Subgroups — C₄×C₇⋊#170 #11764677

Order 1120.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₄	C₇⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₇	C₄×C₈
Frattini	C₂²	C₂×C₇⋊AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₇	C₄
socle, prime radical	C₂²×C₅×C₇	C₂×C₄
cosocle	C₂×C₇×Dic₅	C₂²
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
6. C₇⋊#170	C₄	direct (N × H)
10. C₄×C₅×C₇	C₈	semidirect
11. C₇×Dic₅	C₈	semidirect
28. C₅×C₇	C₄×C₈	semidirect
33. C₄×C₇	#170	semidirect
35. C₄×C₅	C₇⋊C₈	semidirect
36. Dic₅	C₇⋊C₈	semidirect
50. C₇	C₄×#170	semidirect
51. C₅	C₄×C₇⋊C₈	semidirect
55. C₄	C₇⋊#170	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×C₇⋊#170	2
C₄×C₇×Dic₅	2
C₄×C₇⋊C₈	5
C₄×#170	7

Subgroups — C4×C7⋊#170 #11764677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇⋊#170 #11764677