Subgroups — #13104828

Order 1600.

384 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	#11716899	C₂	1	6	yes	maximal
3	C₂²×#1044	C₂²	1	4	yes	—
4	C₂×C₄×C₅⋊D₅	C₂²	1	3	yes	—
5	C₄²×C₅²	C₄	1	1	yes	characteristic, complemented
6	C₄×#1039	C₄	1	3	yes	complemented
7	#65690	—	5	2	no	maximal
8	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
9	C₂×#1044	—	2	12	no	—
10	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
11	C₂×#1039	C₂×C₄	1	3	yes	—
12	C₄×C₅⋊D₅	—	2	6	no	—
13	C₄²×D₅	—	5	2	no	—
14	C₄²×D₅	—	10	2	no	—
15	(C₂×AGL(1,𝔽₅))⋊C₄	—	5	12	no	—
16	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₅⋊D₅	#41	1	6	yes	—
18	#1044	—	4	4	no	—
19	C₄×C₅²	—	2	3	no	—
20	#1039	—	2	3	no	—
21	C₂²×AGL(1,𝔽₅)	—	5	8	no	—
22	C₂×C₄×D₅	—	5	6	no	—
23	C₂×C₄×D₅	—	10	6	no	—
24	C₄²×C₅	—	2	2	no	—
25	C₄²×C₅	AGL(1,𝔽₅)	1	2	yes	complemented
26	C₄×Dic₅	—	5	6	no	—
27	C₄×Dic₅	—	10	6	no	—
28	#383	—	25	1	no	Sylow
29	C₂×C₅²	#117	1	3	yes	—
30	C₅⋊D₅	#126	1	4	yes	—
31	C₂²×D₅	—	5	2	no	—
32	C₂²×D₅	—	10	2	no	—
33	C₂×AGL(1,𝔽₅)	—	10	24	no	—
34	C₂×C₄×C₅	—	2	6	no	—
35	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	6	yes	—
36	C₂×Dic₅	—	5	6	no	—
37	C₂×Dic₅	—	10	6	no	—
38	C₄×D₅	—	10	12	no	—
39	C₄×D₅	—	10	24	no	—
40	C₂×C₄²	—	25	1	no	—
41	#95	—	25	6	no	—
42	C₅²	#383	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	—	2	2	no	—
44	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
45	C₂×D₅	—	5	12	no	—
46	C₂×D₅	—	10	12	no	—
47	AGL(1,𝔽₅)	—	20	8	no	—
48	C₄×C₅	—	2	12	no	—
49	C₄×C₅	—	2	6	no	—
50	Dic₅	—	10	12	no	—
51	Dic₅	—	10	6	no	—
52	C₂²×C₄	—	25	3	no	—
53	C₂²×C₄	—	25	4	no	—
54	C₄²	—	25	3	no	—
55	C₄²	#1044	1	1	yes	characteristic, complemented
56	C₂×C₅	—	2	6	no	—
57	C₂×C₅	#4130	1	6	yes	—
58	D₅	—	5	8	no	—
59	D₅	—	10	8	no	—
60	C₂³	—	25	1	no	—
61	C₂×C₄	—	25	3	no	—
62	C₂×C₄	—	50	6	no	—
63	C₂×C₄	—	50	12	no	—
64	C₂×C₄	C₂×#1044	1	3	yes	—
65	C₅	—	2	2	no	—
66	C₅	#65690	1	2	yes	complemented
67	C₂²	—	25	6	no	—
68	C₂²	C₂²×#1044	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	50	3	no	—
71	C₄	—	100	4	no	—
72	C₂	—	25	4	no	—
73	C₂	#11717451	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1044
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₄²×C₅²	C₄
prime radical, cosocle	C₂²×C₅⋊D₅	C₂³
hypercenter	C₄²	#1044
hypocenter	C₅²	#383
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
5. C₄²×C₅²	C₄	semidirect
6. C₄×#1039	C₄	semidirect
25. C₄²×C₅	AGL(1,𝔽₅)	semidirect
42. C₅²	#383	semidirect
55. C₄²	#1044	semidirect
66. C₅	#65690	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₅⋊D₅	2
#11716899	2
#65690	5