Subgroups — C₂⁶×#1043 #72258463

Order 6400.

444792 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×#1043	C₂	1	126	yes	maximal, direct product factor
3	C₂⁶×C₅²	C₄	1	1	yes	characteristic, complemented
4	C₂⁵×C₅⋊D₅	C₄	1	63	yes	complemented
5	C₂⁵×C₅⋊D₅	C₂²	1	63	yes	—
6	C₂⁴×#1043	C₂²	1	2604	yes	direct product factor
7	C₂⁶×AGL(1,𝔽₅)	—	5	6	no	maximal
8	C₂⁵×C₅²	C₂×C₄	1	63	yes	complemented
9	C₂⁴×C₅⋊D₅	C₂×C₄	1	1953	yes	complemented
10	C₂⁴×C₅⋊D₅	C₂³	1	651	yes	—
11	C₂³×#1043	C₂³	1	11160	yes	direct product factor
12	C₂⁶×D₅	—	5	6	no	—
13	C₂⁵×AGL(1,𝔽₅)	—	5	756	no	—
14	C₂⁴×C₅²	C₂²×C₄	1	651	yes	complemented
15	C₂³×C₅⋊D₅	C₂²×C₄	1	9765	yes	complemented
16	C₂³×C₅⋊D₅	C₂⁴	1	1395	yes	—
17	C₂²×#1043	C₂⁴	1	10416	yes	direct product factor
18	C₂⁶×C₅	AGL(1,𝔽₅)	1	6	yes	complemented
19	C₂⁵×D₅	—	5	378	no	—
20	C₂⁵×D₅	—	5	378	no	—
21	C₂⁴×AGL(1,𝔽₅)	—	5	15624	no	—
22	C₂⁶×C₄	—	25	1	no	Sylow
23	C₂³×C₅²	C₂³×C₄	1	1395	yes	complemented
24	C₂²×C₅⋊D₅	C₂³×C₄	1	9765	yes	complemented
25	C₂²×C₅⋊D₅	C₂⁵	1	651	yes	—
26	C₂×#1043	C₂⁵	1	2016	yes	direct product factor
27	C₂⁵×C₅	C₂×AGL(1,𝔽₅)	1	378	yes	complemented
28	C₂⁴×D₅	—	5	11718	no	—
29	C₂⁴×D₅	—	5	3906	no	—
30	C₂³×AGL(1,𝔽₅)	—	5	66960	no	—
31	C₂⁷	—	25	1	no	—
32	C₂⁵×C₄	—	25	126	no	—
33	C₂²×C₅²	C₂⁴×C₄	1	651	yes	complemented
34	C₂×C₅⋊D₅	C₂⁴×C₄	1	1953	yes	complemented
35	C₂×C₅⋊D₅	C₂⁶	1	63	yes	—
36	#1043	C₂⁶	1	64	yes	direct product factor
37	C₂⁴×C₅	C₂²×AGL(1,𝔽₅)	1	3906	yes	complemented
38	C₂³×D₅	—	5	8370	no	—
39	C₂³×D₅	—	5	58590	no	—
40	C₂²×AGL(1,𝔽₅)	—	5	62496	no	—
41	C₂⁶	—	25	63	no	—
42	C₂⁶	—	25	63	no	—
43	C₂⁶	#1043	1	1	yes	characteristic, direct product factor
44	C₂⁴×C₄	—	25	2604	no	—
45	C₂×C₅²	C₂⁵×C₄	1	63	yes	complemented
46	C₅⋊D₅	C₂⁵×C₄	1	63	yes	complemented
47	C₅⋊D₅	C₂⁷	1	1	yes	characteristic
48	C₂³×C₅	C₂³×AGL(1,𝔽₅)	1	8370	yes	complemented
49	C₂²×D₅	—	5	58590	no	—
50	C₂²×D₅	—	5	3906	no	—
51	C₂×AGL(1,𝔽₅)	—	5	12096	no	—
52	C₂⁵	—	25	651	no	—
53	C₂⁵	—	25	1953	no	—
54	C₂⁵	C₂×#1043	1	63	yes	direct product factor
55	C₂³×C₄	—	25	11160	no	—
56	C₅²	C₂⁶×C₄	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₅	C₂⁴×AGL(1,𝔽₅)	1	3906	yes	complemented
58	C₂×D₅	—	5	11718	no	—
59	C₂×D₅	—	5	378	no	—
60	AGL(1,𝔽₅)	—	5	384	no	—
61	C₂⁴	—	25	1395	no	—
62	C₂⁴	—	25	9765	no	—
63	C₂⁴	C₂²×#1043	1	651	yes	direct product factor
64	C₂²×C₄	—	25	10416	no	—
65	C₂×C₅	C₂⁵×AGL(1,𝔽₅)	1	378	yes	complemented
66	D₅	—	5	378	no	—
67	D₅	—	5	6	no	—
68	C₂³	—	25	651	no	—
69	C₂³	—	25	9765	no	—
70	C₂³	C₂³×#1043	1	1395	yes	direct product factor
71	C₂×C₄	—	25	2016	no	—
72	C₅	C₂⁶×AGL(1,𝔽₅)	1	6	yes	complemented
73	C₂²	—	25	63	no	—
74	C₂²	—	25	1953	no	—
75	C₂²	C₂⁴×#1043	1	651	yes	direct product factor
76	C₄	—	25	64	no	—
77	C₂	—	25	1	no	—
78	C₂	—	25	63	no	—
79	C₂	C₂⁵×#1043	1	63	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁶	#1043
commutator, hypocenter	C₅²	C₂⁶×C₄
Frattini	C₁	G
Fitting, socle	C₂⁶×C₅²	C₄
prime radical	C₂⁶×C₅⋊D₅	C₂
cosocle	C₅⋊D₅	C₂⁷
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂⁶

Complements

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

N	H ≅ G/N	type
2. C₂⁵×#1043	C₂	direct (N × H)
3. C₂⁶×C₅²	C₄	semidirect
4. C₂⁵×C₅⋊D₅	C₄	semidirect
6. C₂⁴×#1043	C₂²	direct (N × H)
8. C₂⁵×C₅²	C₂×C₄	semidirect
9. C₂⁴×C₅⋊D₅	C₂×C₄	semidirect
11. C₂³×#1043	C₂³	direct (N × H)
14. C₂⁴×C₅²	C₂²×C₄	semidirect
15. C₂³×C₅⋊D₅	C₂²×C₄	semidirect
17. C₂²×#1043	C₂⁴	direct (N × H)
18. C₂⁶×C₅	AGL(1,𝔽₅)	semidirect
23. C₂³×C₅²	C₂³×C₄	semidirect
24. C₂²×C₅⋊D₅	C₂³×C₄	semidirect
26. C₂×#1043	C₂⁵	direct (N × H)
27. C₂⁵×C₅	C₂×AGL(1,𝔽₅)	semidirect
33. C₂²×C₅²	C₂⁴×C₄	semidirect
34. C₂×C₅⋊D₅	C₂⁴×C₄	semidirect
36. #1043	C₂⁶	direct (N × H)
37. C₂⁴×C₅	C₂²×AGL(1,𝔽₅)	semidirect
43. C₂⁶	#1043	direct (N × H)
45. C₂×C₅²	C₂⁵×C₄	semidirect
46. C₅⋊D₅	C₂⁵×C₄	semidirect
48. C₂³×C₅	C₂³×AGL(1,𝔽₅)	semidirect
54. C₂⁵	C₂×#1043	direct (N × H)
56. C₅²	C₂⁶×C₄	semidirect
57. C₂²×C₅	C₂⁴×AGL(1,𝔽₅)	semidirect
63. C₂⁴	C₂²×#1043	direct (N × H)
65. C₂×C₅	C₂⁵×AGL(1,𝔽₅)	semidirect
70. C₂³	C₂³×#1043	direct (N × H)
72. C₅	C₂⁶×AGL(1,𝔽₅)	semidirect
75. C₂²	C₂⁴×#1043	direct (N × H)
79. C₂	C₂⁵×#1043	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×C₅⋊D₅	2
C₂⁵×#1043	2
C₂⁶×AGL(1,𝔽₅)	5

Subgroups — C26×#1043 #72258463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁶×#1043 #72258463