Subgroups — #11765206

Order 1120.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#10587902	C₂	1	3	yes	maximal
3	#10587900	C₂	1	3	yes	maximal, complemented
4	C₂²×D₃₅	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
6	C₂×Dic₃₅	C₂²	1	3	yes	—
7	#6439	—	5	1	no	maximal, Hall
8	#4022	—	7	1	no	maximal, Hall
9	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
10	C₂×D₃₅	—	2	3	no	—
11	C₄×C₅×C₇	—	2	3	no	—
12	Dic₃₅	—	2	3	no	—
13	C₄²×C₇	D₅	1	1	yes	characteristic, complemented
14	(C₂×D₇)⋊C₄	—	5	3	no	—
15	Dic₇⋊C₄	—	5	3	no	—
16	C₄²×C₅	D₇	1	1	yes	characteristic, complemented
17	#685	—	7	3	no	—
18	#683	—	7	3	no	—
19	C₂×C₅×C₇	#41	1	3	yes	—
20	D₃₅	—	4	1	no	—
21	C₂²×D₇	—	5	1	no	—
22	C₂×C₄×C₇	C₂×D₅	1	3	yes	—
23	C₂×Dic₇	—	5	3	no	—
24	C₂²×D₅	—	7	1	no	—
25	C₂×C₄×C₅	C₂×D₇	1	3	yes	—
26	C₂×Dic₅	—	7	3	no	—
27	C₅×C₇	#126	1	1	yes	characteristic, complemented, Hall
28	#126	—	35	1	no	Sylow
29	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
30	C₂×D₇	—	10	3	no	—
31	C₄×C₇	—	2	3	no	—
32	Dic₇	—	10	3	no	—
33	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
34	C₂×D₅	—	14	3	no	—
35	C₄×C₅	—	2	3	no	—
36	Dic₅	—	14	3	no	—
37	C₄⋊C₄	—	35	3	no	—
38	C₂²⋊C₄	—	35	3	no	—
39	C₄²	D₃₅	1	1	yes	characteristic, complemented
40	C₂×C₇	#709	1	3	yes	—
41	D₇	—	20	1	no	—
42	C₂×C₅	#1157	1	3	yes	—
43	D₅	—	28	1	no	—
44	C₂³	—	35	1	no	—
45	C₂×C₄	—	35	3	no	—
46	C₂×C₄	C₂×D₃₅	1	3	yes	—
47	C₇	#4022	1	1	yes	characteristic, complemented, Sylow
48	C₅	#6439	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	70	3	no	—
50	C₂²	C₂²×D₃₅	1	1	yes	characteristic
51	C₄	—	2	3	no	—
52	C₄	—	70	3	no	—
53	C₂	—	140	1	no	—
54	C₂	#10587987	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₅
commutator, socle, cosocle	C₂²×C₅×C₇	C₂³
Fitting	C₄²×C₅×C₇	C₂
prime radical	C₂²×D₃₅	C₂²
hypercenter	C₄²	D₃₅
hypocenter	C₅×C₇	#126
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. C₄²×C₅×C₇	C₂	semidirect
3. #10587900	C₂	semidirect
13. C₄²×C₇	D₅	semidirect
16. C₄²×C₅	D₇	semidirect
27. C₅×C₇	#126	semidirect
39. C₄²	D₃₅	semidirect
47. C₇	#4022	semidirect
48. C₅	#6439	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₅×C₇	2
#10587902	2
#10587900	2
#6439	5
#4022	7