Subgroups — #11764681

Order 1120.

76 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	—	2	2	no	—
7	C₄⋊(C₇⋊C₈)	—	5	1	no	maximal, Hall
8	#4004	—	7	1	no	maximal, Hall
9	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
10	C₄×C₅×C₇	—	2	1	no	—
11	C₇×Dic₅	C₈	1	2	yes	complemented
12	C₇×Dic₅	D₄	1	1	yes	characteristic
13	C₇×Dic₅	Q₈	1	1	yes	characteristic
14	C₄²×C₇	—	5	1	no	—
15	C₂×C₇⋊C₈	—	5	2	no	—
16	C₂×#170	—	7	2	no	—
17	C₄×Dic₅	D₇	1	1	yes	characteristic
18	C₂×C₅×C₇	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₅×C₇	C₂×C₈	1	1	yes	characteristic
20	C₂×C₅×C₇	M₁₆	1	1	yes	characteristic
21	C₂×C₄×C₇	—	5	1	no	—
22	C₂×C₄×C₇	—	5	1	no	—
23	C₂×C₄×C₇	AGL(1,𝔽₅)	1	1	yes	characteristic
24	C₇⋊C₈	—	10	2	no	—
25	C₂×C₄×C₅	Dic₇	1	1	yes	characteristic
26	C₂×Dic₅	Dic₇	1	1	yes	characteristic
27	C₂×Dic₅	C₂×D₇	1	1	yes	characteristic
28	#170	—	14	2	no	—
29	C₅×C₇	C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
30	C₄⋊C₈	—	35	1	no	Sylow
31	C₂²×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
32	C₄×C₇	—	2	1	no	—
33	C₄×C₇	—	5	2	no	—
34	C₄×C₇	—	5	1	no	—
35	C₄×C₇	—	5	1	no	—
36	C₂²×C₅	C₂×Dic₇	1	1	yes	characteristic
37	C₄×C₅	—	2	1	no	—
38	Dic₅	C₇⋊C₈	1	2	yes	complemented
39	Dic₅	Dic₁₄	1	1	yes	characteristic
40	Dic₅	D₂₈	1	1	yes	characteristic
41	C₂×C₈	—	35	2	no	—
42	C₄²	—	5	1	no	—
43	C₂×C₇	D₅⋊C₈	1	1	yes	characteristic
44	C₂×C₇	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
45	C₂×C₇	#704	1	1	yes	characteristic
46	C₂×C₅	C₂×C₇⋊C₈	1	1	yes	characteristic
47	C₂×C₅	#1136	1	1	yes	characteristic
48	C₂×C₅	C₄⋊Dic₇	1	1	yes	characteristic
49	C₂×C₄	—	5	1	no	—
50	C₂×C₄	—	5	1	no	—
51	C₂×C₄	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₈	—	70	2	no	—
53	C₇	#4004	1	1	yes	characteristic, complemented, Sylow
54	C₅	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic, complemented, Sylow
55	C₂²	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	5	2	no	—
58	C₄	—	5	1	no	—
59	C₄	—	5	1	no	—
60	C₂	D₅⋊(C₇⋊C₈)	1	1	yes	characteristic
61	C₂	C₄⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic
62	C₂	#10587935	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
11. C₇×Dic₅	C₈	semidirect
29. C₅×C₇	C₄⋊C₈	semidirect
38. Dic₅	C₇⋊C₈	semidirect
53. C₇	#4004	semidirect
54. C₅	C₄⋊(C₇⋊C₈)	semidirect

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
#4004	7