Subgroups — C₂×C₃×C₂₃×D₁₁₅ #44540770

Order 31740.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₃×D₁₁₅	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅×C₂₃²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×D₁₁₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₅×C₂₃²	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₂₃×D₂₃	—	5	1	no	maximal, Hall
6	C₂₃×D₁₁₅	C₂×C₃	1	2	yes	direct product factor
7	C₂×C₅×C₂₃²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₂₃×D₂₃	—	5	2	no	—
9	C₂×C₃×C₂₃²	D₅	1	1	yes	characteristic, complemented
10	C₅×C₂₃²	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₂₃×D₂₃	—	5	1	no	Hall
12	C₃×C₂₃²	C₂×D₅	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×D₁₁₅	C₂₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃×D₅×C₂₃	—	23	1	no	maximal
15	C₂×C₂₃²	C₃×D₅	1	1	yes	characteristic, complemented
16	C₂₃×D₂₃	—	5	2	no	—
17	C₂×C₃×C₅×C₂₃	—	2	11	no	—
18	C₂×C₃×C₅×C₂₃	D₂₃	1	1	yes	characteristic, complemented
19	C₂×C₃×C₅×C₂₃	C₂×C₂₃	1	1	yes	characteristic, complemented
20	C₃×D₁₁₅	C₂×C₂₃	1	2	yes	direct product factor
21	C₃×D₅×C₂₃	—	23	2	no	—
22	C₂₃²	C₂×C₃×D₅	1	1	yes	characteristic, complemented, Sylow
23	C₂×D₁₁₅	C₃×C₂₃	1	1	yes	characteristic, direct product factor
24	C₂×D₅×C₂₃	—	23	1	no	—
25	C₃×C₅×C₂₃	—	2	11	no	—
26	C₃×C₅×C₂₃	C₂×D₂₃	1	1	yes	characteristic, complemented
27	C₃×C₅×C₂₃	C₂²×C₂₃	1	1	yes	characteristic, complemented
28	C₂²×C₃×C₂₃	—	115	1	no	—
29	C₂×C₃×D₂₃	—	5	1	no	—
30	C₂×C₅×C₂₃	—	2	11	no	—
31	C₂×C₅×C₂₃	C₃×D₂₃	1	1	yes	characteristic, complemented
32	C₂×C₅×C₂₃	C₂×C₃×C₂₃	1	1	yes	characteristic, complemented
33	D₁₁₅	C₂×C₃×C₂₃	1	2	yes	direct product factor
34	D₅×C₂₃	—	23	2	no	—
35	C₂×C₃×C₂₃	—	2	11	no	—
36	C₂×C₃×C₂₃	—	115	2	no	—
37	C₂×C₃×C₂₃	D₅×C₂₃	1	1	yes	characteristic, complemented
38	C₂×C₃×C₂₃	D₁₁₅	1	1	yes	characteristic, direct product factor
39	C₃×D₂₃	—	5	2	no	—
40	C₅×C₂₃	—	2	11	no	—
41	C₅×C₂₃	C₂×C₃×D₂₃	1	1	yes	characteristic, complemented
42	C₅×C₂₃	C₂²×C₃×C₂₃	1	1	yes	characteristic, complemented
43	C₂²×C₂₃	—	115	1	no	—
44	C₂×D₂₃	—	5	1	no	—
45	C₃×C₂₃	—	2	11	no	—
46	C₃×C₂₃	C₂×D₅×C₂₃	1	1	yes	characteristic, complemented
47	C₃×C₂₃	C₂×D₁₁₅	1	1	yes	characteristic, direct product factor
48	C₂×C₃×D₅	—	23	1	no	Hall
49	C₂×C₂₃	—	2	11	no	—
50	C₂×C₂₃	—	115	2	no	—
51	C₂×C₂₃	C₃×D₅×C₂₃	1	1	yes	characteristic, complemented
52	C₂×C₂₃	C₃×D₁₁₅	1	1	yes	characteristic, direct product factor
53	D₂₃	—	5	2	no	—
54	C₂×C₃×C₅	C₂₃×D₂₃	1	1	yes	characteristic, complemented
55	C₃×D₅	—	23	2	no	—
56	C₂₃	—	2	11	no	—
57	C₂₃	C₂×C₃×D₅×C₂₃	1	1	yes	characteristic, complemented
58	C₂₃	C₂×C₃×D₁₁₅	1	1	yes	characteristic, direct product factor
59	C₂×D₅	—	23	1	no	Hall
60	C₃×C₅	C₂×C₂₃×D₂₃	1	1	yes	characteristic, complemented, Hall
61	C₂²×C₃	—	115	1	no	Hall
62	C₂×C₅	C₃×C₂₃×D₂₃	1	1	yes	characteristic, complemented
63	D₅	—	23	2	no	—
64	C₂×C₃	—	115	2	no	—
65	C₂×C₃	C₂₃×D₁₁₅	1	1	yes	characteristic, direct product factor
66	C₅	C₂×C₃×C₂₃×D₂₃	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	115	1	no	Sylow
68	C₃	C₂×C₂₃×D₁₁₅	1	1	yes	characteristic, direct product factor, Sylow
69	C₂	—	115	2	no	—
70	C₂	C₃×C₂₃×D₁₁₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₂₃	D₁₁₅
commutator, cosocle, hypocenter	C₅×C₂₃	C₂²×C₃×C₂₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₂₃²	C₂
LCS	G ▷ C₅×C₂₃
UCS	C₁ ◁ C₂×C₃×C₂₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₂₃×D₁₁₅	C₂	direct (N × H)
2. C₂×C₃×C₅×C₂₃²	C₂	semidirect
3. C₂×C₂₃×D₁₁₅	C₃	direct (N × H)
4. C₃×C₅×C₂₃²	C₂²	semidirect
6. C₂₃×D₁₁₅	C₂×C₃	direct (N × H)
7. C₂×C₅×C₂₃²	C₂×C₃	semidirect
9. C₂×C₃×C₂₃²	D₅	semidirect
10. C₅×C₂₃²	C₂²×C₃	semidirect
12. C₃×C₂₃²	C₂×D₅	semidirect
13. C₂×C₃×D₁₁₅	C₂₃	direct (N × H)
15. C₂×C₂₃²	C₃×D₅	semidirect
18. C₂×C₃×C₅×C₂₃	D₂₃	semidirect
19. C₂×C₃×C₅×C₂₃	C₂×C₂₃	semidirect
20. C₃×D₁₁₅	C₂×C₂₃	direct (N × H)
22. C₂₃²	C₂×C₃×D₅	semidirect
23. C₂×D₁₁₅	C₃×C₂₃	direct (N × H)
26. C₃×C₅×C₂₃	C₂×D₂₃	semidirect
27. C₃×C₅×C₂₃	C₂²×C₂₃	semidirect
31. C₂×C₅×C₂₃	C₃×D₂₃	semidirect
32. C₂×C₅×C₂₃	C₂×C₃×C₂₃	semidirect
33. D₁₁₅	C₂×C₃×C₂₃	direct (N × H)
37. C₂×C₃×C₂₃	D₅×C₂₃	semidirect
38. C₂×C₃×C₂₃	D₁₁₅	direct (N × H)
41. C₅×C₂₃	C₂×C₃×D₂₃	semidirect
42. C₅×C₂₃	C₂²×C₃×C₂₃	semidirect
46. C₃×C₂₃	C₂×D₅×C₂₃	semidirect
47. C₃×C₂₃	C₂×D₁₁₅	direct (N × H)
51. C₂×C₂₃	C₃×D₅×C₂₃	semidirect
52. C₂×C₂₃	C₃×D₁₁₅	direct (N × H)
54. C₂×C₃×C₅	C₂₃×D₂₃	semidirect
57. C₂₃	C₂×C₃×D₅×C₂₃	semidirect
58. C₂₃	C₂×C₃×D₁₁₅	direct (N × H)
60. C₃×C₅	C₂×C₂₃×D₂₃	semidirect
62. C₂×C₅	C₃×C₂₃×D₂₃	semidirect
65. C₂×C₃	C₂₃×D₁₁₅	direct (N × H)
66. C₅	C₂×C₃×C₂₃×D₂₃	semidirect
68. C₃	C₂×C₂₃×D₁₁₅	direct (N × H)
70. C₂	C₃×C₂₃×D₁₁₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₂₃×D₁₁₅	2
C₂×C₃×C₅×C₂₃²	2
C₂×C₂₃×D₁₁₅	3
C₂×C₃×C₂₃×D₂₃	5
C₂×C₃×D₁₁₅	23
C₂×C₃×D₅×C₂₃	23

Subgroups — C2×C3×C23×D115 #44540770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₂₃×D₁₁₅ #44540770