Subgroups — C₅×#6511 #11765106

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	C₅×#1136	C₂	1	3	yes	maximal, complemented
3	C₂×C₅×C₇⋊C₈	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	C₅×C₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₅×C₇⋊C₈	C₂²	1	3	yes	—
9	#6511	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₅×#131	—	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	C₂×C₅	1	1	yes	characteristic
15	#1136	C₂×C₅	1	3	yes	complemented
16	C₂×C₇⋊C₈	C₂×C₅	1	3	yes	complemented
17	C₅×#41	D₇	1	1	yes	characteristic
18	C₅×M₁₆	—	7	3	no	—
19	C₂×C₅×C₈	—	7	3	no	—
20	C₂×C₅×C₇	—	2	3	no	—
21	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
22	C₇×Q₈	C₄×C₅	1	1	yes	characteristic
23	C₇×D₄	C₄×C₅	1	3	yes	—
24	C₂×C₄×C₇	C₂²×C₅	1	3	yes	—
25	C₇⋊C₈	C₂²×C₅	1	1	yes	characteristic
26	C₇⋊C₈	C₂²×C₅	1	3	yes	—
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	C₅×C₈	—	7	1	no	—
31	C₅×C₈	—	7	3	no	—
32	C₅×C₇	#131	1	1	yes	characteristic, complemented, Hall
33	#131	—	7	1	no	Sylow
34	C₂²×C₇	C₂×C₄×C₅	1	3	yes	—
35	C₄×C₇	C₂×C₄×C₅	1	3	yes	—
36	C₄×C₇	C₂³×C₅	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	C₅×D₇	1	1	yes	characteristic
41	M₁₆	—	7	3	no	—
42	C₂×C₈	—	7	3	no	—
43	C₂×C₇	—	2	3	no	—
44	C₂×C₇	C₂²×C₄×C₅	1	1	yes	characteristic
45	C₂×C₅	—	2	3	no	—
46	C₂×C₅	C₂²×Dic₇	1	1	yes	characteristic
47	Q₈	C₅×Dic₇	1	1	yes	characteristic
48	D₄	C₅×Dic₇	1	3	yes	—
49	C₂×C₄	C₂×C₅×D₇	1	3	yes	—
50	C₈	—	7	1	no	—
51	C₈	—	7	3	no	—
52	C₇	C₅×#131	1	1	yes	characteristic, complemented, Sylow
53	C₅	#6511	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²	C₂×C₅×Dic₇	1	3	yes	—
55	C₄	C₂×C₅×Dic₇	1	3	yes	—
56	C₄	C₂²×C₅×D₇	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₅×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×D₇
commutator	C₂×C₇	C₂²×C₄×C₅
Frattini	C₄	C₂²×C₅×D₇
Fitting, prime radical	C₅×C₇×#41	C₂
socle	C₂×C₅×C₇	C₂²×C₄
cosocle	C₄×C₇	C₂³×C₅
hypercenter	C₅×#41	D₇
hypocenter	C₇	C₅×#131
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₅×#1136	C₂	semidirect
3. C₂×C₅×C₇⋊C₈	C₂	semidirect
9. #6511	C₅	direct (N × H)
15. #1136	C₂×C₅	semidirect
16. C₂×C₇⋊C₈	C₂×C₅	semidirect
32. C₅×C₇	#131	semidirect
52. C₇	C₅×#131	semidirect
53. C₅	#6511	direct (N × H)

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
C₅×#1136	2
C₂×C₅×C₇⋊C₈	2
#6511	5
C₅×#131	7

Subgroups — C5×#6511 #11765106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#6511 #11765106