Subgroups — Dic₇×#170 #11764631

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	1	yes	maximal, characteristic
2	C₂×C₇×#170	C₂	1	1	yes	maximal, characteristic
3	Dic₅×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₅	C₄	1	1	yes	characteristic
5	C₂×C₇×Dic₅	C₂²	1	1	yes	characteristic
6	C₂×C₅×Dic₇	C₄	1	1	yes	characteristic
7	C₇⋊#170	C₄	1	2	yes	complemented
8	C₇×#170	C₄	1	2	yes	complemented
9	C₈×Dic₇	—	5	1	no	maximal, Hall
10	C₄×#170	—	7	1	no	maximal, Hall
11	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
12	Dic₃₅	C₈	1	2	yes	complemented
13	C₅×Dic₇	C₈	1	2	yes	complemented
14	C₇×Dic₅	C₂×C₄	1	1	yes	characteristic
15	C₇×Dic₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₇×C₈	—	5	1	no	—
17	C₄×Dic₇	—	5	1	no	—
18	C₂×C₇⋊C₈	—	5	1	no	—
19	C₂×#170	—	7	1	no	—
20	C₂×#170	D₇	1	1	yes	characteristic
21	C₄×Dic₅	—	7	1	no	—
22	C₂×C₅×C₇	C₄²	1	1	yes	characteristic
23	C₂×C₅×C₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₅×C₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₇	—	5	1	no	—
26	C₂×Dic₇	—	5	1	no	—
27	C₂×Dic₇	AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₇×C₈	—	5	2	no	—
29	C₇⋊C₈	—	5	2	no	—
30	C₂×C₄×C₅	—	7	1	no	—
31	C₂×Dic₅	—	7	1	no	—
32	C₂×Dic₅	C₂×D₇	1	1	yes	characteristic
33	#170	—	7	2	no	—
34	#170	Dic₇	1	2	yes	direct product factor
35	C₅×C₇	C₄×C₈	1	1	yes	characteristic, complemented, Hall
36	C₄×C₈	—	35	1	no	Sylow
37	C₂²×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
38	C₄×C₇	—	5	1	no	—
39	C₄×C₇	—	5	1	no	—
40	Dic₇	—	5	2	no	—
41	Dic₇	#170	1	2	yes	direct product factor
42	C₂²×C₅	C₄×D₇	1	1	yes	characteristic
43	C₄×C₅	—	7	2	no	—
44	Dic₅	—	7	2	no	—
45	Dic₅	C₄×D₇	1	1	yes	characteristic
46	Dic₅	C₂×Dic₇	1	1	yes	characteristic
47	C₂×C₈	—	5	1	no	—
48	C₂×C₈	—	35	1	no	—
49	C₄²	—	35	1	no	—
50	C₂×C₇	D₅⋊C₈	1	1	yes	characteristic
51	C₂×C₇	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂×C₇	C₂×#170	1	1	yes	characteristic
53	C₂×C₅	C₈×D₇	1	1	yes	characteristic
54	C₂×C₅	C₈×D₇	1	1	yes	characteristic
55	C₂×C₅	C₄×Dic₇	1	1	yes	characteristic
56	C₂×C₄	—	5	1	no	—
57	C₂×C₄	—	7	1	no	—
58	C₂×C₄	—	35	1	no	—
59	C₈	—	5	2	no	—
60	C₈	—	35	2	no	—
61	C₇	C₄×#170	1	1	yes	characteristic, complemented, Sylow
62	C₅	C₈×Dic₇	1	1	yes	characteristic, complemented, Sylow
63	C₂²	D₇×AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₄	—	5	1	no	—
65	C₄	—	5	1	no	—
66	C₄	—	7	2	no	—
67	C₄	—	35	2	no	—
68	C₂	AGL(1,𝔽₅)×Dic₇	1	1	yes	characteristic
69	C₂	D₇×#170	1	1	yes	characteristic
70	C₂	#10587920	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₇×AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₇	C₄×C₈
Fitting, socle, prime radical	C₂²×C₅×C₇	C₂×C₄
cosocle	C₂×C₇×Dic₅	C₂²
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
7. C₇⋊#170	C₄	semidirect
8. C₇×#170	C₄	semidirect
12. Dic₃₅	C₈	semidirect
13. C₅×Dic₇	C₈	semidirect
34. #170	Dic₇	direct (N × H)
35. C₅×C₇	C₄×C₈	semidirect
41. Dic₇	#170	direct (N × H)
61. C₇	C₄×#170	semidirect
62. C₅	C₈×Dic₇	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
C₂×C₇×#170	2
Dic₅×Dic₇	2
C₈×Dic₇	5
C₄×#170	7

Subgroups — Dic7×#170 #11764631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₇×#170 #11764631