Subgroups — #11765204

Order 1120.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₄₀	C₂	1	6	yes	maximal, complemented
2	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₃₅	C₂²	1	4	yes	—
4	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
5	D₁₄₀	—	2	12	no	—
6	#6437	—	5	1	no	maximal, Hall
7	#4020	—	7	1	no	maximal, Hall
8	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
9	C₂×D₃₅	—	2	12	no	—
10	C₄×C₅×C₇	D₄	1	6	yes	complemented
11	C₂×D₂₈	—	5	6	no	—
12	C₄²×C₇	D₅	1	1	yes	characteristic, complemented
13	C₂×D₂₀	—	7	6	no	—
14	C₄²×C₅	D₇	1	1	yes	characteristic, complemented
15	C₂×C₅×C₇	C₂×D₄	1	3	yes	—
16	D₃₅	—	4	4	no	—
17	C₂²×D₇	—	5	4	no	—
18	C₂×C₄×C₇	C₂×D₅	1	3	yes	—
19	D₂₈	—	10	12	no	—
20	C₂²×D₅	—	7	4	no	—
21	C₂×C₄×C₅	C₂×D₇	1	3	yes	—
22	D₂₀	—	14	12	no	—
23	C₅×C₇	#127	1	1	yes	characteristic, complemented, Hall
24	#127	—	35	1	no	Sylow
25	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
26	C₂×D₇	—	10	12	no	—
27	C₄×C₇	D₂₀	1	6	yes	complemented
28	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
29	C₂×D₅	—	14	12	no	—
30	C₄×C₅	D₂₈	1	6	yes	complemented
31	C₂×D₄	—	35	6	no	—
32	C₄²	D₃₅	1	1	yes	characteristic, complemented
33	C₂×C₇	C₂×D₂₀	1	3	yes	—
34	D₇	—	20	4	no	—
35	C₂×C₅	C₂×D₂₈	1	3	yes	—
36	D₅	—	28	4	no	—
37	C₂³	—	35	4	no	—
38	D₄	—	70	12	no	—
39	C₂×C₄	C₂×D₃₅	1	3	yes	—
40	C₇	#4020	1	1	yes	characteristic, complemented, Sylow
41	C₅	#6437	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	70	12	no	—
43	C₂²	C₂²×D₃₅	1	1	yes	characteristic
44	C₄	D₁₄₀	1	6	yes	complemented
45	C₂	—	140	4	no	—
46	C₂	C₂×D₁₄₀	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₅
commutator, socle, cosocle	C₂²×C₅×C₇	C₂³
Fitting	C₄²×C₅×C₇	C₂
hypercenter	C₄²	D₃₅
hypocenter	C₅×C₇	#127
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₄₀	C₂	semidirect
2. C₄²×C₅×C₇	C₂	semidirect
10. C₄×C₅×C₇	D₄	semidirect
12. C₄²×C₇	D₅	semidirect
14. C₄²×C₅	D₇	semidirect
23. C₅×C₇	#127	semidirect
27. C₄×C₇	D₂₀	semidirect
30. C₄×C₅	D₂₈	semidirect
32. C₄²	D₃₅	semidirect
40. C₇	#4020	semidirect
41. C₅	#6437	semidirect
44. C₄	D₁₄₀	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₄₀	2
C₄²×C₅×C₇	2
#6437	5
#4020	7