Subgroups — C₂²×C₇×Dic₁₀ #11765436

Order 1120.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×Dic₅	C₂	1	2	yes	maximal
3	C₂×C₇×Dic₁₀	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅×C₇	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₇	C₂²	1	6	yes	—
6	C₂×C₇×Dic₅	C₂²	1	12	yes	—
7	C₇×Dic₁₀	C₂²	1	16	yes	direct product factor
8	C₂²×C₇×Q₈	—	5	1	no	maximal, Hall
9	C₂²×Dic₁₀	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₅×C₇	Q₈	1	4	yes	complemented
11	C₂²×C₅×C₇	C₂³	1	3	yes	—
12	C₄×C₅×C₇	C₂³	1	4	yes	—
13	C₇×Dic₅	C₂³	1	8	yes	—
14	C₂×C₇×Q₈	—	5	12	no	—
15	C₂²×C₄×C₇	—	5	2	no	—
16	C₂²×C₄×C₇	D₅	1	1	yes	characteristic
17	C₂²×C₄×C₅	C₂×C₇	1	1	yes	characteristic
18	C₂²×Dic₅	C₂×C₇	1	2	yes	—
19	C₂×Dic₁₀	C₂×C₇	1	12	yes	direct product factor
20	C₂×C₅×C₇	C₂×Q₈	1	6	yes	complemented
21	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
22	C₂³×C₇	C₂×D₅	1	1	yes	characteristic
23	C₇×Q₈	—	5	16	no	—
24	C₂×C₄×C₇	—	5	12	no	—
25	C₂×C₄×C₇	C₂×D₅	1	6	yes	—
26	C₂³×C₅	C₂²×C₇	1	1	yes	characteristic
27	C₂×C₄×C₅	C₂²×C₇	1	6	yes	—
28	C₂×Dic₅	C₂²×C₇	1	12	yes	—
29	Dic₁₀	C₂²×C₇	1	16	yes	direct product factor
30	C₅×C₇	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
31	C₂²×Q₈	—	5	1	no	Sylow
32	C₂²×C₇	Dic₁₀	1	4	yes	direct product factor
33	C₂²×C₇	C₂²×D₅	1	3	yes	—
34	C₄×C₇	—	5	8	no	—
35	C₄×C₇	C₂²×D₅	1	4	yes	—
36	C₂²×C₅	C₇×Q₈	1	4	yes	complemented
37	C₂²×C₅	C₂³×C₇	1	3	yes	—
38	C₄×C₅	C₂³×C₇	1	4	yes	—
39	Dic₅	C₂³×C₇	1	8	yes	—
40	C₂×Q₈	—	5	12	no	—
41	C₂²×C₄	—	5	2	no	—
42	C₂²×C₄	C₇×D₅	1	1	yes	characteristic
43	C₂×C₇	C₂×Dic₁₀	1	6	yes	direct product factor
44	C₂×C₇	C₂³×D₅	1	1	yes	characteristic
45	C₂×C₅	C₂×C₇×Q₈	1	6	yes	complemented
46	C₂×C₅	C₂⁴×C₇	1	1	yes	characteristic
47	C₂³	C₂×C₇×D₅	1	1	yes	characteristic
48	Q₈	—	5	16	no	—
49	C₂×C₄	—	5	12	no	—
50	C₂×C₄	C₂×C₇×D₅	1	6	yes	—
51	C₇	C₂²×Dic₁₀	1	1	yes	characteristic, direct product factor, Sylow
52	C₅	C₂²×C₇×Q₈	1	1	yes	characteristic, complemented, Sylow
53	C₂²	C₇×Dic₁₀	1	4	yes	direct product factor
54	C₂²	C₂²×C₇×D₅	1	3	yes	—
55	C₄	—	5	8	no	—
56	C₄	C₂²×C₇×D₅	1	4	yes	—
57	C₂	C₂×C₇×Dic₁₀	1	6	yes	direct product factor
58	C₂	C₂³×C₇×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₇	C₂×D₅
commutator, cosocle	C₂×C₅	C₂⁴×C₇
Frattini	C₂	C₂³×C₇×D₅
Fitting	C₂²×C₄×C₅×C₇	C₂
socle, prime radical	C₂³×C₅×C₇	C₂²
hypercenter	C₂²×C₄×C₇	D₅
hypocenter	C₅	C₂²×C₇×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×Dic₁₀	C₂	direct (N × H)
7. C₇×Dic₁₀	C₂²	direct (N × H)
9. C₂²×Dic₁₀	C₇	direct (N × H)
10. C₂²×C₅×C₇	Q₈	semidirect
19. C₂×Dic₁₀	C₂×C₇	direct (N × H)
20. C₂×C₅×C₇	C₂×Q₈	semidirect
29. Dic₁₀	C₂²×C₇	direct (N × H)
30. C₅×C₇	C₂²×Q₈	semidirect
32. C₂²×C₇	Dic₁₀	direct (N × H)
36. C₂²×C₅	C₇×Q₈	semidirect
43. C₂×C₇	C₂×Dic₁₀	direct (N × H)
45. C₂×C₅	C₂×C₇×Q₈	semidirect
51. C₇	C₂²×Dic₁₀	direct (N × H)
52. C₅	C₂²×C₇×Q₈	semidirect
53. C₂²	C₇×Dic₁₀	direct (N × H)
57. C₂	C₂×C₇×Dic₁₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₇	2
C₂²×C₇×Dic₅	2
C₂×C₇×Dic₁₀	2
C₂²×C₇×Q₈	5
C₂²×Dic₁₀	7

Subgroups — C22×C7×Dic10 #11765436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×Dic₁₀ #11765436