Subgroups — C₂²×C₅×#41 #66293

Order 320.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#41	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×D₄	C₂	1	3	yes	maximal, complemented
4	C₂³×C₄×C₅	C₂	1	3	yes	maximal, complemented
5	C₂⁴×C₅	C₂²	1	3	yes	—
6	C₅×#41	C₂²	1	64	yes	direct product factor
7	C₂×C₅×Q₈	C₂²	1	12	yes	complemented
8	C₂×C₅×D₄	C₂²	1	36	yes	complemented
9	C₂²×C₄×C₅	C₂²	1	3	yes	—
10	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅	C₂²	1	36	yes	complemented
12	C₂²×#41	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₅	—	2	12	no	—
14	C₂³×C₅	C₂³	1	18	yes	—
15	C₂³×C₅	C₂³	1	1	yes	characteristic
16	C₅×Q₈	C₂³	1	16	yes	complemented
17	C₅×D₄	C₂³	1	48	yes	complemented
18	C₂×C₄×C₅	C₂³	1	18	yes	—
19	C₂×C₄×C₅	C₂³	1	6	yes	—
20	C₂×C₄×C₅	C₂³	1	48	yes	complemented
21	C₂×#41	C₂×C₅	1	24	yes	direct product factor
22	C₂²×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
23	C₂²×D₄	C₂×C₅	1	3	yes	complemented
24	C₂³×C₄	C₂×C₅	1	3	yes	complemented
25	C₂²×C₅	—	2	36	no	—
26	C₂²×C₅	#41	1	4	yes	direct product factor
27	C₂²×C₅	C₂⁴	1	12	yes	—
28	C₂²×C₅	C₂⁴	1	3	yes	—
29	C₄×C₅	C₂⁴	1	4	yes	—
30	C₄×C₅	C₂⁴	1	12	yes	—
31	C₂⁴	C₂²×C₅	1	3	yes	—
32	#41	C₂²×C₅	1	64	yes	direct product factor
33	C₂×Q₈	C₂²×C₅	1	12	yes	complemented
34	C₂×D₄	C₂²×C₅	1	36	yes	complemented
35	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic
36	C₂²×C₄	C₂²×C₅	1	3	yes	—
37	C₂²×C₄	C₂²×C₅	1	36	yes	complemented
38	C₂×C₅	—	2	12	no	—
39	C₂×C₅	C₂×#41	1	6	yes	direct product factor
40	C₂×C₅	C₂⁵	1	1	yes	characteristic
41	C₂³	—	2	12	no	—
42	C₂³	C₂³×C₅	1	18	yes	—
43	C₂³	C₂³×C₅	1	1	yes	characteristic
44	Q₈	C₂³×C₅	1	16	yes	complemented
45	D₄	C₂³×C₅	1	48	yes	complemented
46	C₂×C₄	C₂³×C₅	1	6	yes	—
47	C₂×C₄	C₂³×C₅	1	18	yes	—
48	C₂×C₄	C₂³×C₅	1	48	yes	complemented
49	C₅	C₂²×#41	1	1	yes	characteristic, direct product factor, Sylow
50	C₂²	—	2	36	no	—
51	C₂²	C₅×#41	1	4	yes	direct product factor
52	C₂²	C₂⁴×C₅	1	12	yes	—
53	C₂²	C₂⁴×C₅	1	3	yes	—
54	C₄	C₂⁴×C₅	1	4	yes	—
55	C₄	C₂⁴×C₅	1	12	yes	—
56	C₂	—	2	12	no	—
57	C₂	C₂×C₅×#41	1	6	yes	direct product factor
58	C₂	C₂⁵×C₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²
commutator, Frattini, cosocle	C₂	C₂⁵×C₅
socle	C₂³×C₅	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#41	C₂	direct (N × H)
2. C₂²×C₅×Q₈	C₂	semidirect
3. C₂²×C₅×D₄	C₂	semidirect
4. C₂³×C₄×C₅	C₂	semidirect
6. C₅×#41	C₂²	direct (N × H)
7. C₂×C₅×Q₈	C₂²	semidirect
8. C₂×C₅×D₄	C₂²	semidirect
11. C₂²×C₄×C₅	C₂²	semidirect
12. C₂²×#41	C₅	direct (N × H)
16. C₅×Q₈	C₂³	semidirect
17. C₅×D₄	C₂³	semidirect
20. C₂×C₄×C₅	C₂³	semidirect
21. C₂×#41	C₂×C₅	direct (N × H)
22. C₂²×Q₈	C₂×C₅	semidirect
23. C₂²×D₄	C₂×C₅	semidirect
24. C₂³×C₄	C₂×C₅	semidirect
26. C₂²×C₅	#41	direct (N × H)
32. #41	C₂²×C₅	direct (N × H)
33. C₂×Q₈	C₂²×C₅	semidirect
34. C₂×D₄	C₂²×C₅	semidirect
37. C₂²×C₄	C₂²×C₅	semidirect
39. C₂×C₅	C₂×#41	direct (N × H)
44. Q₈	C₂³×C₅	semidirect
45. D₄	C₂³×C₅	semidirect
48. C₂×C₄	C₂³×C₅	semidirect
49. C₅	C₂²×#41	direct (N × H)
51. C₂²	C₅×#41	direct (N × H)
57. C₂	C₂×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
C₂×C₅×#41	2
C₂²×C₅×Q₈	2
C₂²×C₅×D₄	2
C₂³×C₄×C₅	2
C₂²×#41	5

Subgroups — C22×C5×#41 #66293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#41 #66293