Subgroups — #72269605

Order 3240.

432 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃⋊D₃²	C₂	1	1	yes	maximal, characteristic
2	#13108537	C₂	1	2	yes	maximal, complemented
3	C₅×C₃⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic
4	C₃²×C₅×C₃⋊D₃	C₄	1	2	yes	complemented
5	#10620198	—	5	1	no	maximal, Hall
6	C₅×D₃×C₃⋊D₃	—	6	4	no	—
7	C₃⁴×C₅	C₂×C₄	1	1	yes	characteristic, complemented, Hall
8	C₂×#4379	—	9	2	no	maximal
9	C₃⋊D₃²	D₅	1	1	yes	characteristic
10	#66472	—	5	2	no	—
11	C₅×C₃⋊(C₃⋊D₃)	—	6	4	no	—
12	C₅×C₃⋊(C₃⋊D₃)	—	12	8	no	—
13	C₃×C₅×C₃⋊D₃	—	2	4	no	—
14	C₃²×C₅×D₃	—	6	4	no	—
15	C₂×C₅×C₃⋊D₃	—	9	2	no	—
16	C₅×D₃²	—	18	8	no	—
17	#4379	—	9	4	no	—
18	#4379	—	18	8	no	—
19	C₃⋊(C₃⋊(C₃⋊D₃))	C₂×D₅	1	1	yes	characteristic
20	C₃²×C₃⋊D₃	Dic₅	1	2	yes	complemented
21	C₃³×C₅	—	2	4	no	—
22	C₃³×C₅	—	4	8	no	—
23	D₃×C₃⋊D₃	—	6	4	no	—
24	C₂×C₃²×C₅	—	9	2	no	—
25	C₅×C₃⋊D₃	—	9	2	no	—
26	C₅×C₃⋊D₃	—	18	8	no	—
27	C₅×C₃⋊D₃	—	18	8	no	—
28	C₅×C₃⋊D₃	—	36	16	no	—
29	C₅×C₃⋊D₃	—	36	8	no	—
30	C₅×C₃⋊D₃	#157	1	2	yes	complemented
31	C₃×C₅×D₃	—	6	16	no	—
32	C₃⁴	C₂×Dic₅	1	1	yes	characteristic, complemented, Sylow
33	C₂×#157	—	45	2	no	—
34	C₂×C₅×D₃	—	54	4	no	—
35	C₃⋊(C₃⋊D₃)	—	6	4	no	—
36	C₃⋊(C₃⋊D₃)	—	12	8	no	—
37	C₃×C₃⋊D₃	—	2	4	no	—
38	C₃²×D₃	—	6	4	no	—
39	C₃²×C₅	—	2	8	no	—
40	C₃²×C₅	—	2	8	no	—
41	C₃²×C₅	—	4	16	no	—
42	C₃²×C₅	—	4	8	no	—
43	C₃²×C₅	C₂×#157	1	2	yes	complemented
44	C₂×Dic₅	—	81	1	no	Hall
45	C₂×C₃⋊D₃	—	9	2	no	—
46	D₃²	—	18	8	no	—
47	#157	—	45	4	no	—
48	#157	—	90	8	no	—
49	C₂×C₃×C₅	—	18	4	no	—
50	C₅×D₃	—	6	4	no	—
51	C₅×D₃	—	54	4	no	—
52	C₅×D₃	—	108	8	no	—
53	C₃³	—	2	4	no	—
54	C₃³	—	4	8	no	—
55	C₂²×C₅	—	81	1	no	—
56	Dic₅	—	81	2	no	—
57	C₂×C₃²	—	9	2	no	—
58	C₃⋊D₃	—	9	2	no	—
59	C₃⋊D₃	—	18	8	no	—
60	C₃⋊D₃	—	18	8	no	—
61	C₃⋊D₃	—	36	16	no	—
62	C₃⋊D₃	—	36	8	no	—
63	C₃⋊D₃	#4379	1	2	yes	complemented
64	C₃×D₃	—	6	16	no	—
65	C₃×C₅	—	2	4	no	—
66	C₃×C₅	—	4	8	no	—
67	C₂×D₃	—	54	4	no	—
68	C₂×C₅	—	9	2	no	—
69	C₂×C₅	—	81	1	no	—
70	C₃²	—	2	8	no	—
71	C₃²	—	2	8	no	—
72	C₃²	—	4	16	no	—
73	C₃²	—	4	8	no	—
74	C₃²	C₂×#4379	1	2	yes	complemented
75	C₂×C₄	—	405	1	no	Sylow
76	C₂×C₃	—	18	4	no	—
77	D₃	—	6	4	no	—
78	D₃	—	54	4	no	—
79	D₃	—	108	8	no	—
80	C₅	#10620198	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	81	1	no	—
82	C₄	—	405	2	no	—
83	C₃	—	2	4	no	—
84	C₃	—	4	8	no	—
85	C₂	—	9	2	no	—
86	C₂	—	81	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃⁴×C₅	C₂×C₄
prime radical	C₅×C₃⋊D₃²	C₂
cosocle	C₅×C₃⋊(C₃⋊(C₃⋊D₃))	C₂²
LCS	G ▷ C₃⁴×C₅
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. #13108537	C₂	semidirect
4. C₃²×C₅×C₃⋊D₃	C₄	semidirect
7. C₃⁴×C₅	C₂×C₄	semidirect
20. C₃²×C₃⋊D₃	Dic₅	semidirect
30. C₅×C₃⋊D₃	#157	semidirect
32. C₃⁴	C₂×Dic₅	semidirect
43. C₃²×C₅	C₂×#157	semidirect
63. C₃⋊D₃	#4379	semidirect
74. C₃²	C₂×#4379	semidirect
80. C₅	#10620198	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₃⋊D₃²	2
#13108537	2
#10620198	5
C₂×#4379	9