Subgroups — C₂³×C₅×Dic₇ #11765434

Order 1120.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₇	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₅×C₇	C₄	1	8	yes	complemented
4	C₂³×C₅×C₇	C₂²	1	7	yes	—
5	C₂×C₅×Dic₇	C₂²	1	28	yes	direct product factor
6	C₂³×Dic₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₄×C₅	—	7	1	no	maximal, Hall
8	C₂²×C₅×C₇	C₂×C₄	1	28	yes	complemented
9	C₂²×C₅×C₇	C₂³	1	7	yes	—
10	C₅×Dic₇	C₂³	1	8	yes	direct product factor
11	C₂⁴×C₇	C₂×C₅	1	1	yes	characteristic
12	C₂²×Dic₇	C₂×C₅	1	14	yes	direct product factor
13	C₂⁴×C₅	D₇	1	1	yes	characteristic
14	C₂²×C₄×C₅	—	7	14	no	—
15	C₂×C₅×C₇	C₂²×C₄	1	14	yes	complemented
16	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
17	C₂³×C₇	C₄×C₅	1	8	yes	complemented
18	C₂³×C₇	C₂²×C₅	1	7	yes	—
19	C₂×Dic₇	C₂²×C₅	1	28	yes	direct product factor
20	C₂³×C₅	Dic₇	1	8	yes	direct product factor
21	C₂³×C₅	C₂×D₇	1	7	yes	—
22	C₂×C₄×C₅	—	7	28	no	—
23	C₅×C₇	C₂³×C₄	1	1	yes	characteristic, complemented, Hall
24	C₂³×C₄	—	7	1	no	Sylow
25	C₂²×C₇	C₂×C₄×C₅	1	28	yes	complemented
26	C₂²×C₇	C₂³×C₅	1	7	yes	—
27	Dic₇	C₂³×C₅	1	8	yes	direct product factor
28	C₂²×C₅	C₂×Dic₇	1	28	yes	direct product factor
29	C₂²×C₅	C₂²×D₇	1	7	yes	—
30	C₄×C₅	—	7	8	no	—
31	C₂⁴	C₅×D₇	1	1	yes	characteristic
32	C₂²×C₄	—	7	14	no	—
33	C₂×C₇	C₂²×C₄×C₅	1	14	yes	complemented
34	C₂×C₇	C₂⁴×C₅	1	1	yes	characteristic
35	C₂×C₅	C₂²×Dic₇	1	14	yes	direct product factor
36	C₂×C₅	C₂³×D₇	1	1	yes	characteristic
37	C₂³	C₅×Dic₇	1	8	yes	direct product factor
38	C₂³	C₂×C₅×D₇	1	7	yes	—
39	C₂×C₄	—	7	28	no	—
40	C₇	C₂³×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
41	C₅	C₂³×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²	C₂×C₅×Dic₇	1	28	yes	direct product factor
43	C₂²	C₂²×C₅×D₇	1	7	yes	—
44	C₄	—	7	8	no	—
45	C₂	C₂²×C₅×Dic₇	1	14	yes	direct product factor
46	C₂	C₂³×C₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₅	D₇
commutator, hypocenter	C₇	C₂³×C₄×C₅
Frattini	C₂	C₂³×C₅×D₇
Fitting, socle, prime radical	C₂⁴×C₅×C₇	C₂
cosocle	C₂×C₇	C₂⁴×C₅
LCS	G ▷ C₇
UCS	C₁ ◁ C₂⁴×C₅

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×Dic₇	C₂	direct (N × H)
3. C₂³×C₅×C₇	C₄	semidirect
5. C₂×C₅×Dic₇	C₂²	direct (N × H)
6. C₂³×Dic₇	C₅	direct (N × H)
8. C₂²×C₅×C₇	C₂×C₄	semidirect
10. C₅×Dic₇	C₂³	direct (N × H)
12. C₂²×Dic₇	C₂×C₅	direct (N × H)
15. C₂×C₅×C₇	C₂²×C₄	semidirect
17. C₂³×C₇	C₄×C₅	semidirect
19. C₂×Dic₇	C₂²×C₅	direct (N × H)
20. C₂³×C₅	Dic₇	direct (N × H)
23. C₅×C₇	C₂³×C₄	semidirect
25. C₂²×C₇	C₂×C₄×C₅	semidirect
27. Dic₇	C₂³×C₅	direct (N × H)
28. C₂²×C₅	C₂×Dic₇	direct (N × H)
33. C₂×C₇	C₂²×C₄×C₅	semidirect
35. C₂×C₅	C₂²×Dic₇	direct (N × H)
37. C₂³	C₅×Dic₇	direct (N × H)
40. C₇	C₂³×C₄×C₅	semidirect
41. C₅	C₂³×Dic₇	direct (N × H)
42. C₂²	C₂×C₅×Dic₇	direct (N × H)
45. C₂	C₂²×C₅×Dic₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×C₇	2
C₂²×C₅×Dic₇	2
C₂³×Dic₇	5
C₂³×C₄×C₅	7

Subgroups — C23×C5×Dic7 #11765434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×Dic₇ #11765434