Subgroups — #13102930

Order 1600.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717194	C₂	1	1	yes	maximal, characteristic
3	#11716676	C₂	1	4	yes	maximal, complemented
4	#11716670	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
6	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
7	C₅²×C₄⋊C₄	—	2	2	no	—
8	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄²×C₅²	C₂²	1	1	yes	characteristic
10	C₂×#6029	C₂²	1	2	yes	—
11	#65356	—	5	6	no	maximal
12	C₂²×C₅⋊D₅	—	2	2	no	—
13	C₂×C₄×C₅²	—	2	2	no	—
14	C₂×C₄×C₅²	D₄	1	2	yes	—
15	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
16	C₄⋊(C₅⋊D₅)	—	2	4	no	—
17	C₄⋊(C₅⋊D₅)	—	4	4	no	—
18	#6029	—	2	2	no	—
19	C₅×#125	D₅	1	6	yes	complemented
20	#4020	—	5	6	no	—
21	#3941	—	5	24	no	—
22	#3935	—	5	6	no	—
23	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
24	C₂×C₅⋊D₅	—	4	2	no	—
25	C₂×C₅⋊D₅	—	4	4	no	—
26	C₄×C₅²	—	2	2	no	—
27	C₄×C₅²	—	4	2	no	—
28	C₄×C₅²	#41	1	2	yes	—
29	C₂×D₂₀	—	5	12	no	—
30	C₂×D₂₀	—	10	12	no	—
31	C₅×C₄⋊C₄	—	2	12	no	—
32	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
33	C₄²×C₅	C₂×D₅	1	6	yes	—
34	C₂×#168	—	5	12	no	—
35	#490	—	25	1	no	Sylow
36	C₂×C₅²	#124	1	1	yes	characteristic
37	C₂×C₅²	AGL(1,ℤ/8)	1	2	yes	—
38	C₅⋊D₅	—	8	2	no	—
39	C₂²×D₅	—	10	12	no	—
40	C₂×C₄×C₅	—	2	12	no	—
41	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
42	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
43	D₂₀	—	10	24	no	—
44	D₂₀	—	20	24	no	—
45	#168	—	10	12	no	—
46	#127	—	25	1	no	—
47	#125	C₅⋊D₅	1	1	yes	characteristic, complemented
48	#102	—	25	4	no	—
49	#97	—	25	1	no	—
50	C₅²	#490	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
52	C₂×D₅	—	20	12	no	—
53	C₂×D₅	—	20	24	no	—
54	C₄×C₅	—	2	12	no	—
55	C₄×C₅	—	4	12	no	—
56	C₄×C₅	#713	1	12	yes	—
57	C₂×D₄	—	25	2	no	—
58	C₂×D₄	—	50	2	no	—
59	C₂×C₈	—	25	2	no	—
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
62	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
63	C₂×C₅	#4093	1	6	yes	—
64	C₂×C₅	#4095	1	12	yes	—
65	D₅	—	40	12	no	—
66	C₂³	—	50	2	no	—
67	D₄	—	50	4	no	—
68	D₄	—	100	4	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
71	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
72	C₈	—	50	2	no	—
73	C₅	#65356	1	6	yes	complemented
74	C₂²	—	100	2	no	—
75	C₂²	—	100	4	no	—
76	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	4	2	no	—
79	C₄	#87904	1	2	yes	—
80	C₂	—	200	2	no	—
81	C₂	#11717267	1	1	yes	characteristic
82	C₂	#11717269	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#125	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	#11717194	C₂
hypercenter	#125	C₅⋊D₅
hypocenter	C₅²	#490
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₅²×#125	C₂	semidirect
3. #11716676	C₂	semidirect
4. #11716670	C₂	semidirect
19. C₅×#125	D₅	semidirect
47. #125	C₅⋊D₅	semidirect
50. C₅²	#490	semidirect
73. C₅	#65356	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#125	2
#11717194	2
#11716676	2
#11716670	2
#65356	5