Subgroups — C₂×C₅²×2₋¹⁺⁴ #13107900

Order 1600.

3176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×2₋¹⁺⁴	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₅²×#41	C₂	1	10	yes	maximal, complemented
3	C₂²×C₅²×Q₈	C₂	1	5	yes	maximal, complemented
4	C₅²×#41	C₂²	1	80	yes	complemented
5	C₂×C₅²×Q₈	C₂²	1	10	yes	—
6	C₂×C₅²×Q₈	C₂²	1	40	yes	complemented
7	C₂×C₅²×D₄	C₂²	1	10	yes	—
8	C₂²×C₄×C₅²	C₂²	1	15	yes	—
9	C₂×C₅×2₋¹⁺⁴	C₅	1	6	yes	maximal, direct product factor
10	C₂³×C₅²	C₂³	1	5	yes	—
11	C₅²×Q₈	C₂³	1	40	yes	—
12	C₅²×D₄	C₂³	1	40	yes	—
13	C₂×C₄×C₅²	C₂³	1	60	yes	—
14	C₂×C₄×C₅²	C₂³	1	10	yes	—
15	C₅×2₋¹⁺⁴	C₂×C₅	1	96	yes	direct product factor
16	C₂×C₅×#41	C₂×C₅	1	60	yes	complemented
17	C₂²×C₅×Q₈	C₂×C₅	1	30	yes	complemented
18	C₂²×C₅²	—	2	10	no	—
19	C₂²×C₅²	C₂⁴	1	10	yes	—
20	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
21	C₄×C₅²	C₂⁴	1	20	yes	—
22	C₅×#41	C₂²×C₅	1	480	yes	complemented
23	C₂×C₅×Q₈	C₂²×C₅	1	60	yes	—
24	C₂×C₅×Q₈	C₂²×C₅	1	240	yes	complemented
25	C₂×C₅×D₄	C₂²×C₅	1	60	yes	—
26	C₂²×C₄×C₅	C₂²×C₅	1	90	yes	—
27	C₂×2₋¹⁺⁴	C₅²	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×C₅²	—	2	10	no	—
29	C₂×C₅²	2₋¹⁺⁴	1	2	yes	direct product factor
30	C₂×C₅²	C₂⁵	1	1	yes	characteristic
31	C₂³×C₅	C₂³×C₅	1	30	yes	—
32	C₅×Q₈	C₂³×C₅	1	240	yes	—
33	C₅×D₄	C₂³×C₅	1	240	yes	—
34	C₂×C₄×C₅	C₂³×C₅	1	60	yes	—
35	C₂×C₄×C₅	C₂³×C₅	1	360	yes	—
36	2₋¹⁺⁴	C₂×C₅²	1	16	yes	direct product factor
37	C₂×#41	C₂×C₅²	1	10	yes	complemented
38	C₂²×Q₈	C₂×C₅²	1	5	yes	complemented
39	C₅²	C₂×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²×C₅	—	2	60	no	—
41	C₂²×C₅	C₂⁴×C₅	1	60	yes	—
42	C₂²×C₅	C₂⁴×C₅	1	6	yes	—
43	C₄×C₅	C₂⁴×C₅	1	120	yes	—
44	#41	C₂²×C₅²	1	80	yes	complemented
45	C₂×Q₈	C₂²×C₅²	1	10	yes	—
46	C₂×Q₈	C₂²×C₅²	1	40	yes	complemented
47	C₂×D₄	C₂²×C₅²	1	10	yes	—
48	C₂²×C₄	C₂²×C₅²	1	15	yes	—
49	C₂×C₅	—	2	60	no	—
50	C₂×C₅	C₅×2₋¹⁺⁴	1	12	yes	direct product factor
51	C₂×C₅	C₂⁵×C₅	1	6	yes	—
52	C₂³	C₂³×C₅²	1	5	yes	—
53	Q₈	C₂³×C₅²	1	40	yes	—
54	D₄	C₂³×C₅²	1	40	yes	—
55	C₂×C₄	C₂³×C₅²	1	10	yes	—
56	C₂×C₄	C₂³×C₅²	1	60	yes	—
57	C₅	C₂×C₅×2₋¹⁺⁴	1	6	yes	direct product factor
58	C₂²	—	2	10	no	—
59	C₂²	C₂⁴×C₅²	1	10	yes	—
60	C₂²	C₂⁴×C₅²	1	1	yes	characteristic
61	C₄	C₂⁴×C₅²	1	20	yes	—
62	C₂	—	2	10	no	—
63	C₂	C₅²×2₋¹⁺⁴	1	2	yes	direct product factor
64	C₂	C₂⁵×C₅²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅²	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₅²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅²×2₋¹⁺⁴	C₂	direct (N × H)
2. C₂×C₅²×#41	C₂	semidirect
3. C₂²×C₅²×Q₈	C₂	semidirect
4. C₅²×#41	C₂²	semidirect
6. C₂×C₅²×Q₈	C₂²	semidirect
9. C₂×C₅×2₋¹⁺⁴	C₅	direct (N × H)
15. C₅×2₋¹⁺⁴	C₂×C₅	direct (N × H)
16. C₂×C₅×#41	C₂×C₅	semidirect
17. C₂²×C₅×Q₈	C₂×C₅	semidirect
22. C₅×#41	C₂²×C₅	semidirect
24. C₂×C₅×Q₈	C₂²×C₅	semidirect
27. C₂×2₋¹⁺⁴	C₅²	direct (N × H)
29. C₂×C₅²	2₋¹⁺⁴	direct (N × H)
36. 2₋¹⁺⁴	C₂×C₅²	direct (N × H)
37. C₂×#41	C₂×C₅²	semidirect
38. C₂²×Q₈	C₂×C₅²	semidirect
39. C₅²	C₂×2₋¹⁺⁴	direct (N × H)
44. #41	C₂²×C₅²	semidirect
46. C₂×Q₈	C₂²×C₅²	semidirect
50. C₂×C₅	C₅×2₋¹⁺⁴	direct (N × H)
57. C₅	C₂×C₅×2₋¹⁺⁴	direct (N × H)
63. C₂	C₅²×2₋¹⁺⁴	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅²×2₋¹⁺⁴	2
C₂×C₅²×#41	2
C₂²×C₅²×Q₈	2
C₂×C₅×2₋¹⁺⁴	5

Subgroups — C2×C52×2−1+4 #13107900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×2₋¹⁺⁴ #13107900