1	1	yes	characteristic, complemented
51	C₇×D₅	—	15	8	no	—
52	C₃²×C₇	C₇×D₂₅	1	8	yes	complemented
53	D₂₅	—	3	1	no	Hall
54	C₇²	C₃×D₇₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₃²×C₅	C₇²×D₅	1	1	yes	characteristic
56	C₂×C₃×C₇	—	75	8	no	—
57	D₃×C₇	—	25	8	no	—
58	C₅×C₇	C₃×C₇×D₁₅	1	8	yes	—
59	D₁₅	—	5	1	no	—
60	C₃×D₅	—	15	1	no	—
61	C₂₅	C₃×D₃×C₇²	1	1	yes	characteristic, complemented, Sylow
62	C₃×C₇	—	2	8	no	—
63	C₃×C₇	C₃×C₇×D₂₅	1	8	yes	complemented
64	C₃×C₇	C₇×D₇₅	1	8	yes	direct product factor
65	C₃×D₃	—	25	1	no	Hall
66	C₃×C₅	—	2	1	no	—
67	C₃×C₅	C₃×C₇²×D₅	1	1	yes	characteristic
68	C₃×C₅	C₇²×D₁₅	1	1	yes	characteristic
69	C₂×C₇	—	75	8	no	—
70	D₅	—	15	1	no	—
71	C₃²	C₇²×D₂₅	1	1	yes	characteristic, complemented, Sylow
72	C₇	C₃×C₇×D₇₅	1	8	yes	direct product factor
73	C₂×C₃	—	75	1	no	—
74	D₃	—	25	1	no	—
75	C₅	C₃×C₇²×D₁₅	1	1	yes	characteristic
76	C₃	—	2	1	no	—
77	C₃	C₃×C₇²×D₂₅	1	1	yes	characteristic, complemented
78	C₃	C₇²×D₇₅	1	1	yes	characteristic, direct product factor
79	C₂	—	75	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₇²	D₇₅
commutator, cosocle, hypocenter	C₃×C₂₅	C₂×C₃×C₇²
Frattini	C₅	C₃×C₇²×D₁₅
Fitting	C₃²×C₇²×C₂₅	C₂
socle	C₃²×C₅×C₇²	D₅
LCS	G ▷ C₃×C₂₅
UCS	C₁ ◁ C₃×C₇²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₇²×C₂₅	C₂	semidirect
2. C₇²×D₇₅	C₃	direct (N × H)
6. C₃×C₇²×C₂₅	D₃	semidirect
7. C₃×C₇²×C₂₅	C₂×C₃	semidirect
8. C₃×C₇×D₇₅	C₇	direct (N × H)
11. C₃²×C₇×C₂₅	C₂×C₇	semidirect
14. C₇²×C₂₅	C₃×D₃	semidirect
15. C₇×D₇₅	C₃×C₇	direct (N × H)
23. C₃×C₇×C₂₅	D₃×C₇	semidirect
24. C₃×C₇×C₂₅	C₂×C₃×C₇	semidirect
26. C₃×D₇₅	C₇²	direct (N × H)
27. C₃²×C₇²	D₂₅	semidirect
33. C₃²×C₂₅	C₂×C₇²	semidirect
36. C₇×C₂₅	C₃×D₃×C₇	semidirect
37. D₇₅	C₃×C₇²	direct (N × H)
40. C₃×C₇²	C₃×D₂₅	semidirect
41. C₃×C₇²	D₇₅	direct (N × H)
49. C₃×C₂₅	D₃×C₇²	semidirect
50. C₃×C₂₅	C₂×C₃×C₇²	semidirect
52. C₃²×C₇	C₇×D₂₅	semidirect
54. C₇²	C₃×D₇₅	direct (N × H)
61. C₂₅	C₃×D₃×C₇²	semidirect
63. C₃×C₇	C₃×C₇×D₂₅	semidirect
64. C₃×C₇	C₇×D₇₅	direct (N × H)
71. C₃²	C₇²×D₂₅	semidirect
72. C₇	C₃×C₇×D₇₅	direct (N × H)
77. C₃	C₃×C₇²×D₂₅	semidirect
78. C₃	C₇²×D₇₅	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×C₇²×C₂₅	2
C₇²×D₇₅	3
C₃×C₇²×D₂₅	3
C₃×C₇²×D₁₅	5
C₃×C₇×D₇₅	7

Subgroups — C3×C72×D75 #36347775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇²×D₇₅ #36347775