Subgroups — C₅²×#627 #14268276

Order 1800.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅²×Dic₃	C₂	1	2	yes	maximal
3	C₅²×Dic₆	—	3	2	no	maximal
4	C₂×C₃²×C₅²	C₂²	1	1	yes	characteristic
5	C₅×#627	C₅	1	6	yes	maximal, direct product factor
6	C₃×C₄×C₅²	—	3	2	no	—
7	C₅²×Dic₃	—	3	2	no	—
8	C₅²×Dic₃	—	6	1	no	—
9	C₅²×Dic₃	D₃	1	2	yes	—
10	C₃²×C₅²	Q₈	1	1	yes	characteristic, complemented, Hall
11	C₅²×Q₈	—	9	1	no	Hall
12	C₅×C₃⋊Dic₃	C₂×C₅	1	6	yes	—
13	C₃×C₅×Dic₃	C₂×C₅	1	12	yes	—
14	C₂×C₃×C₅²	—	2	1	no	—
15	C₂×C₃×C₅²	C₂×D₃	1	2	yes	—
16	C₅×Dic₆	—	3	12	no	—
17	C₄×C₅²	—	3	2	no	—
18	C₄×C₅²	—	9	1	no	—
19	C₂×C₃²×C₅	C₂²×C₅	1	6	yes	—
20	C₃×C₅²	—	2	1	no	—
21	C₃×C₅²	Dic₆	1	2	yes	complemented
22	#627	C₅²	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₄×C₅	—	3	12	no	—
24	C₅×Dic₃	—	3	12	no	—
25	C₅×Dic₃	—	6	6	no	—
26	C₅×Dic₃	C₅×D₃	1	12	yes	—
27	C₂×C₅²	D₃²	1	1	yes	characteristic
28	C₃²×C₅	C₅×Q₈	1	6	yes	complemented
29	C₅×Q₈	—	9	6	no	—
30	C₃⋊Dic₃	C₂×C₅²	1	1	yes	characteristic
31	C₃×Dic₃	C₂×C₅²	1	2	yes	—
32	C₂×C₃×C₅	—	2	6	no	—
33	C₂×C₃×C₅	C₂×C₅×D₃	1	12	yes	—
34	C₅²	#627	1	1	yes	characteristic, direct product factor, Sylow
35	Dic₆	—	3	2	no	—
36	C₄×C₅	—	3	12	no	—
37	C₄×C₅	—	9	6	no	—
38	C₂×C₃²	C₂²×C₅²	1	1	yes	characteristic
39	C₃×C₅	—	2	6	no	—
40	C₃×C₅	C₅×Dic₆	1	12	yes	complemented
41	C₃×C₄	—	3	2	no	—
42	Dic₃	—	3	2	no	—
43	Dic₃	—	6	1	no	—
44	Dic₃	C₅²×D₃	1	2	yes	—
45	C₂×C₅	C₅×D₃²	1	6	yes	—
46	C₃²	C₅²×Q₈	1	1	yes	characteristic, complemented, Sylow
47	Q₈	—	9	1	no	Sylow
48	C₂×C₃	—	2	1	no	—
49	C₂×C₃	C₂×C₅²×D₃	1	2	yes	—
50	C₅	C₅×#627	1	6	yes	direct product factor
51	C₄	—	3	2	no	—
52	C₄	—	9	1	no	—
53	C₃	—	2	1	no	—
54	C₃	C₅²×Dic₆	1	2	yes	complemented
55	C₂	C₅²×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅²	D₃²
commutator, cosocle	C₂×C₃²	C₂²×C₅²
Frattini	C₂	C₅²×D₃²
Fitting, socle, prime radical	C₂×C₃²×C₅²	C₂²
hypocenter	C₃²	C₅²×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅²

Complements

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

N	H ≅ G/N	type
5. C₅×#627	C₅	direct (N × H)
10. C₃²×C₅²	Q₈	semidirect
21. C₃×C₅²	Dic₆	semidirect
22. #627	C₅²	direct (N × H)
28. C₃²×C₅	C₅×Q₈	semidirect
34. C₅²	#627	direct (N × H)
40. C₃×C₅	C₅×Dic₆	semidirect
46. C₃²	C₅²×Q₈	semidirect
50. C₅	C₅×#627	direct (N × H)
54. C₃	C₅²×Dic₆	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₃⋊Dic₃	2
C₃×C₅²×Dic₃	2
C₅²×Dic₆	3
C₅×#627	5

Subgroups — C52×#627 #14268276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#627 #14268276