Subgroups — #13104576

Order 1600.

291 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	#11716849	C₂	1	6	yes	maximal
3	C₂²×#1042	C₂²	1	4	yes	—
4	C₂×C₄×C₅×D₅	C₂²	1	3	yes	—
5	C₄²×C₅²	C₄	1	1	yes	characteristic, complemented
6	C₄×C₅×Dic₅	C₄	1	3	yes	complemented
7	#65690	—	5	1	no	maximal
8	#65226	—	5	1	no	maximal
9	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
10	C₂×#1042	—	2	12	no	—
11	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
12	C₂×C₅×Dic₅	C₂×C₄	1	3	yes	—
13	C₄×C₅×D₅	—	2	6	no	—
14	C₂×C₄²×C₅	—	5	1	no	—
15	C₄²×D₅	D₅	1	1	yes	characteristic
16	(C₂×AGL(1,𝔽₅))⋊C₄	—	5	6	no	—
17	#3963	—	5	6	no	—
18	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₅×D₅	#41	1	6	yes	—
20	#1042	—	4	4	no	—
21	C₄×C₅²	—	2	3	no	—
22	C₅×Dic₅	—	2	3	no	—
23	C₂²×AGL(1,𝔽₅)	—	5	4	no	—
24	C₂²×C₄×C₅	—	5	3	no	—
25	C₂²×Dic₅	—	5	4	no	—
26	C₂×C₄×D₅	C₂×D₅	1	3	yes	—
27	C₄²×C₅	—	4	1	no	—
28	C₄²×C₅	—	5	3	no	—
29	C₄²×C₅	Dic₅	1	1	yes	characteristic, complemented
30	C₄²×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
31	C₄×Dic₅	Dic₅	1	3	yes	complemented
32	#383	—	25	1	no	Sylow
33	C₂×C₅²	#117	1	3	yes	—
34	C₅×D₅	#126	1	4	yes	—
35	C₂³×C₅	—	5	1	no	—
36	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
37	C₂×AGL(1,𝔽₅)	—	10	12	no	—
38	C₂×C₄×C₅	—	4	3	no	—
39	C₂×C₄×C₅	—	5	3	no	—
40	C₂×C₄×C₅	—	10	6	no	—
41	C₂×C₄×C₅	C₂×Dic₅	1	3	yes	—
42	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	—
43	C₂×Dic₅	—	10	12	no	—
44	C₂×Dic₅	C₂×Dic₅	1	3	yes	—
45	C₄×D₅	—	2	6	no	—
46	C₂×C₄²	—	5	1	no	—
47	#95	—	25	6	no	—
48	C₅²	#383	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₅	—	4	1	no	—
50	C₂²×C₅	—	5	6	no	—
51	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
52	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₂×D₅	#709	1	6	yes	—
54	AGL(1,𝔽₅)	—	20	4	no	—
55	C₄×C₅	—	2	3	no	—
56	C₄×C₅	—	2	3	no	—
57	C₄×C₅	—	4	6	no	—
58	C₄×C₅	—	10	3	no	—
59	Dic₅	—	2	3	no	—
60	Dic₅	—	20	4	no	—
61	C₂²×C₄	—	5	3	no	—
62	C₂²×C₄	—	25	4	no	—
63	C₄²	—	5	3	no	—
64	C₄²	#1042	1	1	yes	characteristic, complemented
65	C₂×C₅	—	4	3	no	—
66	C₂×C₅	—	5	4	no	—
67	C₂×C₅	#4072	1	3	yes	—
68	C₂×C₅	#4130	1	3	yes	—
69	D₅	#4022	1	4	yes	—
70	C₂³	—	5	1	no	—
71	C₂×C₄	—	5	3	no	—
72	C₂×C₄	—	10	6	no	—
73	C₂×C₄	—	50	12	no	—
74	C₂×C₄	C₂×#1042	1	3	yes	—
75	C₅	—	4	1	no	—
76	C₅	#65226	1	1	yes	characteristic, complemented
77	C₅	#65690	1	1	yes	characteristic, complemented
78	C₂²	—	5	6	no	—
79	C₂²	C₂²×#1042	1	1	yes	characteristic
80	C₄	—	2	3	no	—
81	C₄	—	10	3	no	—
82	C₄	—	100	4	no	—
83	C₂	—	5	4	no	—
84	C₂	#11717425	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1042
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₄²×C₅²	C₄
prime radical, cosocle	C₂²×C₅×D₅	C₂³
hypercenter	C₄²	#1042
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×C₅×Dic₅	C₄	semidirect
29. C₄²×C₅	Dic₅	semidirect
30. C₄²×C₅	AGL(1,𝔽₅)	semidirect
31. C₄×Dic₅	Dic₅	semidirect
48. C₅²	#383	semidirect
64. C₄²	#1042	semidirect
76. C₅	#65226	semidirect
77. 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
#11716849	2
#65690	5
#65226	5