Subgroups — D₆₅×D₁₂₃ #44544476

Order 31980.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇₉₉₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₁₃×D₁₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄₁×D₆₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄₁×D₆₅	—	3	1	no	maximal, Hall
5	C₃×C₅×C₁₃×C₄₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₁₃×D₁₂₃	—	5	1	no	maximal, Hall
7	D₂₆₆₅	—	3	1	no	—
8	C₄₁×D₆₅	D₃	1	1	yes	characteristic, complemented
9	C₅×C₁₃×D₄₁	—	3	1	no	—
10	D₁₅₉₉	—	5	1	no	—
11	C₁₃×D₁₂₃	D₅	1	1	yes	characteristic, complemented
12	C₃×D₁₃×C₄₁	—	5	1	no	—
13	C₅×C₁₃×C₄₁	C₂×D₃	1	1	yes	characteristic, complemented, Hall
14	D₅×D₁₂₃	—	13	1	no	maximal, Hall
15	D₁₃×D₄₁	—	15	1	no	Hall
16	C₃×C₁₃×C₄₁	C₂×D₅	1	1	yes	characteristic, complemented, Hall
17	D₆₁₅	—	13	1	no	—
18	C₅×D₁₂₃	D₁₃	1	1	yes	characteristic, complemented
19	C₃×D₅×C₄₁	—	13	1	no	—
20	D₅₃₃	—	15	1	no	—
21	C₁₃×D₄₁	—	3	1	no	—
22	D₁₃×C₄₁	—	5	1	no	—
23	D₅×D₄₁	—	39	1	no	Hall
24	D₃×D₆₅	—	41	1	no	maximal, Hall
25	C₃×C₅×C₄₁	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
26	C₁₃×C₄₁	D₃×D₅	1	1	yes	characteristic, complemented, Hall
27	C₂×D₁₂₃	—	65	1	no	Hall
28	D₂₀₅	—	39	1	no	—
29	C₅×D₄₁	—	3	1	no	—
30	D₅×C₄₁	—	13	1	no	—
31	D₁₉₅	—	41	1	no	—
32	C₅×D₃×C₁₃	—	41	1	no	—
33	C₃×D₆₅	D₄₁	1	1	yes	characteristic, complemented
34	C₂×D₆₅	—	123	1	no	Hall
35	C₂×C₃×C₄₁	—	65	1	no	—
36	D₁₂₃	—	65	1	no	—
37	D₁₂₃	D₆₅	1	1	yes	characteristic, direct product factor
38	C₅×C₄₁	D₃×D₁₃	1	1	yes	characteristic, complemented, Hall
39	C₃×C₅×C₁₃	C₂×D₄₁	1	1	yes	characteristic, complemented, Hall
40	C₂×D₄₁	—	195	1	no	Hall
41	D₃×D₁₃	—	205	1	no	Hall
42	C₂×C₅×C₁₃	—	123	1	no	—
43	D₆₅	—	123	1	no	—
44	D₆₅	D₁₂₃	1	1	yes	characteristic, direct product factor
45	C₃×C₄₁	C₂×D₆₅	1	1	yes	characteristic, complemented, Hall
46	C₂×C₄₁	—	65	1	no	—
47	D₄₁	—	3	1	no	—
48	D₄₁	—	195	1	no	—
49	D₃₉	—	205	1	no	—
50	C₃×D₁₃	—	5	1	no	—
51	D₃×C₁₃	—	41	1	no	—
52	C₅×C₁₃	C₂×D₁₂₃	1	1	yes	characteristic, complemented, Hall
53	D₃×D₅	—	533	1	no	Hall
54	C₂×D₁₃	—	615	1	no	Hall
55	C₄₁	D₃×D₆₅	1	1	yes	characteristic, complemented, Sylow
56	C₃×C₁₃	D₅×D₄₁	1	1	yes	characteristic, complemented, Hall
57	D₁₅	—	533	1	no	—
58	C₃×D₅	—	13	1	no	—
59	C₅×D₃	—	41	1	no	—
60	C₂×C₁₃	—	123	1	no	—
61	D₁₃	—	5	1	no	—
62	D₁₃	—	615	1	no	—
63	C₂×D₅	—	1599	1	no	Hall
64	C₃×C₅	D₁₃×D₄₁	1	1	yes	characteristic, complemented, Hall
65	C₁₃	D₅×D₁₂₃	1	1	yes	characteristic, complemented, Sylow
66	C₂×D₃	—	2665	1	no	Hall
67	C₂×C₅	—	123	1	no	—
68	D₅	—	13	1	no	—
69	D₅	—	1599	1	no	—
70	C₂×C₃	—	65	1	no	—
71	D₃	—	41	1	no	—
72	D₃	—	2665	1	no	—
73	C₅	D₁₃×D₁₂₃	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	7995	1	no	Sylow
75	C₃	D₄₁×D₆₅	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	7995	1	no	—
77	C₂	—	65	1	no	—
78	C₂	—	123	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₃×C₄₁	C₂²
LCS	G ▷ C₃×C₅×C₁₃×C₄₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. D₇₉₉₅	C₂	semidirect
2. C₅×C₁₃×D₁₂₃	C₂	semidirect
3. C₃×C₄₁×D₆₅	C₂	semidirect
5. C₃×C₅×C₁₃×C₄₁	C₂²	semidirect
8. C₄₁×D₆₅	D₃	semidirect
11. C₁₃×D₁₂₃	D₅	semidirect
13. C₅×C₁₃×C₄₁	C₂×D₃	semidirect
16. C₃×C₁₃×C₄₁	C₂×D₅	semidirect
18. C₅×D₁₂₃	D₁₃	semidirect
25. C₃×C₅×C₄₁	C₂×D₁₃	semidirect
26. C₁₃×C₄₁	D₃×D₅	semidirect
33. C₃×D₆₅	D₄₁	semidirect
37. D₁₂₃	D₆₅	direct (N × H)
38. C₅×C₄₁	D₃×D₁₃	semidirect
39. C₃×C₅×C₁₃	C₂×D₄₁	semidirect
44. D₆₅	D₁₂₃	direct (N × H)
45. C₃×C₄₁	C₂×D₆₅	semidirect
52. C₅×C₁₃	C₂×D₁₂₃	semidirect
55. C₄₁	D₃×D₆₅	semidirect
56. C₃×C₁₃	D₅×D₄₁	semidirect
64. C₃×C₅	D₁₃×D₄₁	semidirect
65. C₁₃	D₅×D₁₂₃	semidirect
73. C₅	D₁₃×D₁₂₃	semidirect
75. C₃	D₄₁×D₆₅	semidirect

Maximal subgroups

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

subgroup	index
D₇₉₉₅	2
C₅×C₁₃×D₁₂₃	2
C₃×C₄₁×D₆₅	2
D₄₁×D₆₅	3
D₁₃×D₁₂₃	5
D₅×D₁₂₃	13
D₃×D₆₅	41

Subgroups — D65×D123 #44544476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₅×D₁₂₃ #44544476