Subgroups — C₅²×#13093287 #49147059

Order 36900.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₃⋊D₁₂₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₄₁	C₄	1	1	yes	characteristic, complemented, Hall
3	C₅×#13093287	C₅	1	6	yes	maximal, direct product factor
4	C₅²×D₁₂₃	—	6	2	no	—
5	C₅²×#4223	—	9	1	no	maximal, Hall
6	C₅×C₃⋊D₁₂₃	C₂×C₅	1	6	yes	—
7	C₃×C₅²×C₄₁	—	2	2	no	—
8	C₅²×D₄₁	—	9	1	no	—
9	C₃²×C₅×C₄₁	C₄×C₅	1	6	yes	complemented
10	#13093287	C₅²	1	1	yes	characteristic, direct product factor, Hall
11	C₅×D₁₂₃	—	6	12	no	—
12	C₅²×C₄₁	#157	1	1	yes	characteristic, complemented, Hall
13	C₅²×#157	—	41	1	no	maximal, Hall
14	C₅×#4223	—	9	6	no	—
15	C₃⋊D₁₂₃	C₂×C₅²	1	1	yes	characteristic
16	C₃×C₅×C₄₁	—	2	12	no	—
17	C₅²×C₃⋊D₃	—	41	1	no	—
18	C₅×D₄₁	—	9	6	no	—
19	C₃²×C₄₁	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
20	D₁₂₃	—	6	2	no	—
21	C₃²×C₅²	#4223	1	1	yes	characteristic, complemented, Hall
22	C₅×C₄₁	C₅×#157	1	6	yes	complemented
23	C₅×#157	—	41	6	no	—
24	#4223	—	9	1	no	Hall
25	C₅²×D₃	—	246	2	no	—
26	C₃×C₄₁	—	2	2	no	—
27	C₄×C₅²	—	369	1	no	Hall
28	C₅×C₃⋊D₃	—	41	6	no	—
29	D₄₁	—	9	1	no	—
30	C₃×C₅²	—	2	2	no	—
31	C₂×C₅²	—	369	1	no	—
32	C₃²×C₅	C₅×#4223	1	6	yes	complemented
33	C₄₁	C₅²×#157	1	1	yes	characteristic, complemented, Sylow
34	#157	—	41	1	no	Hall
35	C₅×D₃	—	246	12	no	—
36	C₅²	#13093287	1	1	yes	characteristic, direct product factor, Sylow
37	C₄×C₅	—	369	6	no	—
38	C₃⋊D₃	—	41	1	no	—
39	C₃×C₅	—	2	12	no	—
40	C₂×C₅	—	369	6	no	—
41	C₃²	C₅²×#4223	1	1	yes	characteristic, complemented, Sylow
42	D₃	—	246	2	no	—
43	C₅	C₅×#13093287	1	6	yes	direct product factor
44	C₄	—	369	1	no	Sylow
45	C₃	—	2	2	no	—
46	C₂	—	369	1	no	—

	subgroup	G/H
center, hypercenter	C₅²	#13093287
commutator, hypocenter	C₃²×C₄₁	C₄×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₄₁	C₄
prime radical	C₅²×C₃⋊D₁₂₃	C₂
cosocle	C₃⋊D₁₂₃	C₂×C₅²
LCS	G ▷ C₃²×C₄₁
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₄₁	C₄	semidirect
3. C₅×#13093287	C₅	direct (N × H)
9. C₃²×C₅×C₄₁	C₄×C₅	semidirect
10. #13093287	C₅²	direct (N × H)
12. C₅²×C₄₁	#157	semidirect
19. C₃²×C₄₁	C₄×C₅²	semidirect
21. C₃²×C₅²	#4223	semidirect
22. C₅×C₄₁	C₅×#157	semidirect
32. C₃²×C₅	C₅×#4223	semidirect
33. C₄₁	C₅²×#157	semidirect
36. C₅²	#13093287	direct (N × H)
41. C₃²	C₅²×#4223	semidirect
43. C₅	C₅×#13093287	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×C₃⋊D₁₂₃	2
C₅×#13093287	5
C₅²×#4223	9
C₅²×#157	41

Subgroups — C52×#13093287 #49147059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#13093287 #49147059