Subgroups — C₂×C₇²×D₆₅ #27164974

Order 12740.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×D₆₅	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₅×C₇²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₇²×C₁₃	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇²×D₁₃	—	5	1	no	maximal, Hall
5	C₂×C₇×D₆₅	C₇	1	8	yes	maximal, direct product factor
6	C₂×C₇²×C₁₃	D₅	1	1	yes	characteristic, complemented
7	C₇²×D₁₃	—	5	2	no	—
8	C₂×C₇²×D₅	—	13	1	no	maximal, Hall
9	C₂×C₅×C₇×C₁₃	C₂×C₇	1	8	yes	complemented
10	C₇×D₆₅	C₂×C₇	1	16	yes	direct product factor
11	C₇²×C₁₃	C₂×D₅	1	1	yes	characteristic, complemented, Hall
12	C₂×C₅×C₇²	D₁₃	1	1	yes	characteristic, complemented
13	C₇²×D₅	—	13	2	no	—
14	C₅×C₇×C₁₃	C₂²×C₇	1	8	yes	complemented
15	C₂×C₇×D₁₃	—	5	8	no	—
16	C₂×D₆₅	C₇²	1	1	yes	characteristic, direct product factor, Hall
17	C₅×C₇²	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
18	C₂²×C₇²	—	65	1	no	Hall
19	C₂×C₇×C₁₃	C₇×D₅	1	8	yes	complemented
20	C₇×D₁₃	—	5	16	no	—
21	C₂×C₇×D₅	—	13	8	no	—
22	C₂×C₅×C₁₃	C₂×C₇²	1	1	yes	characteristic, complemented
23	D₆₅	C₂×C₇²	1	2	yes	direct product factor
24	C₂×C₇²	—	65	2	no	—
25	C₂×C₇²	D₆₅	1	1	yes	characteristic, direct product factor
26	C₇×C₁₃	C₂×C₇×D₅	1	8	yes	complemented
27	C₂×C₅×C₇	C₇×D₁₃	1	8	yes	complemented
28	C₇×D₅	—	13	16	no	—
29	C₅×C₁₃	C₂²×C₇²	1	1	yes	characteristic, complemented, Hall
30	C₂×D₁₃	—	5	1	no	Hall
31	C₇²	C₂×D₆₅	1	1	yes	characteristic, direct product factor, Sylow
32	C₅×C₇	C₂×C₇×D₁₃	1	8	yes	complemented
33	C₂²×C₇	—	65	8	no	—
34	C₂×C₁₃	C₇²×D₅	1	1	yes	characteristic, complemented
35	D₁₃	—	5	2	no	—
36	C₂×D₅	—	13	1	no	Hall
37	C₂×C₇	—	65	16	no	—
38	C₂×C₇	C₇×D₆₅	1	8	yes	direct product factor
39	C₁₃	C₂×C₇²×D₅	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₅	C₇²×D₁₃	1	1	yes	characteristic, complemented
41	D₅	—	13	2	no	—
42	C₇	C₂×C₇×D₆₅	1	8	yes	direct product factor
43	C₅	C₂×C₇²×D₁₃	1	1	yes	characteristic, complemented, Sylow
44	C₂²	—	65	1	no	Sylow
45	C₂	—	65	2	no	—
46	C₂	C₇²×D₆₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇²	D₆₅
commutator, cosocle, hypocenter	C₅×C₁₃	C₂²×C₇²
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇²×C₁₃	C₂
LCS	G ▷ C₅×C₁₃
UCS	C₁ ◁ C₂×C₇²

Complements

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

N	H ≅ G/N	type
1. C₇²×D₆₅	C₂	direct (N × H)
2. C₂×C₅×C₇²×C₁₃	C₂	semidirect
3. C₅×C₇²×C₁₃	C₂²	semidirect
5. C₂×C₇×D₆₅	C₇	direct (N × H)
6. C₂×C₇²×C₁₃	D₅	semidirect
9. C₂×C₅×C₇×C₁₃	C₂×C₇	semidirect
10. C₇×D₆₅	C₂×C₇	direct (N × H)
11. C₇²×C₁₃	C₂×D₅	semidirect
12. C₂×C₅×C₇²	D₁₃	semidirect
14. C₅×C₇×C₁₃	C₂²×C₇	semidirect
16. C₂×D₆₅	C₇²	direct (N × H)
17. C₅×C₇²	C₂×D₁₃	semidirect
19. C₂×C₇×C₁₃	C₇×D₅	semidirect
22. C₂×C₅×C₁₃	C₂×C₇²	semidirect
23. D₆₅	C₂×C₇²	direct (N × H)
25. C₂×C₇²	D₆₅	direct (N × H)
26. C₇×C₁₃	C₂×C₇×D₅	semidirect
27. C₂×C₅×C₇	C₇×D₁₃	semidirect
29. C₅×C₁₃	C₂²×C₇²	semidirect
31. C₇²	C₂×D₆₅	direct (N × H)
32. C₅×C₇	C₂×C₇×D₁₃	semidirect
34. C₂×C₁₃	C₇²×D₅	semidirect
38. C₂×C₇	C₇×D₆₅	direct (N × H)
39. C₁₃	C₂×C₇²×D₅	semidirect
40. C₂×C₅	C₇²×D₁₃	semidirect
42. C₇	C₂×C₇×D₆₅	direct (N × H)
43. C₅	C₂×C₇²×D₁₃	semidirect
46. C₂	C₇²×D₆₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇²×D₆₅	2
C₂×C₅×C₇²×C₁₃	2
C₂×C₇²×D₁₃	5
C₂×C₇×D₆₅	7
C₂×C₇²×D₅	13

Subgroups — C2×C72×D65 #27164974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇²×D₆₅ #27164974