Subgroups — #11764667

Order 1120.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#674	C₂	1	2	yes	maximal, complemented
3	C₄×C₇×D₅	C₄	1	1	yes	characteristic
4	C₅×C₇×C₈	C₄	1	1	yes	characteristic
5	C₇×#168	C₂²	1	1	yes	characteristic
6	#6386	—	5	1	no	maximal, Hall
7	#3990	—	7	1	no	maximal, Hall
8	C₂×C₇×D₅	C₈	1	1	yes	characteristic
9	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
10	C₇×Dic₅	C₈	1	1	yes	characteristic
11	C₂×C₇×C₈	—	5	1	no	—
12	C₇⋊C₁₆	—	5	2	no	—
13	C₈×D₅	D₇	1	1	yes	characteristic
14	#674	—	7	2	no	—
15	C₂×C₅×C₇	C₂×C₈	1	1	yes	characteristic
16	C₇×D₅	—	2	1	no	—
17	C₂×C₄×C₇	—	5	1	no	—
18	C₇×C₈	—	5	1	no	—
19	C₇×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
20	C₄×D₅	Dic₇	1	1	yes	characteristic
21	C₅×C₈	Dic₇	1	1	yes	characteristic
22	#168	C₂×D₇	1	1	yes	characteristic
23	C₅×C₇	M₃₂	1	1	yes	characteristic, complemented, Hall
24	M₃₂	—	35	1	no	Sylow
25	C₂²×C₇	—	5	1	no	—
26	C₄×C₇	—	5	1	no	—
27	C₄×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₂×D₅	C₇⋊C₈	1	1	yes	characteristic
29	C₄×C₅	C₂×Dic₇	1	1	yes	characteristic
30	Dic₅	C₇⋊C₈	1	1	yes	characteristic
31	C₂×C₈	—	5	1	no	—
32	C₁₆	—	35	2	no	—
33	C₂×C₇	—	10	1	no	—
34	C₂×C₇	D₅⋊C₈	1	1	yes	characteristic
35	C₂×C₅	C₂×C₇⋊C₈	1	1	yes	characteristic
36	D₅	—	2	1	no	—
37	C₂×C₄	—	5	1	no	—
38	C₈	—	5	1	no	—
39	C₈	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₇	#3990	1	1	yes	characteristic, complemented, Sylow
41	C₅	#6386	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	5	1	no	—
43	C₄	—	5	1	no	—
44	C₄	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
45	C₂	—	10	1	no	—
46	C₂	D₅⋊(C₇⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×C₇⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₅×C₇	C₂×C₈
Fitting	C₅×C₇×C₈	C₄
prime radical	C₂×C₇×D₅	C₈
cosocle	C₇×#168	C₂²
hypercenter	C₈	C₇⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	M₃₂
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
2. C₇⋊#674	C₂	semidirect
23. C₅×C₇	M₃₂	semidirect
40. C₇	#3990	semidirect
41. C₅	#6386	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₈×D₅	2
C₇⋊#674	2
#6386	5
#3990	7