Subgroups — #11764625

Order 1120.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10587931	C₂	1	1	yes	maximal, characteristic
2	C₇×#700	C₂	1	1	yes	maximal, characteristic
3	D₅×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄×C₇×D₅	C₂²	1	1	yes	characteristic
5	C₇⋊#170	—	2	1	no	—
6	C₇×#170	—	2	1	no	—
7	C₇⋊#168	C₄	1	1	yes	characteristic
8	C₅×C₇⋊C₈	C₄	1	1	yes	characteristic
9	#6396	—	5	1	no	maximal, Hall
10	#3995	—	7	1	no	maximal, Hall
11	C₂×C₇×D₅	Q₈	1	1	yes	characteristic
12	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
13	C₇×Dic₅	D₄	1	1	yes	characteristic
14	C₇×M₁₆	—	5	1	no	—
15	#1136	—	5	1	no	—
16	C₂×C₇⋊C₈	—	5	1	no	—
17	#700	—	7	1	no	—
18	#700	D₇	1	1	yes	characteristic
19	C₈×D₅	—	7	1	no	—
20	C₂×C₅×C₇	C₄⋊C₄	1	1	yes	characteristic
21	C₇×D₅	—	2	1	no	—
22	C₂×C₄×C₇	—	5	1	no	—
23	C₇×C₈	—	10	1	no	—
24	C₇⋊C₈	—	5	1	no	—
25	C₇⋊C₈	—	10	1	no	—
26	C₇⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
27	C₄×D₅	C₂×D₇	1	1	yes	characteristic
28	#170	—	2	1	no	—
29	#170	—	14	1	no	—
30	C₅×C₈	—	7	1	no	—
31	#168	—	7	1	no	—
32	C₅×C₇	#108	1	1	yes	characteristic, complemented, Hall
33	#108	—	35	1	no	Sylow
34	C₂²×C₇	—	5	1	no	—
35	C₄×C₇	—	5	1	no	—
36	C₄×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
37	C₂×D₅	Dic₁₄	1	1	yes	characteristic
38	C₄×C₅	C₄×D₇	1	1	yes	characteristic
39	Dic₅	C₂²⋊D₇	1	1	yes	characteristic
40	M₁₆	—	5	1	no	—
41	M₁₆	—	35	1	no	—
42	C₂×C₈	—	35	1	no	—
43	C₂×C₇	—	10	1	no	—
44	C₂×C₇	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
45	C₂×C₅	Dic₇⋊C₄	1	1	yes	characteristic
46	D₅	—	2	1	no	—
47	C₂×C₄	—	5	1	no	—
48	C₈	—	7	1	no	—
49	C₈	—	10	1	no	—
50	C₈	—	35	1	no	—
51	C₈	—	70	1	no	—
52	C₇	#3995	1	1	yes	characteristic, complemented, Sylow
53	C₅	#6396	1	1	yes	characteristic, complemented, Sylow
54	C₂²	—	5	1	no	—
55	C₄	—	5	1	no	—
56	C₄	D₇×AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂	—	10	1	no	—
58	C₂	Dic₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	Dic₇⋊AGL(1,𝔽₅)
commutator, Fitting	C₄×C₅×C₇	C₂×C₄
Frattini, hypercenter	C₄	D₇×AGL(1,𝔽₅)
socle	C₂×C₅×C₇	C₄⋊C₄
prime radical	C₂×C₇×D₅	Q₈
cosocle	C₄×C₇×D₅	C₂²
hypocenter	C₅×C₇	#108
LCS	G ▷ C₄×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
32. C₅×C₇	#108	semidirect
52. C₇	#3995	semidirect
53. C₅	#6396	semidirect

Maximal subgroups

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

subgroup	index
#10587931	2
C₇×#700	2
D₅×C₇⋊C₈	2
#6396	5
#3995	7