Subgroups — C₃²×C₅×#41 #10624293

Order 720.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₅×D₄	C₂	1	3	yes	maximal, complemented
3	C₂×C₃²×C₄×C₅	C₂	1	3	yes	maximal, complemented
4	C₃×C₅×#41	C₃	1	4	yes	maximal, direct product factor
5	C₂²×C₃²×C₅	C₂²	1	3	yes	—
6	C₃²×C₄×C₅	C₂²	1	3	yes	—
7	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃²×#41	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×Q₈	C₂×C₃	1	4	yes	complemented
10	C₃×C₅×D₄	C₂×C₃	1	12	yes	complemented
11	C₂×C₃×C₄×C₅	C₂×C₃	1	12	yes	complemented
12	C₂×C₃²×C₅	—	2	3	no	—
13	C₂×C₃²×C₅	C₂³	1	1	yes	characteristic
14	C₅×#41	C₃²	1	1	yes	characteristic, direct product factor, Hall
15	C₃²×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
16	C₃²×D₄	C₂×C₅	1	3	yes	complemented
17	C₂×C₃²×C₄	C₂×C₅	1	3	yes	complemented
18	C₂²×C₃×C₅	C₂²×C₃	1	12	yes	—
19	C₃×C₄×C₅	C₂²×C₃	1	12	yes	—
20	C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
21	C₃×#41	C₃×C₅	1	4	yes	direct product factor
22	C₃²×C₅	#41	1	1	yes	characteristic, direct product factor, Hall
23	C₅×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
24	C₅×D₄	C₂×C₃²	1	3	yes	complemented
25	C₂×C₄×C₅	C₂×C₃²	1	3	yes	complemented
26	C₂²×C₃²	C₂²×C₅	1	3	yes	—
27	C₃²×C₄	C₂²×C₅	1	3	yes	—
28	C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
29	C₂×C₃×C₅	—	2	12	no	—
30	C₂×C₃×C₅	C₂³×C₃	1	4	yes	—
31	C₃×Q₈	C₂×C₃×C₅	1	4	yes	complemented
32	C₃×D₄	C₂×C₃×C₅	1	12	yes	complemented
33	C₂×C₃×C₄	C₂×C₃×C₅	1	12	yes	complemented
34	C₂²×C₅	C₂²×C₃²	1	3	yes	—
35	C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
36	C₄×C₅	C₂²×C₃²	1	3	yes	—
37	C₂×C₃²	—	2	3	no	—
38	C₂×C₃²	C₂³×C₅	1	1	yes	characteristic
39	#41	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
40	C₃×C₅	C₃×#41	1	4	yes	direct product factor
41	C₂²×C₃	C₂²×C₃×C₅	1	12	yes	—
42	C₃×C₄	C₂²×C₃×C₅	1	4	yes	—
43	C₃×C₄	C₂²×C₃×C₅	1	12	yes	—
44	C₂×C₅	—	2	3	no	—
45	C₂×C₅	C₂³×C₃²	1	1	yes	characteristic
46	C₃²	C₅×#41	1	1	yes	characteristic, direct product factor, Sylow
47	Q₈	C₂×C₃²×C₅	1	1	yes	characteristic, complemented
48	D₄	C₂×C₃²×C₅	1	3	yes	complemented
49	C₂×C₄	C₂×C₃²×C₅	1	3	yes	complemented
50	C₂×C₃	—	2	12	no	—
51	C₂×C₃	C₂³×C₃×C₅	1	4	yes	—
52	C₅	C₃²×#41	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	C₂²×C₃²×C₅	1	3	yes	—
54	C₄	C₂²×C₃²×C₅	1	1	yes	characteristic
55	C₄	C₂²×C₃²×C₅	1	3	yes	—
56	C₃	C₃×C₅×#41	1	4	yes	direct product factor
57	C₂	—	2	3	no	—
58	C₂	C₂³×C₃²×C₅	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄×C₅	C₂²
commutator, Frattini, cosocle	C₂	C₂³×C₃²×C₅
socle	C₂×C₃²×C₅	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×Q₈	C₂	semidirect
2. C₃²×C₅×D₄	C₂	semidirect
3. C₂×C₃²×C₄×C₅	C₂	semidirect
4. C₃×C₅×#41	C₃	direct (N × H)
8. C₃²×#41	C₅	direct (N × H)
9. C₃×C₅×Q₈	C₂×C₃	semidirect
10. C₃×C₅×D₄	C₂×C₃	semidirect
11. C₂×C₃×C₄×C₅	C₂×C₃	semidirect
14. C₅×#41	C₃²	direct (N × H)
15. C₃²×Q₈	C₂×C₅	semidirect
16. C₃²×D₄	C₂×C₅	semidirect
17. C₂×C₃²×C₄	C₂×C₅	semidirect
21. C₃×#41	C₃×C₅	direct (N × H)
22. C₃²×C₅	#41	direct (N × H)
23. C₅×Q₈	C₂×C₃²	semidirect
24. C₅×D₄	C₂×C₃²	semidirect
25. C₂×C₄×C₅	C₂×C₃²	semidirect
31. C₃×Q₈	C₂×C₃×C₅	semidirect
32. C₃×D₄	C₂×C₃×C₅	semidirect
33. C₂×C₃×C₄	C₂×C₃×C₅	semidirect
39. #41	C₃²×C₅	direct (N × H)
40. C₃×C₅	C₃×#41	direct (N × H)
46. C₃²	C₅×#41	direct (N × H)
47. Q₈	C₂×C₃²×C₅	semidirect
48. D₄	C₂×C₃²×C₅	semidirect
49. C₂×C₄	C₂×C₃²×C₅	semidirect
52. C₅	C₃²×#41	direct (N × H)
56. C₃	C₃×C₅×#41	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×Q₈	2
C₃²×C₅×D₄	2
C₂×C₃²×C₄×C₅	2
C₃×C₅×#41	3
C₃²×#41	5

Subgroups — C32×C5×#41 #10624293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×#41 #10624293