Subgroups — #11764682

Order 1120.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	#10587935	C₂	1	2	yes	maximal
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×Dic₁₀	—	2	2	no	—
7	C₇⋊#170	—	2	2	no	—
8	#6410	—	5	1	no	maximal, Hall
9	#4005	—	7	1	no	maximal, Hall
10	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
11	C₄×C₅×C₇	—	2	1	no	—
12	C₇×Dic₅	—	2	1	no	—
13	C₇×Dic₅	D₄	1	2	yes	—
14	C₂×C₇×Q₈	—	5	1	no	—
15	#1136	—	5	2	no	—
16	C₂×Dic₁₀	D₇	1	1	yes	characteristic
17	#704	—	7	2	no	—
18	C₂×C₅×C₇	—	2	1	no	—
19	C₂×C₅×C₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₇×Q₈	—	10	2	no	—
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	Dic₁₀	—	2	2	no	—
29	#170	—	14	2	no	—
30	C₅×C₇	#101	1	1	yes	characteristic, complemented, Hall
31	#101	—	35	1	no	Sylow
32	C₂²×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
33	C₄×C₇	—	2	1	no	—
34	C₄×C₇	—	5	2	no	—
35	C₄×C₇	—	10	1	no	—
36	C₂²×C₅	C₂×Dic₇	1	1	yes	characteristic
37	C₄×C₅	—	2	1	no	—
38	Dic₅	—	2	1	no	—
39	Dic₅	C₂²⋊D₇	1	2	yes	—
40	C₂×Q₈	—	5	1	no	—
41	M₁₆	—	35	2	no	—
42	C₂×C₇	—	2	1	no	—
43	C₂×C₇	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₂×C₅	—	2	1	no	—
45	C₂×C₅	C₂²⋊Dic₇	1	1	yes	characteristic
46	Q₈	—	10	2	no	—
47	C₂×C₄	—	5	1	no	—
48	C₂×C₄	—	5	1	no	—
49	C₂×C₄	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₈	—	70	2	no	—
51	C₇	#4005	1	1	yes	characteristic, complemented, Sylow
52	C₅	#6410	1	1	yes	characteristic, complemented, Sylow
53	C₂²	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₄	—	2	1	no	—
55	C₄	—	5	2	no	—
56	C₄	—	10	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₂²⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊(C₇⋊AGL(1,𝔽₅))
commutator, prime radical	C₂²×C₅×C₇	C₂×C₄
Frattini	C₂²	C₂×C₇⋊AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₇	C₄
socle	C₂×C₅×C₇	C₂²⋊C₄
cosocle	C₂×C₇×Dic₅	C₂²
hypercenter	C₂×C₄	C₇⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	#101
LCS	G ▷ C₂²×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
30. C₅×C₇	#101	semidirect
51. C₇	#4005	semidirect
52. C₅	#6410	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Dic₁₀	2
#10587935	2
#6410	5
#4005	7