Subgroups — C₄²×#1044 #13104823

Order 1600.

624 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#1044	C₂	1	6	yes	maximal
2	C₄²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×#1044	C₂²	1	4	yes	—
4	C₂×C₄×C₅⋊D₅	C₂²	1	3	yes	—
5	C₄×#1044	C₄	1	24	yes	direct product factor
6	C₄²×C₅²	C₄	1	1	yes	characteristic, complemented
7	C₄×#1039	C₄	1	3	yes	complemented
8	C₄²×AGL(1,𝔽₅)	—	5	2	no	maximal
9	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
10	C₂×#1044	C₂×C₄	1	24	yes	—
11	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
12	C₂×#1039	C₂×C₄	1	3	yes	—
13	C₄×C₅⋊D₅	C₂×C₄	1	12	yes	—
14	C₂×C₄×AGL(1,𝔽₅)	—	5	12	no	—
15	C₄²×D₅	—	5	2	no	—
16	C₄²×D₅	—	10	2	no	—
17	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₅⋊D₅	C₂²×C₄	1	6	yes	—
19	#1044	C₄²	1	16	yes	direct product factor
20	C₄×C₅²	C₄²	1	6	yes	complemented
21	#1039	C₄²	1	6	yes	complemented
22	C₂²×AGL(1,𝔽₅)	—	5	8	no	—
23	C₂×C₄×D₅	—	5	6	no	—
24	C₂×C₄×D₅	—	10	6	no	—
25	C₄×AGL(1,𝔽₅)	—	5	48	no	—
26	C₄²×C₅	—	2	2	no	—
27	C₄²×C₅	AGL(1,𝔽₅)	1	2	yes	complemented
28	C₄×Dic₅	—	5	6	no	—
29	C₄×Dic₅	—	10	6	no	—
30	C₄³	—	25	1	no	Sylow
31	C₂×C₅²	C₂×C₄²	1	3	yes	—
32	C₅⋊D₅	C₂×C₄²	1	4	yes	—
33	C₂²×D₅	—	5	2	no	—
34	C₂²×D₅	—	10	2	no	—
35	C₂×AGL(1,𝔽₅)	—	5	48	no	—
36	C₂×C₄×C₅	—	2	6	no	—
37	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	6	yes	—
38	C₂×Dic₅	—	5	6	no	—
39	C₂×Dic₅	—	10	6	no	—
40	C₄×D₅	—	5	24	no	—
41	C₄×D₅	—	10	24	no	—
42	C₂×C₄²	—	25	6	no	—
43	C₂×C₄²	—	25	1	no	—
44	C₅²	C₄³	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₅	—	2	2	no	—
46	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
47	C₂×D₅	—	5	12	no	—
48	C₂×D₅	—	10	12	no	—
49	AGL(1,𝔽₅)	—	5	32	no	—
50	C₄×C₅	—	2	12	no	—
51	C₄×C₅	C₄×AGL(1,𝔽₅)	1	12	yes	complemented
52	Dic₅	—	5	12	no	—
53	Dic₅	—	10	12	no	—
54	C₂²×C₄	—	25	4	no	—
55	C₂²×C₄	—	25	3	no	—
56	C₄²	—	25	24	no	—
57	C₄²	—	25	3	no	—
58	C₄²	#1044	1	1	yes	characteristic, direct product factor
59	C₂×C₅	—	2	6	no	—
60	C₂×C₅	C₂×C₄×AGL(1,𝔽₅)	1	6	yes	—
61	D₅	—	5	8	no	—
62	D₅	—	10	8	no	—
63	C₂³	—	25	1	no	—
64	C₂×C₄	—	25	24	no	—
65	C₂×C₄	—	25	3	no	—
66	C₂×C₄	—	25	12	no	—
67	C₂×C₄	C₂×#1044	1	3	yes	—
68	C₅	—	2	2	no	—
69	C₅	C₄²×AGL(1,𝔽₅)	1	2	yes	complemented
70	C₂²	—	25	6	no	—
71	C₂²	C₂²×#1044	1	1	yes	characteristic
72	C₄	—	25	16	no	—
73	C₄	—	25	6	no	—
74	C₄	C₄×#1044	1	6	yes	direct product factor
75	C₂	—	25	4	no	—
76	C₂	C₂×C₄×#1044	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²	#1044
commutator, hypocenter	C₅²	C₄³
Frattini	C₂²	C₂²×#1044
Fitting	C₄²×C₅²	C₄
socle	C₂²×C₅²	C₂²×C₄
prime radical, cosocle	C₂²×C₅⋊D₅	C₂³
LCS	G ▷ C₅²
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
5. C₄×#1044	C₄	direct (N × H)
6. C₄²×C₅²	C₄	semidirect
7. C₄×#1039	C₄	semidirect
19. #1044	C₄²	direct (N × H)
20. C₄×C₅²	C₄²	semidirect
21. #1039	C₄²	semidirect
27. C₄²×C₅	AGL(1,𝔽₅)	semidirect
44. C₅²	C₄³	semidirect
51. C₄×C₅	C₄×AGL(1,𝔽₅)	semidirect
58. C₄²	#1044	direct (N × H)
69. C₅	C₄²×AGL(1,𝔽₅)	semidirect
74. C₄	C₄×#1044	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×C₄×#1044	2
C₄²×C₅⋊D₅	2
C₄²×AGL(1,𝔽₅)	5

Subgroups — C42×#1044 #13104823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#1044 #13104823