Subgroups — C₅×#6551 #11765430

Order 1120.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#1161	C₂	1	4	yes	maximal, complemented
3	C₅×Q₈×D₇	C₂	1	4	yes	maximal, complemented
4	C₅×#1157	C₂	1	6	yes	maximal, complemented
5	C₅×C₇×Q₈	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
7	C₅×C₂²⋊D₇	C₂²	1	4	yes	—
8	C₅×D₂₈	C₂²	1	6	yes	—
9	C₄×C₅×D₇	C₂²	1	12	yes	—
10	C₅×Dic₁₄	C₂²	1	6	yes	—
11	#6551	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₅×2₋¹⁺⁴	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₂×C₅×D₇	C₂³	1	4	yes	—
15	C₄×C₅×C₇	C₂³	1	6	yes	—
16	C₅×Dic₇	C₂³	1	4	yes	—
17	C₂×C₇×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
18	#1161	C₂×C₅	1	4	yes	complemented
19	Q₈×D₇	C₂×C₅	1	4	yes	complemented
20	#1157	C₂×C₅	1	6	yes	complemented
21	C₅×#41	—	7	4	no	—
22	C₅×#41	—	7	6	no	—
23	C₂×C₅×Q₈	—	7	4	no	—
24	C₂×C₅×Q₈	D₇	1	1	yes	characteristic, complemented
25	C₂×C₅×C₇	—	2	1	no	—
26	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
27	C₅×D₇	—	2	4	no	—
28	C₇×Q₈	C₂²×C₅	1	4	yes	—
29	C₂×C₄×C₇	C₂²×C₅	1	3	yes	—
30	C₂²⋊D₇	C₂²×C₅	1	4	yes	—
31	D₂₈	C₂²×C₅	1	6	yes	—
32	C₄×D₇	C₂²×C₅	1	12	yes	—
33	Dic₁₄	C₂²×C₅	1	6	yes	—
34	C₅×Q₈	—	7	6	no	—
35	C₅×Q₈	C₂×D₇	1	4	yes	—
36	C₅×D₄	—	7	4	no	—
37	C₅×D₄	—	7	6	no	—
38	C₂×C₄×C₅	—	7	12	no	—
39	C₂×C₄×C₅	C₂×D₇	1	3	yes	—
40	C₅×C₇	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
41	2₋¹⁺⁴	—	7	1	no	Sylow
42	C₂²×C₇	C₂³×C₅	1	1	yes	characteristic
43	C₂×D₇	C₂³×C₅	1	4	yes	—
44	C₄×C₇	C₂³×C₅	1	6	yes	—
45	Dic₇	C₂³×C₅	1	4	yes	—
46	C₂²×C₅	—	7	4	no	—
47	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
48	C₄×C₅	—	7	4	no	—
49	C₄×C₅	C₂²×D₇	1	6	yes	—
50	#41	—	7	4	no	—
51	#41	—	7	6	no	—
52	C₂×Q₈	—	7	4	no	—
53	C₂×Q₈	C₅×D₇	1	1	yes	characteristic, complemented
54	C₂×C₇	—	2	1	no	—
55	C₂×C₇	C₂⁴×C₅	1	1	yes	characteristic
56	D₇	—	2	4	no	—
57	C₂×C₅	—	2	1	no	—
58	C₂×C₅	—	14	4	no	—
59	C₂×C₅	C₂³×D₇	1	1	yes	characteristic
60	Q₈	—	7	6	no	—
61	Q₈	C₂×C₅×D₇	1	4	yes	—
62	D₄	—	7	4	no	—
63	D₄	—	7	6	no	—
64	C₂×C₄	—	7	12	no	—
65	C₂×C₄	C₂×C₅×D₇	1	3	yes	—
66	C₇	C₅×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
67	C₅	#6551	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²	—	7	4	no	—
69	C₂²	C₂²×C₅×D₇	1	1	yes	characteristic
70	C₄	—	7	4	no	—
71	C₄	C₂²×C₅×D₇	1	6	yes	—
72	C₂	—	2	1	no	—
73	C₂	—	14	4	no	—
74	C₂	C₂³×C₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂³×D₇
commutator, cosocle	C₂×C₇	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×D₇
Fitting	C₂×C₅×C₇×Q₈	C₂
socle	C₂×C₅×C₇	C₂⁴
hypercenter	C₂×C₅×Q₈	D₇
hypocenter	C₇	C₅×2₋¹⁺⁴
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₇×Q₈	C₂	semidirect
2. C₅×#1161	C₂	semidirect
3. C₅×Q₈×D₇	C₂	semidirect
4. C₅×#1157	C₂	semidirect
11. #6551	C₅	direct (N × H)
17. C₂×C₇×Q₈	C₂×C₅	semidirect
18. #1161	C₂×C₅	semidirect
19. Q₈×D₇	C₂×C₅	semidirect
20. #1157	C₂×C₅	semidirect
24. C₂×C₅×Q₈	D₇	semidirect
40. C₅×C₇	2₋¹⁺⁴	semidirect
53. C₂×Q₈	C₅×D₇	semidirect
66. C₇	C₅×2₋¹⁺⁴	semidirect
67. C₅	#6551	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₇×Q₈	2
C₅×#1161	2
C₅×Q₈×D₇	2
C₅×#1157	2
#6551	5
C₅×2₋¹⁺⁴	7

Subgroups — C5×#6551 #11765430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#6551 #11765430