Subgroups — #72221128

Order 3200.

1120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13102558	C₂	1	6	yes	maximal, complemented
2	C₅²×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87801	—	2	12	no	—
4	#11717194	C₂²	1	4	yes	—
5	C₄×C₅²×C₈	C₂²	1	3	yes	—
6	#10598855	—	5	6	no	maximal
7	C₂×C₄⋊(C₅⋊D₅)	—	2	12	no	—
8	C₂×C₅²×C₈	D₄	1	6	yes	—
9	C₄²×C₅²	C₂³	1	1	yes	characteristic
10	#87801	—	4	24	no	—
11	#64984	—	5	36	no	—
12	C₅×C₈²	D₅	1	6	yes	complemented
13	C₂²×C₅⋊D₅	—	4	4	no	—
14	C₂×C₄×C₅²	C₂×D₄	1	3	yes	—
15	C₄⋊(C₅⋊D₅)	—	4	24	no	—
16	C₅²×C₈	D₈	1	12	yes	complemented
17	C₂×D₄₀	—	10	72	no	—
18	#4020	—	5	24	no	—
19	C₄×C₅×C₈	C₂×D₅	1	18	yes	—
20	#1712	—	25	1	no	Sylow
21	C₂²×C₅²	#127	1	1	yes	characteristic
22	C₂×C₅⋊D₅	—	8	12	no	—
23	C₄×C₅²	C₂×D₈	1	6	yes	—
24	C₂×D₂₀	—	10	72	no	—
25	C₂×C₅×C₈	D₂₀	1	36	yes	—
26	C₄²×C₅	C₂²×D₅	1	6	yes	—
27	D₄₀	—	20	144	no	—
28	#493	—	25	6	no	—
29	C₈²	C₅⋊D₅	1	1	yes	characteristic, complemented
30	C₂×C₅²	#493	1	3	yes	—
31	C₅⋊D₅	—	16	4	no	—
32	C₂²×D₅	—	20	24	no	—
33	C₂×C₄×C₅	C₂×D₂₀	1	18	yes	—
34	D₂₀	—	20	144	no	—
35	C₅×C₈	D₄₀	1	72	yes	complemented
36	C₂×D₈	—	50	12	no	—
37	#127	—	25	4	no	—
38	C₄×C₈	C₂×C₅⋊D₅	1	3	yes	—
39	C₅²	#1712	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₅	#4020	1	6	yes	—
41	C₂×D₅	—	40	72	no	—
42	C₄×C₅	C₂×D₄₀	1	36	yes	—
43	C₂×D₄	—	50	12	no	—
44	D₈	—	100	24	no	—
45	C₂×C₈	C₄⋊(C₅⋊D₅)	1	6	yes	—
46	C₄²	C₂²×C₅⋊D₅	1	1	yes	characteristic
47	C₂×C₅	#64984	1	18	yes	—
48	D₅	—	80	24	no	—
49	C₂³	—	100	4	no	—
50	D₄	—	100	24	no	—
51	C₂×C₄	C₂×C₄⋊(C₅⋊D₅)	1	3	yes	—
52	C₈	#87801	1	12	yes	complemented
53	C₅	#10598855	1	6	yes	complemented
54	C₂²	—	200	12	no	—
55	C₂²	#11717194	1	1	yes	characteristic
56	C₄	C₂×#87801	1	6	yes	—
57	C₂	—	400	4	no	—
58	C₂	#13102558	1	3	yes	—

	subgroup	G/H
center	C₂²	#11717194
commutator, cosocle	C₄²×C₅²	C₂³
Frattini	C₄²	C₂²×C₅⋊D₅
Fitting	C₅²×C₈²	C₂
socle	C₂²×C₅²	#127
hypercenter	C₈²	C₅⋊D₅
hypocenter	C₅²	#1712
LCS	G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #13102558	C₂	semidirect
2. C₅²×C₈²	C₂	semidirect
12. C₅×C₈²	D₅	semidirect
16. C₅²×C₈	D₈	semidirect
29. C₈²	C₅⋊D₅	semidirect
35. C₅×C₈	D₄₀	semidirect
39. C₅²	#1712	semidirect
52. C₈	#87801	semidirect
53. C₅	#10598855	semidirect

Maximal subgroups

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

subgroup	index
#13102558	2
C₅²×C₈²	2
#10598855	5