Subgroups — #11765278

Order 1120.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	#10587898	C₂	1	3	yes	maximal, complemented
3	C₂×C₇⋊#168	C₂	1	3	yes	maximal, complemented
4	C₅×C₇×Q₈	C₄	1	1	yes	characteristic
5	C₅×C₇×D₄	C₄	1	3	yes	—
6	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
7	C₇⋊#168	C₂²	1	1	yes	characteristic
8	C₇⋊#168	C₂²	1	3	yes	—
9	#6511	—	5	1	no	maximal, Hall
10	#4094	—	7	1	no	maximal, Hall
11	C₂²×C₅×C₇	C₂×C₄	1	3	yes	—
12	C₄×C₅×C₇	C₂×C₄	1	3	yes	—
13	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₇×#41	D₅	1	1	yes	characteristic
15	#1136	—	5	3	no	—
16	C₂×C₇⋊C₈	—	5	3	no	—
17	C₅×#41	D₇	1	1	yes	characteristic
18	#681	—	7	3	no	—
19	C₂×#168	—	7	3	no	—
20	C₂×C₅×C₇	—	2	3	no	—
21	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
22	C₇×Q₈	Dic₅	1	1	yes	characteristic
23	C₇×D₄	Dic₅	1	3	yes	—
24	C₂×C₄×C₇	C₂×D₅	1	3	yes	—
25	C₇⋊C₈	—	5	1	no	—
26	C₇⋊C₈	—	5	3	no	—
27	C₅×Q₈	Dic₇	1	1	yes	characteristic
28	C₅×D₄	Dic₇	1	3	yes	—
29	C₂×C₄×C₅	C₂×D₇	1	3	yes	—
30	#168	—	7	1	no	—
31	#168	—	7	3	no	—
32	C₅×C₇	#131	1	1	yes	characteristic, complemented, Hall
33	#131	—	35	1	no	Sylow
34	C₂²×C₇	C₂×Dic₅	1	3	yes	—
35	C₄×C₇	C₂×Dic₅	1	3	yes	—
36	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
37	C₂²×C₅	C₂×Dic₇	1	3	yes	—
38	C₄×C₅	C₂×Dic₇	1	3	yes	—
39	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
40	#41	D₃₅	1	1	yes	characteristic
41	M₁₆	—	35	3	no	—
42	C₂×C₈	—	35	3	no	—
43	C₂×C₇	—	2	3	no	—
44	C₂×C₇	C₂²×Dic₅	1	1	yes	characteristic
45	C₂×C₅	—	2	3	no	—
46	C₂×C₅	C₂²×Dic₇	1	1	yes	characteristic
47	Q₈	Dic₃₅	1	1	yes	characteristic
48	D₄	Dic₃₅	1	3	yes	—
49	C₂×C₄	C₂×D₃₅	1	3	yes	—
50	C₈	—	35	1	no	—
51	C₈	—	35	3	no	—
52	C₇	#4094	1	1	yes	characteristic, complemented, Sylow
53	C₅	#6511	1	1	yes	characteristic, complemented, Sylow
54	C₂²	C₂×Dic₃₅	1	3	yes	—
55	C₄	C₂×Dic₃₅	1	3	yes	—
56	C₄	C₂²×D₃₅	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×Dic₃₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₃₅
commutator, socle	C₂×C₅×C₇	C₂²×C₄
Fitting, prime radical	C₅×C₇×#41	C₂
cosocle	C₄×C₅×C₇	C₂³
hypercenter	#41	D₃₅
hypocenter	C₅×C₇	#131
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #10587898	C₂	semidirect
3. C₂×C₇⋊#168	C₂	semidirect
32. C₅×C₇	#131	semidirect
52. C₇	#4094	semidirect
53. C₅	#6511	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₇×#41	2
#10587898	2
C₂×C₇⋊#168	2
#6511	5
#4094	7