Subgroups — #11764675

Order 1120.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#168	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#674	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×C₅×C₇	C₄	1	1	yes	characteristic
4	C₇×#168	C₄	1	1	yes	characteristic
5	C₇×#168	C₂²	1	1	yes	characteristic
6	#6386	—	5	1	no	maximal, Hall
7	#3998	—	7	1	no	maximal, Hall
8	C₂²×C₅×C₇	C₈	1	1	yes	characteristic
9	C₄×C₅×C₇	C₈	1	1	yes	characteristic
10	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₇×C₈	—	5	1	no	—
12	C₇⋊C₁₆	—	5	2	no	—
13	C₂×#168	D₇	1	1	yes	characteristic
14	#674	—	7	2	no	—
15	C₂×C₅×C₇	—	2	1	no	—
16	C₂×C₅×C₇	C₂×C₈	1	1	yes	characteristic
17	C₂×C₄×C₇	AGL(1,𝔽₅)	1	1	yes	characteristic
18	C₇×C₈	—	5	1	no	—
19	C₇×C₈	—	5	1	no	—
20	C₂×C₄×C₅	Dic₇	1	1	yes	characteristic
21	#168	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	#170	1	1	yes	characteristic
26	C₄×C₇	#170	1	1	yes	characteristic
27	C₄×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₂²×C₅	C₇⋊C₈	1	1	yes	characteristic
29	C₄×C₅	C₇⋊C₈	1	1	yes	characteristic
30	C₄×C₅	C₂×Dic₇	1	1	yes	characteristic
31	C₂×C₈	—	5	1	no	—
32	C₁₆	—	35	2	no	—
33	C₂×C₇	—	2	1	no	—
34	C₂×C₇	C₂×#170	1	1	yes	characteristic
35	C₂×C₅	—	2	1	no	—
36	C₂×C₅	C₂×C₇⋊C₈	1	1	yes	characteristic
37	C₂×C₄	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
38	C₈	—	5	1	no	—
39	C₈	—	5	1	no	—
40	C₇	#3998	1	1	yes	characteristic, complemented, Sylow
41	C₅	#6386	1	1	yes	characteristic, complemented, Sylow
42	C₂²	C₇⋊#170	1	1	yes	characteristic
43	C₄	C₇⋊#170	1	1	yes	characteristic
44	C₄	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	C₂×C₇⋊#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×C₇⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₅×C₇	C₂×C₈
Fitting	C₂×C₄×C₅×C₇	C₄
prime radical	C₂²×C₅×C₇	C₈
cosocle	C₇×#168	C₂²
hypercenter	C₂×C₄	C₇⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	M₃₂
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄

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	#3998	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
C₂×C₇×#168	2
C₇⋊#674	2
#6386	5
#3998	7