Subgroups — C₁₃×D₇×D₆₉ #38737431

Order 25116.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×D₄₈₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₁₃×D₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₁₃×D₇×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×D₇×D₂₃	—	3	1	no	maximal, Hall
5	C₃×C₇×C₁₃×C₂₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₁₃×D₁₆₁	—	3	1	no	—
7	C₁₃×D₇×C₂₃	D₃	1	1	yes	characteristic, complemented
8	C₇×C₁₃×D₂₃	—	3	1	no	—
9	C₂×C₁₃×D₆₉	—	7	1	no	maximal, Hall
10	C₇×C₁₃×C₂₃	C₂×D₃	1	1	yes	characteristic, complemented, Hall
11	D₇×D₆₉	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₃×C₁₃×C₂₃	—	7	1	no	—
13	C₁₃×D₆₉	—	7	1	no	—
14	C₁₃×D₆₉	D₇	1	1	yes	characteristic, direct product factor
15	C₂×C₁₃×D₂₃	—	21	1	no	Hall
16	D₃×C₁₃×D₇	—	23	1	no	maximal, Hall
17	D₄₈₃	C₂×C₁₃	1	1	yes	characteristic, complemented
18	C₇×D₆₉	C₂×C₁₃	1	1	yes	characteristic, complemented
19	C₃×D₇×C₂₃	C₂×C₁₃	1	1	yes	characteristic, complemented
20	C₃×C₁₃×C₂₃	C₂×D₇	1	1	yes	characteristic, complemented, Hall
21	D₇×D₂₃	—	3	1	no	Hall
22	C₂×C₁₃×C₂₃	—	7	1	no	—
23	C₁₃×D₂₃	—	3	1	no	—
24	C₁₃×D₂₃	—	21	1	no	—
25	C₁₃×D₂₁	—	23	1	no	—
26	D₃×C₇×C₁₃	—	23	1	no	—
27	C₃×C₁₃×D₇	D₂₃	1	1	yes	characteristic, complemented
28	C₃×C₇×C₂₃	C₂²×C₁₃	1	1	yes	characteristic, complemented, Hall
29	C₂×C₁₃×D₇	—	69	1	no	Hall
30	D₁₆₁	—	3	1	no	—
31	C₇×D₂₃	—	3	1	no	—
32	D₇×C₂₃	D₃×C₁₃	1	1	yes	characteristic, complemented
33	C₁₃×C₂₃	D₃×D₇	1	1	yes	characteristic, complemented, Hall
34	C₂×D₆₉	—	7	1	no	Hall
35	C₃×C₇×C₁₃	C₂×D₂₃	1	1	yes	characteristic, complemented, Hall
36	C₂×C₇×C₁₃	—	69	1	no	—
37	C₁₃×D₇	—	69	1	no	—
38	C₁₃×D₇	D₆₉	1	1	yes	characteristic, direct product factor
39	C₇×C₂₃	C₂×D₃×C₁₃	1	1	yes	characteristic, complemented, Hall
40	C₂×D₃×C₁₃	—	161	1	no	Hall
41	C₂×C₃×C₂₃	—	7	1	no	—
42	D₆₉	—	7	1	no	—
43	D₆₉	C₁₃×D₇	1	1	yes	characteristic, direct product factor
44	C₂×D₂₃	—	21	1	no	Hall
45	C₇×C₁₃	C₂×D₆₉	1	1	yes	characteristic, complemented, Hall
46	D₃×D₇	—	23	1	no	Hall
47	C₂×C₃×C₁₃	—	7	1	no	—
48	D₃×C₁₃	—	23	1	no	—
49	D₃×C₁₃	—	161	1	no	—
50	C₃×C₂₃	C₂×C₁₃×D₇	1	1	yes	characteristic, complemented, Hall
51	C₂²×C₁₃	—	483	1	no	Hall
52	C₂×C₂₃	—	7	1	no	—
53	D₂₃	—	3	1	no	—
54	D₂₃	—	21	1	no	—
55	D₂₁	—	23	1	no	—
56	C₃×D₇	C₁₃×D₂₃	1	1	yes	characteristic, complemented
57	D₃×C₇	—	23	1	no	—
58	C₃×C₁₃	D₇×D₂₃	1	1	yes	characteristic, complemented, Hall
59	C₂×D₇	—	69	1	no	Hall
60	C₂×C₁₃	—	7	1	no	—
61	C₂×C₁₃	—	69	1	no	—
62	C₂×C₁₃	—	483	1	no	—
63	C₂₃	D₃×C₁₃×D₇	1	1	yes	characteristic, complemented, Sylow
64	C₃×C₇	C₂×C₁₃×D₂₃	1	1	yes	characteristic, complemented, Hall
65	C₂×C₇	—	69	1	no	—
66	D₇	—	69	1	no	—
67	D₇	C₁₃×D₆₉	1	1	yes	characteristic, direct product factor
68	C₁₃	D₇×D₆₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×D₃	—	161	1	no	Hall
70	C₇	C₂×C₁₃×D₆₉	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	7	1	no	—
72	D₃	—	23	1	no	—
73	D₃	—	161	1	no	—
74	C₂²	—	483	1	no	Sylow
75	C₃	C₁₃×D₇×D₂₃	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	7	1	no	—
77	C₂	—	69	1	no	—
78	C₂	—	483	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃	D₇×D₆₉
commutator, cosocle, hypocenter	C₃×C₇×C₂₃	C₂²×C₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₃×C₂₃	C₂²
LCS	G ▷ C₃×C₇×C₂₃
UCS	C₁ ◁ C₁₃

Complements

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

N	H ≅ G/N	type
1. C₁₃×D₄₈₃	C₂	semidirect
2. C₇×C₁₃×D₆₉	C₂	semidirect
3. C₃×C₁₃×D₇×C₂₃	C₂	semidirect
5. C₃×C₇×C₁₃×C₂₃	C₂²	semidirect
7. C₁₃×D₇×C₂₃	D₃	semidirect
10. C₇×C₁₃×C₂₃	C₂×D₃	semidirect
11. D₇×D₆₉	C₁₃	direct (N × H)
14. C₁₃×D₆₉	D₇	direct (N × H)
17. D₄₈₃	C₂×C₁₃	semidirect
18. C₇×D₆₉	C₂×C₁₃	semidirect
19. C₃×D₇×C₂₃	C₂×C₁₃	semidirect
20. C₃×C₁₃×C₂₃	C₂×D₇	semidirect
27. C₃×C₁₃×D₇	D₂₃	semidirect
28. C₃×C₇×C₂₃	C₂²×C₁₃	semidirect
32. D₇×C₂₃	D₃×C₁₃	semidirect
33. C₁₃×C₂₃	D₃×D₇	semidirect
35. C₃×C₇×C₁₃	C₂×D₂₃	semidirect
38. C₁₃×D₇	D₆₉	direct (N × H)
39. C₇×C₂₃	C₂×D₃×C₁₃	semidirect
43. D₆₉	C₁₃×D₇	direct (N × H)
45. C₇×C₁₃	C₂×D₆₉	semidirect
50. C₃×C₂₃	C₂×C₁₃×D₇	semidirect
56. C₃×D₇	C₁₃×D₂₃	semidirect
58. C₃×C₁₃	D₇×D₂₃	semidirect
63. C₂₃	D₃×C₁₃×D₇	semidirect
64. C₃×C₇	C₂×C₁₃×D₂₃	semidirect
67. D₇	C₁₃×D₆₉	direct (N × H)
68. C₁₃	D₇×D₆₉	direct (N × H)
70. C₇	C₂×C₁₃×D₆₉	semidirect
75. C₃	C₁₃×D₇×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×D₄₈₃	2
C₇×C₁₃×D₆₉	2
C₃×C₁₃×D₇×C₂₃	2
C₁₃×D₇×D₂₃	3
C₂×C₁₃×D₆₉	7
D₇×D₆₉	13
D₃×C₁₃×D₇	23

Subgroups — C13×D7×D69 #38737431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×D₇×D₆₉ #38737431