Subgroups — #13183173

Order 1760.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₂₀	C₂	1	6	yes	maximal, complemented
2	C₄²×C₅×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₅₅	C₂²	1	4	yes	—
4	C₂×C₄×C₅×C₁₁	C₂²	1	3	yes	—
5	D₂₂₀	—	2	12	no	—
6	#66927	—	5	1	no	maximal, Hall
7	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
8	C₂×D₅₅	—	2	12	no	—
9	C₄×C₅×C₁₁	D₄	1	6	yes	complemented
10	C₂×D₄₄	—	5	6	no	—
11	C₄²×C₁₁	D₅	1	1	yes	characteristic, complemented
12	#4020	—	11	1	no	maximal, Hall
13	C₂×C₅×C₁₁	C₂×D₄	1	3	yes	—
14	D₅₅	—	4	4	no	—
15	C₂²×D₁₁	—	5	4	no	—
16	C₂×C₄×C₁₁	C₂×D₅	1	3	yes	—
17	D₄₄	—	10	12	no	—
18	C₂×D₂₀	—	11	6	no	—
19	C₄²×C₅	D₁₁	1	1	yes	characteristic, complemented
20	C₅×C₁₁	#127	1	1	yes	characteristic, complemented, Hall
21	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
22	C₂×D₁₁	—	10	12	no	—
23	C₄×C₁₁	D₂₀	1	6	yes	complemented
24	C₂²×D₅	—	11	4	no	—
25	C₂×C₄×C₅	C₂×D₁₁	1	3	yes	—
26	D₂₀	—	22	12	no	—
27	#127	—	55	1	no	Sylow
28	C₂×C₁₁	C₂×D₂₀	1	3	yes	—
29	D₁₁	—	20	4	no	—
30	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
31	C₂×D₅	—	22	12	no	—
32	C₄×C₅	D₄₄	1	6	yes	complemented
33	C₂×D₄	—	55	6	no	—
34	C₄²	D₅₅	1	1	yes	characteristic, complemented
35	C₁₁	#4020	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₅	C₂×D₄₄	1	3	yes	—
37	D₅	—	44	4	no	—
38	C₂³	—	55	4	no	—
39	D₄	—	110	12	no	—
40	C₂×C₄	C₂×D₅₅	1	3	yes	—
41	C₅	#66927	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	110	12	no	—
43	C₂²	C₂²×D₅₅	1	1	yes	characteristic
44	C₄	D₂₂₀	1	6	yes	complemented
45	C₂	—	220	4	no	—
46	C₂	C₂×D₂₂₀	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅₅
commutator, socle, cosocle	C₂²×C₅×C₁₁	C₂³
Fitting	C₄²×C₅×C₁₁	C₂
hypercenter	C₄²	D₅₅
hypocenter	C₅×C₁₁	#127
LCS	G ▷ C₂²×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₂₀	C₂	semidirect
2. C₄²×C₅×C₁₁	C₂	semidirect
9. C₄×C₅×C₁₁	D₄	semidirect
11. C₄²×C₁₁	D₅	semidirect
19. C₄²×C₅	D₁₁	semidirect
20. C₅×C₁₁	#127	semidirect
23. C₄×C₁₁	D₂₀	semidirect
32. C₄×C₅	D₄₄	semidirect
34. C₄²	D₅₅	semidirect
35. C₁₁	#4020	semidirect
41. C₅	#66927	semidirect
44. C₄	D₂₂₀	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₂₀	2
C₄²×C₅×C₁₁	2
#66927	5
#4020	11