Subgroups — C₂²×C₅²×#41 #13107898

Order 1600.

3560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅²×D₄	C₂	1	3	yes	maximal, complemented
4	C₂³×C₄×C₅²	C₂	1	3	yes	maximal, complemented
5	C₂⁴×C₅²	C₂²	1	3	yes	—
6	C₅²×#41	C₂²	1	64	yes	direct product factor
7	C₂×C₅²×Q₈	C₂²	1	12	yes	complemented
8	C₂×C₅²×D₄	C₂²	1	36	yes	complemented
9	C₂²×C₄×C₅²	C₂²	1	3	yes	—
10	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅²	C₂²	1	36	yes	complemented
12	C₂²×C₅×#41	C₅	1	6	yes	maximal, direct product factor
13	C₂³×C₅²	—	2	12	no	—
14	C₂³×C₅²	C₂³	1	18	yes	—
15	C₂³×C₅²	C₂³	1	1	yes	characteristic
16	C₅²×Q₈	C₂³	1	16	yes	complemented
17	C₅²×D₄	C₂³	1	48	yes	complemented
18	C₂×C₄×C₅²	C₂³	1	18	yes	—
19	C₂×C₄×C₅²	C₂³	1	6	yes	—
20	C₂×C₄×C₅²	C₂³	1	48	yes	complemented
21	C₂×C₅×#41	C₂×C₅	1	144	yes	direct product factor
22	C₂²×C₅×Q₈	C₂×C₅	1	6	yes	complemented
23	C₂²×C₅×D₄	C₂×C₅	1	18	yes	complemented
24	C₂³×C₄×C₅	C₂×C₅	1	18	yes	complemented
25	C₂²×C₅²	—	2	36	no	—
26	C₂²×C₅²	#41	1	4	yes	direct product factor
27	C₂²×C₅²	C₂⁴	1	12	yes	—
28	C₂²×C₅²	C₂⁴	1	3	yes	—
29	C₄×C₅²	C₂⁴	1	4	yes	—
30	C₄×C₅²	C₂⁴	1	12	yes	—
31	C₂⁴×C₅	C₂²×C₅	1	18	yes	—
32	C₅×#41	C₂²×C₅	1	384	yes	direct product factor
33	C₂×C₅×Q₈	C₂²×C₅	1	72	yes	complemented
34	C₂×C₅×D₄	C₂²×C₅	1	216	yes	complemented
35	C₂²×C₄×C₅	C₂²×C₅	1	6	yes	—
36	C₂²×C₄×C₅	C₂²×C₅	1	18	yes	—
37	C₂²×C₄×C₅	C₂²×C₅	1	216	yes	complemented
38	C₂²×#41	C₅²	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₅²	—	2	12	no	—
40	C₂×C₅²	C₂×#41	1	6	yes	direct product factor
41	C₂×C₅²	C₂⁵	1	1	yes	characteristic
42	C₂³×C₅	—	2	72	no	—
43	C₂³×C₅	C₂³×C₅	1	108	yes	—
44	C₂³×C₅	C₂³×C₅	1	6	yes	—
45	C₅×Q₈	C₂³×C₅	1	96	yes	complemented
46	C₅×D₄	C₂³×C₅	1	288	yes	complemented
47	C₂×C₄×C₅	C₂³×C₅	1	36	yes	—
48	C₂×C₄×C₅	C₂³×C₅	1	108	yes	—
49	C₂×C₄×C₅	C₂³×C₅	1	288	yes	complemented
50	C₂×#41	C₂×C₅²	1	24	yes	direct product factor
51	C₂²×Q₈	C₂×C₅²	1	1	yes	characteristic, complemented
52	C₂²×D₄	C₂×C₅²	1	3	yes	complemented
53	C₂³×C₄	C₂×C₅²	1	3	yes	complemented
54	C₅²	C₂²×#41	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₅	—	2	216	no	—
56	C₂²×C₅	C₅×#41	1	24	yes	direct product factor
57	C₂²×C₅	C₂⁴×C₅	1	72	yes	—
58	C₂²×C₅	C₂⁴×C₅	1	18	yes	—
59	C₄×C₅	C₂⁴×C₅	1	24	yes	—
60	C₄×C₅	C₂⁴×C₅	1	72	yes	—
61	C₂⁴	C₂²×C₅²	1	3	yes	—
62	#41	C₂²×C₅²	1	64	yes	direct product factor
63	C₂×Q₈	C₂²×C₅²	1	12	yes	complemented
64	C₂×D₄	C₂²×C₅²	1	36	yes	complemented
65	C₂²×C₄	C₂²×C₅²	1	1	yes	characteristic
66	C₂²×C₄	C₂²×C₅²	1	3	yes	—
67	C₂²×C₄	C₂²×C₅²	1	36	yes	complemented
68	C₂×C₅	—	2	72	no	—
69	C₂×C₅	C₂×C₅×#41	1	36	yes	direct product factor
70	C₂×C₅	C₂⁵×C₅	1	6	yes	—
71	C₂³	—	2	12	no	—
72	C₂³	C₂³×C₅²	1	18	yes	—
73	C₂³	C₂³×C₅²	1	1	yes	characteristic
74	Q₈	C₂³×C₅²	1	16	yes	complemented
75	D₄	C₂³×C₅²	1	48	yes	complemented
76	C₂×C₄	C₂³×C₅²	1	6	yes	—
77	C₂×C₄	C₂³×C₅²	1	18	yes	—
78	C₂×C₄	C₂³×C₅²	1	48	yes	complemented
79	C₅	C₂²×C₅×#41	1	6	yes	direct product factor
80	C₂²	—	2	36	no	—
81	C₂²	C₅²×#41	1	4	yes	direct product factor
82	C₂²	C₂⁴×C₅²	1	12	yes	—
83	C₂²	C₂⁴×C₅²	1	3	yes	—
84	C₄	C₂⁴×C₅²	1	4	yes	—
85	C₄	C₂⁴×C₅²	1	12	yes	—
86	C₂	—	2	12	no	—
87	C₂	C₂×C₅²×#41	1	6	yes	direct product factor
88	C₂	C₂⁵×C₅²	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅²	C₂²
commutator, Frattini, cosocle	C₂	C₂⁵×C₅²
socle	C₂³×C₅²	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#41	C₂	direct (N × H)
2. C₂²×C₅²×Q₈	C₂	semidirect
3. C₂²×C₅²×D₄	C₂	semidirect
4. C₂³×C₄×C₅²	C₂	semidirect
6. C₅²×#41	C₂²	direct (N × H)
7. C₂×C₅²×Q₈	C₂²	semidirect
8. C₂×C₅²×D₄	C₂²	semidirect
11. C₂²×C₄×C₅²	C₂²	semidirect
12. C₂²×C₅×#41	C₅	direct (N × H)
16. C₅²×Q₈	C₂³	semidirect
17. C₅²×D₄	C₂³	semidirect
20. C₂×C₄×C₅²	C₂³	semidirect
21. C₂×C₅×#41	C₂×C₅	direct (N × H)
22. C₂²×C₅×Q₈	C₂×C₅	semidirect
23. C₂²×C₅×D₄	C₂×C₅	semidirect
24. C₂³×C₄×C₅	C₂×C₅	semidirect
26. C₂²×C₅²	#41	direct (N × H)
32. C₅×#41	C₂²×C₅	direct (N × H)
33. C₂×C₅×Q₈	C₂²×C₅	semidirect
34. C₂×C₅×D₄	C₂²×C₅	semidirect
37. C₂²×C₄×C₅	C₂²×C₅	semidirect
38. C₂²×#41	C₅²	direct (N × H)
40. C₂×C₅²	C₂×#41	direct (N × H)
45. C₅×Q₈	C₂³×C₅	semidirect
46. C₅×D₄	C₂³×C₅	semidirect
49. C₂×C₄×C₅	C₂³×C₅	semidirect
50. C₂×#41	C₂×C₅²	direct (N × H)
51. C₂²×Q₈	C₂×C₅²	semidirect
52. C₂²×D₄	C₂×C₅²	semidirect
53. C₂³×C₄	C₂×C₅²	semidirect
54. C₅²	C₂²×#41	direct (N × H)
56. C₂²×C₅	C₅×#41	direct (N × H)
62. #41	C₂²×C₅²	direct (N × H)
63. C₂×Q₈	C₂²×C₅²	semidirect
64. C₂×D₄	C₂²×C₅²	semidirect
67. C₂²×C₄	C₂²×C₅²	semidirect
69. C₂×C₅	C₂×C₅×#41	direct (N × H)
74. Q₈	C₂³×C₅²	semidirect
75. D₄	C₂³×C₅²	semidirect
78. C₂×C₄	C₂³×C₅²	semidirect
79. C₅	C₂²×C₅×#41	direct (N × H)
81. C₂²	C₅²×#41	direct (N × H)
87. C₂	C₂×C₅²×#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×C₅²×#41	2
C₂²×C₅²×Q₈	2
C₂²×C₅²×D₄	2
C₂³×C₄×C₅²	2
C₂²×C₅×#41	5

Subgroups — C22×C52×#41 #13107898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅²×#41 #13107898