Subgroups — #11764669

Order 1120.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	D₅⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₇×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇⋊AGL(1,𝔽₅)	C₄	1	2	yes	—
5	C₄×C₇×D₅	C₂²	1	1	yes	characteristic
6	C₇⋊#170	C₄	1	2	yes	complemented
7	C₅×C₇×C₈	C₄	1	1	yes	characteristic, complemented
8	C₇×#168	C₄	1	1	yes	characteristic, complemented
9	#6389	—	5	1	no	maximal, Hall
10	#3992	—	7	1	no	maximal, Hall
11	C₂×C₇×D₅	C₂×C₄	1	1	yes	characteristic
12	C₇⋊AGL(1,𝔽₅)	—	2	2	no	—
13	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
14	C₇×Dic₅	C₂×C₄	1	1	yes	characteristic
15	C₂×C₇×C₈	—	5	1	no	—
16	C₄×Dic₇	—	5	1	no	—
17	C₂×C₇⋊C₈	—	5	1	no	—
18	C₄×AGL(1,𝔽₅)	—	7	1	no	—
19	D₅⋊C₈	—	7	1	no	—
20	C₈×D₅	D₇	1	1	yes	characteristic
21	C₂×C₅×C₇	C₄²	1	1	yes	characteristic
22	C₇×D₅	M₁₆	1	2	yes	—
23	C₂×C₄×C₇	—	5	1	no	—
24	C₂×Dic₇	—	5	2	no	—
25	C₇×C₈	—	5	1	no	—
26	C₇×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
27	C₇⋊C₈	—	5	2	no	—
28	C₂×AGL(1,𝔽₅)	—	7	2	no	—
29	C₄×D₅	C₂×D₇	1	1	yes	characteristic
30	#170	—	7	2	no	—
31	C₅×C₈	Dic₇	1	1	yes	characteristic, complemented
32	#168	Dic₇	1	1	yes	characteristic, complemented
33	C₅×C₇	#97	1	1	yes	characteristic, complemented, Hall
34	#97	—	35	1	no	Sylow
35	C₂²×C₇	—	5	1	no	—
36	C₄×C₇	—	5	1	no	—
37	C₄×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
38	Dic₇	—	10	2	no	—
39	C₂×D₅	C₄×D₇	1	1	yes	characteristic
40	AGL(1,𝔽₅)	—	14	2	no	—
41	C₄×C₅	C₂×Dic₇	1	1	yes	characteristic
42	Dic₅	C₄×D₇	1	1	yes	characteristic
43	C₂×C₈	—	5	1	no	—
44	C₂×C₈	—	35	1	no	—
45	C₄²	—	35	1	no	—
46	C₂×C₇	—	5	2	no	—
47	C₂×C₇	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
48	C₂×C₅	C₄×Dic₇	1	1	yes	characteristic
49	D₅	#1131	1	2	yes	—
50	C₂×C₄	—	5	1	no	—
51	C₂×C₄	—	35	2	no	—
52	C₈	—	5	1	no	—
53	C₈	—	35	2	no	—
54	C₈	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
55	C₇	#3992	1	1	yes	characteristic, complemented, Sylow
56	C₅	#6389	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	5	1	no	—
58	C₄	—	5	1	no	—
59	C₄	—	70	2	no	—
60	C₄	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₂	—	5	2	no	—
62	C₂	C₄×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×C₇⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₅×C₇	C₄²
Fitting	C₅×C₇×C₈	C₄
prime radical	C₂×C₇×D₅	C₂×C₄
cosocle	C₄×C₇×D₅	C₂²
hypercenter	C₈	C₇⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	#97
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
6. C₇⋊#170	C₄	semidirect
7. C₅×C₇×C₈	C₄	semidirect
8. C₇×#168	C₄	semidirect
26. C₇×C₈	AGL(1,𝔽₅)	semidirect
31. C₅×C₈	Dic₇	semidirect
32. #168	Dic₇	semidirect
33. C₅×C₇	#97	semidirect
54. C₈	C₇⋊AGL(1,𝔽₅)	semidirect
55. C₇	#3992	semidirect
56. C₅	#6389	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇⋊AGL(1,𝔽₅)	2
D₅⋊(C₇⋊C₈)	2
C₇×C₈×D₅	2
#6389	5
#3992	7