Subgroups — #11764628

Order 1120.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇×#168	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#674	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₇	C₄	1	1	yes	characteristic
5	C₇⋊#168	C₄	1	1	yes	characteristic
6	C₇×#168	C₂²	1	1	yes	characteristic
7	#6372	—	5	1	no	maximal, Hall
8	#3998	—	7	1	no	maximal, Hall
9	C₂×C₅×D₇	C₈	1	1	yes	characteristic
10	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
11	C₅×Dic₇	C₈	1	1	yes	characteristic
12	C₈×D₇	—	5	1	no	—
13	C₇×C₁₆	—	5	1	no	—
14	C₇⋊C₁₆	—	5	1	no	—
15	C₂×#168	—	7	1	no	—
16	#674	—	7	1	no	—
17	#674	D₇	1	1	yes	characteristic, complemented
18	C₂×C₅×C₇	C₂×C₈	1	1	yes	characteristic
19	C₅×D₇	—	2	1	no	—
20	C₄×D₇	AGL(1,𝔽₅)	1	1	yes	characteristic
21	C₇×C₈	—	5	1	no	—
22	C₇⋊C₈	—	5	1	no	—
23	C₂×C₄×C₅	—	7	1	no	—
24	#168	—	7	1	no	—
25	#168	C₂×D₇	1	1	yes	characteristic
26	C₅×C₇	M₃₂	1	1	yes	characteristic, complemented, Hall
27	M₃₂	—	35	1	no	Sylow
28	C₂×D₇	#170	1	1	yes	characteristic
29	C₄×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
30	Dic₇	#170	1	1	yes	characteristic
31	C₂²×C₅	—	7	1	no	—
32	C₄×C₅	—	7	1	no	—
33	C₄×C₅	C₄×D₇	1	1	yes	characteristic
34	C₂×C₈	—	35	1	no	—
35	C₁₆	—	5	1	no	—
36	C₁₆	—	35	1	no	—
37	C₂×C₇	C₂×#170	1	1	yes	characteristic
38	D₇	—	2	1	no	—
39	C₂×C₅	—	14	1	no	—
40	C₂×C₅	C₈×D₇	1	1	yes	characteristic
41	C₂×C₄	—	7	1	no	—
42	C₈	—	5	1	no	—
43	C₈	—	35	1	no	—
44	C₇	#3998	1	1	yes	characteristic, complemented, Sylow
45	C₅	#6372	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	7	1	no	—
47	C₄	—	7	1	no	—
48	C₄	D₇×AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₂	—	14	1	no	—
50	C₂	D₇×#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₇×AGL(1,𝔽₅)
commutator, socle	C₂×C₅×C₇	C₂×C₈
Fitting	C₄×C₅×C₇	C₂×C₄
prime radical	C₂×C₅×D₇	C₈
cosocle	C₇×#168	C₂²
hypocenter	C₅×C₇	M₃₂
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₇⋊#674	C₂	semidirect
3. C₇×#674	C₂	semidirect
17. #674	D₇	semidirect
26. C₅×C₇	M₃₂	semidirect
44. C₇	#3998	semidirect
45. C₅	#6372	semidirect

Maximal subgroups

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

subgroup	index
D₇×#168	2
C₇⋊#674	2
C₇×#674	2
#6372	5
#3998	7