Subgroups — #13108557

Order 1620.

712 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₃⋊D₁₅	C₂	1	2	yes	maximal, complemented
3	D₁₅⋊(C₃⋊D₃)	—	3	8	no	maximal
4	C₃⁴×C₅	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃⋊D₃²	—	5	1	no	maximal, Hall
6	C₅×C₃⋊(C₃⋊D₃)	—	3	8	no	—
7	C₅×C₃⋊(C₃⋊D₃)	—	6	16	no	—
8	C₃×C₃⋊D₁₅	D₃	1	8	yes	complemented
9	C₃²×D₁₅	—	3	8	no	—
10	(C₃⋊D₃)⋊D₅	—	9	16	no	—
11	D₅×C₃⋊D₃	—	9	2	no	—
12	C₃⋊(C₃⋊(C₃⋊D₃))	D₅	1	1	yes	characteristic, complemented
13	C₃²×C₃⋊D₃	—	5	2	no	—
14	C₃³×C₅	—	2	16	no	—
15	C₃³×C₅	C₂×D₃	1	8	yes	complemented
16	D₃×C₃⋊D₃	—	15	8	no	—
17	C₃⋊D₁₅	C₃⋊D₃	1	2	yes	complemented
18	C₅×C₃⋊D₃	—	9	2	no	—
19	C₅×C₃⋊D₃	—	9	16	no	—
20	C₅×C₃⋊D₃	—	18	32	no	—
21	C₅×C₃⋊D₃	—	18	24	no	—
22	C₃×D₁₅	—	3	32	no	—
23	C₃²×D₅	—	9	2	no	—
24	C₃⁴	C₂×D₅	1	1	yes	characteristic, complemented, Sylow
25	D₃×D₅	—	27	8	no	—
26	C₃⋊(C₃⋊D₃)	—	3	8	no	—
27	C₃⋊(C₃⋊D₃)	—	6	16	no	—
28	C₃×C₃⋊D₃	—	5	8	no	—
29	C₃²×D₃	—	15	8	no	—
30	C₃²×C₅	—	2	24	no	—
31	C₃²×C₅	—	2	32	no	—
32	C₃²×C₅	D₃²	1	16	yes	complemented
33	C₃²×C₅	C₂×C₃⋊D₃	1	2	yes	complemented
34	C₂×C₃⋊D₃	—	45	2	no	—
35	D₃²	—	45	16	no	—
36	D₁₅	—	3	8	no	—
37	C₃×D₅	—	9	8	no	—
38	C₅×D₃	—	27	8	no	—
39	C₅×D₃	—	54	16	no	—
40	C₃³	—	2	16	no	—
41	C₃³	D₃×D₅	1	8	yes	complemented
42	C₂×D₅	—	81	1	no	Hall
43	C₂×C₃²	—	45	2	no	—
44	C₃⋊D₃	—	5	2	no	—
45	C₃⋊D₃	—	9	2	no	—
46	C₃⋊D₃	—	9	16	no	—
47	C₃⋊D₃	—	18	32	no	—
48	C₃⋊D₃	—	18	24	no	—
49	C₃×D₃	—	15	32	no	—
50	C₃×C₅	—	2	16	no	—
51	C₃×C₅	D₃×C₃⋊D₃	1	8	yes	complemented
52	C₂×D₃	—	135	8	no	—
53	C₂×C₅	—	81	1	no	—
54	D₅	—	9	2	no	—
55	C₃²	—	2	24	no	—
56	C₃²	—	2	32	no	—
57	C₃²	D₅×C₃⋊D₃	1	2	yes	complemented
58	C₃²	(C₃⋊D₃)⋊D₅	1	16	yes	complemented
59	C₂×C₃	—	45	8	no	—
60	D₃	—	15	8	no	—
61	D₃	—	27	8	no	—
62	D₃	—	54	16	no	—
63	C₅	C₃⋊D₃²	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	405	1	no	Sylow
65	C₃	—	2	16	no	—
66	C₃	D₁₅⋊(C₃⋊D₃)	1	8	yes	complemented
67	C₂	—	45	2	no	—
68	C₂	—	81	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃⁴×C₅	C₂²
LCS	G ▷ C₃⁴×C₅
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅×C₃⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
2. C₃²×C₃⋊D₁₅	C₂	semidirect
4. C₃⁴×C₅	C₂²	semidirect
8. C₃×C₃⋊D₁₅	D₃	semidirect
12. C₃⋊(C₃⋊(C₃⋊D₃))	D₅	semidirect
15. C₃³×C₅	C₂×D₃	semidirect
17. C₃⋊D₁₅	C₃⋊D₃	semidirect
24. C₃⁴	C₂×D₅	semidirect
32. C₃²×C₅	D₃²	semidirect
33. C₃²×C₅	C₂×C₃⋊D₃	semidirect
41. C₃³	D₃×D₅	semidirect
51. C₃×C₅	D₃×C₃⋊D₃	semidirect
57. C₃²	D₅×C₃⋊D₃	semidirect
58. C₃²	(C₃⋊D₃)⋊D₅	semidirect
63. C₅	C₃⋊D₃²	semidirect
66. C₃	D₁₅⋊(C₃⋊D₃)	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₃⋊(C₃⋊(C₃⋊D₃))	2
C₃²×C₃⋊D₁₅	2
D₁₅⋊(C₃⋊D₃)	3
C₃⋊D₃²	5