Subgroups — C₃×C₂₃×Dic₁₁₅ #44540705

Order 31740.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₂₃²	C₂	1	1	yes	maximal, characteristic
2	C₂₃×Dic₁₁₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₂₃²	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₂₃×Dic₂₃	—	5	1	no	maximal, Hall
5	C₂×C₅×C₂₃²	C₂×C₃	1	1	yes	characteristic
6	C₂×C₃×C₂₃²	D₅	1	1	yes	characteristic
7	C₅×C₂₃²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₂₃×Dic₂₃	—	5	1	no	Hall
9	C₃×C₂₃²	Dic₅	1	1	yes	characteristic, complemented, Hall
10	C₃×Dic₁₁₅	C₂₃	1	1	yes	maximal, characteristic, direct product factor
11	C₃×Dic₅×C₂₃	—	23	1	no	maximal
12	C₂×C₂₃²	C₃×D₅	1	1	yes	characteristic
13	C₂×C₃×C₅×C₂₃	—	2	11	no	—
14	C₂×C₃×C₅×C₂₃	D₂₃	1	1	yes	characteristic
15	C₂×C₃×C₅×C₂₃	C₂×C₂₃	1	1	yes	characteristic
16	C₂₃²	C₃×Dic₅	1	1	yes	characteristic, complemented, Sylow
17	Dic₁₁₅	C₃×C₂₃	1	1	yes	characteristic, direct product factor
18	Dic₅×C₂₃	—	23	1	no	—
19	C₃×C₅×C₂₃	—	2	11	no	—
20	C₃×C₅×C₂₃	Dic₂₃	1	1	yes	characteristic, complemented
21	C₃×C₅×C₂₃	C₄×C₂₃	1	1	yes	characteristic, complemented
22	C₃×C₄×C₂₃	—	115	1	no	—
23	C₃×Dic₂₃	—	5	1	no	—
24	C₂×C₅×C₂₃	—	2	11	no	—
25	C₂×C₅×C₂₃	C₃×D₂₃	1	1	yes	characteristic
26	C₂×C₅×C₂₃	C₂×C₃×C₂₃	1	1	yes	characteristic
27	C₂×C₃×C₂₃	—	2	11	no	—
28	C₂×C₃×C₂₃	D₅×C₂₃	1	1	yes	characteristic
29	C₂×C₃×C₂₃	D₁₁₅	1	1	yes	characteristic
30	C₅×C₂₃	—	2	11	no	—
31	C₅×C₂₃	C₃×Dic₂₃	1	1	yes	characteristic, complemented
32	C₅×C₂₃	C₃×C₄×C₂₃	1	1	yes	characteristic, complemented
33	C₄×C₂₃	—	115	1	no	—
34	Dic₂₃	—	5	1	no	—
35	C₃×C₂₃	—	2	11	no	—
36	C₃×C₂₃	Dic₅×C₂₃	1	1	yes	characteristic, complemented
37	C₃×C₂₃	Dic₁₁₅	1	1	yes	characteristic, direct product factor
38	C₃×Dic₅	—	23	1	no	Hall
39	C₂×C₂₃	—	2	11	no	—
40	C₂×C₂₃	C₃×D₅×C₂₃	1	1	yes	characteristic
41	C₂×C₂₃	C₃×D₁₁₅	1	1	yes	characteristic
42	C₂×C₃×C₅	C₂₃×D₂₃	1	1	yes	characteristic
43	C₂₃	—	2	11	no	—
44	C₂₃	C₃×Dic₅×C₂₃	1	1	yes	characteristic, complemented
45	C₂₃	C₃×Dic₁₁₅	1	1	yes	characteristic, direct product factor
46	Dic₅	—	23	1	no	Hall
47	C₃×C₅	C₂₃×Dic₂₃	1	1	yes	characteristic, complemented, Hall
48	C₃×C₄	—	115	1	no	Hall
49	C₂×C₅	C₃×C₂₃×D₂₃	1	1	yes	characteristic
50	C₂×C₃	C₂₃×D₁₁₅	1	1	yes	characteristic
51	C₅	C₃×C₂₃×Dic₂₃	1	1	yes	characteristic, complemented, Sylow
52	C₄	—	115	1	no	Sylow
53	C₃	C₂₃×Dic₁₁₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂	C₃×C₂₃×D₁₁₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₂₃	D₁₁₅
commutator, hypocenter	C₅×C₂₃	C₃×C₄×C₂₃
Frattini	C₂	C₃×C₂₃×D₁₁₅
Fitting, socle, prime radical	C₂×C₃×C₅×C₂₃²	C₂
cosocle	C₂×C₅×C₂₃	C₂×C₃×C₂₃
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
2. C₂₃×Dic₁₁₅	C₃	direct (N × H)
3. C₃×C₅×C₂₃²	C₄	semidirect
7. C₅×C₂₃²	C₃×C₄	semidirect
9. C₃×C₂₃²	Dic₅	semidirect
10. C₃×Dic₁₁₅	C₂₃	direct (N × H)
16. C₂₃²	C₃×Dic₅	semidirect
17. Dic₁₁₅	C₃×C₂₃	direct (N × H)
20. C₃×C₅×C₂₃	Dic₂₃	semidirect
21. C₃×C₅×C₂₃	C₄×C₂₃	semidirect
31. C₅×C₂₃	C₃×Dic₂₃	semidirect
32. C₅×C₂₃	C₃×C₄×C₂₃	semidirect
36. C₃×C₂₃	Dic₅×C₂₃	semidirect
37. C₃×C₂₃	Dic₁₁₅	direct (N × H)
44. C₂₃	C₃×Dic₅×C₂₃	semidirect
45. C₂₃	C₃×Dic₁₁₅	direct (N × H)
47. C₃×C₅	C₂₃×Dic₂₃	semidirect
51. C₅	C₃×C₂₃×Dic₂₃	semidirect
53. C₃	C₂₃×Dic₁₁₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₂₃²	2
C₂₃×Dic₁₁₅	3
C₃×C₂₃×Dic₂₃	5
C₃×Dic₁₁₅	23
C₃×Dic₅×C₂₃	23

Subgroups — C3×C23×Dic115 #44540705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂₃×Dic₁₁₅ #44540705