Subgroups — C₂²×C₃×D₆₉ #13110296

Order 1656.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₆₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×D₆₉	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂²×C₃×D₂₃	—	3	1	no	maximal
5	C₂×C₃²×C₂₃	C₂²	1	3	yes	complemented
6	C₃×D₆₉	C₂²	1	4	yes	direct product factor
7	C₂²×C₃×C₂₃	—	2	1	no	—
8	C₂²×C₃×C₂₃	D₃	1	1	yes	characteristic, complemented
9	C₂²×C₃×C₂₃	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×D₆₉	C₂×C₃	1	6	yes	direct product factor
11	C₂×C₃×D₂₃	—	3	6	no	—
12	C₃²×C₂₃	C₂³	1	1	yes	characteristic, complemented, Hall
13	C₂²×D₂₃	—	3	1	no	Hall
14	C₂×C₃×C₂₃	—	2	3	no	—
15	C₂×C₃×C₂₃	C₂×D₃	1	3	yes	complemented
16	C₂×C₃×C₂₃	C₂²×C₃	1	3	yes	complemented
17	D₆₉	C₂²×C₃	1	4	yes	direct product factor
18	C₃×D₂₃	—	3	4	no	—
19	C₂²×C₂₃	C₃×D₃	1	1	yes	characteristic, complemented
20	C₂×D₂₃	—	3	6	no	—
21	C₂²×C₃×D₃	—	23	1	no	maximal, Hall
22	C₃×C₂₃	—	2	1	no	—
23	C₃×C₂₃	C₂²×D₃	1	1	yes	characteristic, complemented
24	C₃×C₂₃	C₂³×C₃	1	1	yes	characteristic, complemented
25	C₂×C₂₃	C₂×C₃×D₃	1	3	yes	complemented
26	D₂₃	—	3	4	no	—
27	C₂²×C₃²	D₂₃	1	1	yes	characteristic, complemented
28	C₂×C₃×D₃	—	23	6	no	—
29	C₂³×C₃	—	69	1	no	—
30	C₂²×D₃	—	23	1	no	—
31	C₂₃	C₂²×C₃×D₃	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₃²	C₂×D₂₃	1	3	yes	complemented
33	C₃×D₃	—	23	4	no	—
34	C₂²×C₃	—	2	1	no	—
35	C₂²×C₃	—	69	6	no	—
36	C₂²×C₃	C₃×D₂₃	1	1	yes	characteristic, complemented
37	C₂²×C₃	D₆₉	1	1	yes	characteristic, direct product factor
38	C₂×D₃	—	23	6	no	—
39	C₃²	C₂²×D₂₃	1	1	yes	characteristic, complemented, Sylow
40	C₂³	—	69	1	no	Sylow
41	C₂×C₃	—	2	3	no	—
42	C₂×C₃	—	69	4	no	—
43	C₂×C₃	C₂×C₃×D₂₃	1	3	yes	complemented
44	C₂×C₃	C₂×D₆₉	1	3	yes	direct product factor
45	D₃	—	23	4	no	—
46	C₂²	—	69	6	no	—
47	C₂²	C₃×D₆₉	1	1	yes	characteristic, direct product factor
48	C₃	—	2	1	no	—
49	C₃	C₂²×C₃×D₂₃	1	1	yes	characteristic, complemented
50	C₃	C₂²×D₆₉	1	1	yes	characteristic, direct product factor
51	C₂	—	69	4	no	—
52	C₂	C₂×C₃×D₆₉	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	D₆₉
commutator, cosocle, hypocenter	C₃×C₂₃	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₂₃	C₂
LCS	G ▷ C₃×C₂₃
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₂₃	C₂	semidirect
2. C₂×C₃×D₆₉	C₂	direct (N × H)
3. C₂²×D₆₉	C₃	direct (N × H)
5. C₂×C₃²×C₂₃	C₂²	semidirect
6. C₃×D₆₉	C₂²	direct (N × H)
8. C₂²×C₃×C₂₃	D₃	semidirect
9. C₂²×C₃×C₂₃	C₂×C₃	semidirect
10. C₂×D₆₉	C₂×C₃	direct (N × H)
12. C₃²×C₂₃	C₂³	semidirect
15. C₂×C₃×C₂₃	C₂×D₃	semidirect
16. C₂×C₃×C₂₃	C₂²×C₃	semidirect
17. D₆₉	C₂²×C₃	direct (N × H)
19. C₂²×C₂₃	C₃×D₃	semidirect
23. C₃×C₂₃	C₂²×D₃	semidirect
24. C₃×C₂₃	C₂³×C₃	semidirect
25. C₂×C₂₃	C₂×C₃×D₃	semidirect
27. C₂²×C₃²	D₂₃	semidirect
31. C₂₃	C₂²×C₃×D₃	semidirect
32. C₂×C₃²	C₂×D₂₃	semidirect
36. C₂²×C₃	C₃×D₂₃	semidirect
37. C₂²×C₃	D₆₉	direct (N × H)
39. C₃²	C₂²×D₂₃	semidirect
43. C₂×C₃	C₂×C₃×D₂₃	semidirect
44. C₂×C₃	C₂×D₆₉	direct (N × H)
47. C₂²	C₃×D₆₉	direct (N × H)
49. C₃	C₂²×C₃×D₂₃	semidirect
50. C₃	C₂²×D₆₉	direct (N × H)
52. C₂	C₂×C₃×D₆₉	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×C₃²×C₂₃	2
C₂×C₃×D₆₉	2
C₂²×D₆₉	3
C₂²×C₃×D₂₃	3
C₂²×C₃×D₃	23

Subgroups — C22×C3×D69 #13110296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×D₆₉ #13110296