Subgroups — #11764678

Order 1120.

80 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	complemented
7	#6377	—	5	1	no	maximal, Hall
8	#4002	—	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	—	2	1	no	—
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	M₁₆	1	1	yes	characteristic
19	C₂×C₅×C₇	M₁₆	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	#97	1	1	yes	characteristic, complemented, Hall
29	#97	—	35	1	no	Sylow
30	C₂²×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
31	C₄×C₇	—	2	1	no	—
32	C₄×C₇	—	5	2	no	—
33	C₄×C₇	—	10	1	no	—
34	C₂²×C₅	C₂×Dic₇	1	1	yes	characteristic
35	C₄×C₅	—	2	1	no	—
36	Dic₅	—	2	1	no	—
37	Dic₅	C₄×D₇	1	2	yes	—
38	C₂×C₈	—	35	2	no	—
39	C₄²	—	5	1	no	—
40	C₂×C₇	#700	1	1	yes	characteristic
41	C₂×C₇	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₂×C₇	#704	1	1	yes	characteristic
43	C₂×C₅	#1136	1	1	yes	characteristic
44	C₂×C₅	#1136	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	#4002	1	1	yes	characteristic, complemented, Sylow
51	C₅	#6377	1	1	yes	characteristic, complemented, Sylow
52	C₂²	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	5	2	no	—
55	C₄	—	10	1	no	—
56	C₂	#10587931	1	1	yes	characteristic
57	C₂	C₄×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₂	#10587935	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×C₇⋊AGL(1,𝔽₅)
commutator	C₂×C₅×C₇	C₄²
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	#97
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
6. C₇⋊#170	C₄	semidirect
28. C₅×C₇	#97	semidirect
50. C₇	#4002	semidirect
51. C₅	#6377	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
#6377	5
#4002	7