Subgroups — C₃×C₅²×C₃⋊#3866 #45686406

Order 33300.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×D₃₇	C₂	1	1	yes	maximal, characteristic
2	C₅²×C₃⋊#3866	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅²×#3866	—	3	1	no	maximal
4	C₃²×C₅²×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₅×C₃⋊#3866	C₅	1	6	yes	maximal, direct product factor
6	C₃×C₅²×D₃₇	—	2	1	no	—
7	C₃×C₅²×D₃₇	D₃	1	1	yes	characteristic
8	C₃×C₅²×D₃₇	C₂×C₃	1	1	yes	characteristic
9	C₅²×#3866	—	3	1	no	Hall
10	C₃²×C₅×D₃₇	C₂×C₅	1	6	yes	—
11	C₃×C₅²×C₃₇	—	2	1	no	—
12	C₃×C₅²×C₃₇	Dic₃	1	1	yes	characteristic, complemented
13	C₃×C₅²×C₃₇	C₃×C₄	1	1	yes	characteristic, complemented
14	C₅×C₃⋊#3866	C₃×C₅	1	6	yes	direct product factor
15	C₃×C₅×#3866	—	3	6	no	—
16	C₅²×D₃₇	C₃×D₃	1	1	yes	characteristic
17	C₃²×C₅×C₃₇	C₄×C₅	1	6	yes	complemented
18	C₃×C₃⋊#3866	C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₃×C₅×D₃₇	—	2	6	no	—
20	C₃×C₅×D₃₇	C₅×D₃	1	6	yes	—
21	C₃×C₅×D₃₇	C₂×C₃×C₅	1	6	yes	—
22	C₅²×C₃₇	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
23	C₃×C₅²×Dic₃	—	37	1	no	maximal, Hall
24	C₅×#3866	—	3	6	no	—
25	C₃²×D₃₇	C₂×C₅²	1	1	yes	characteristic
26	C₃×C₅×C₃₇	—	2	6	no	—
27	C₃×C₅×C₃₇	C₅×Dic₃	1	6	yes	complemented
28	C₃×C₅×C₃₇	C₃×C₄×C₅	1	6	yes	complemented
29	C₂×C₃²×C₅²	—	37	1	no	—
30	C₃⋊#3866	C₃×C₅²	1	1	yes	characteristic, direct product factor
31	C₃×#3866	—	3	1	no	—
32	C₅×D₃₇	C₃×C₅×D₃	1	6	yes	—
33	C₃²×C₃₇	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
34	C₃×C₄×C₅²	—	111	1	no	—
35	C₅²×Dic₃	—	37	1	no	—
36	C₃²×C₅²	#3866	1	1	yes	characteristic, complemented, Hall
37	C₃×D₃₇	—	2	1	no	—
38	C₃×D₃₇	C₅²×D₃	1	1	yes	characteristic
39	C₃×D₃₇	C₂×C₃×C₅²	1	1	yes	characteristic
40	C₅×C₃₇	C₃×C₅×Dic₃	1	6	yes	complemented
41	C₃×C₅×Dic₃	—	37	6	no	—
42	C₂×C₃×C₅²	—	37	1	no	—
43	C₂×C₃×C₅²	—	37	1	no	—
44	C₂×C₃×C₅²	—	74	1	no	—
45	#3866	—	3	1	no	Hall
46	C₃×C₃₇	—	2	1	no	—
47	C₃×C₃₇	C₅²×Dic₃	1	1	yes	characteristic, complemented
48	C₃×C₃₇	C₃×C₄×C₅²	1	1	yes	characteristic, complemented
49	C₄×C₅²	—	111	1	no	Hall
50	C₂×C₃²×C₅	—	37	6	no	—
51	C₃×C₅²	—	2	1	no	—
52	C₃×C₅²	C₃×#3866	1	1	yes	characteristic, complemented
53	C₃×C₅²	C₃⋊#3866	1	1	yes	characteristic, direct product factor
54	D₃₇	C₃×C₅²×D₃	1	1	yes	characteristic
55	C₃×C₄×C₅	—	111	6	no	—
56	C₅×Dic₃	—	37	6	no	—
57	C₂×C₅²	—	37	1	no	—
58	C₃²×C₅	C₅×#3866	1	6	yes	complemented
59	C₃₇	C₃×C₅²×Dic₃	1	1	yes	characteristic, complemented, Sylow
60	C₃×Dic₃	—	37	1	no	Hall
61	C₂×C₃×C₅	—	37	6	no	—
62	C₂×C₃×C₅	—	37	6	no	—
63	C₂×C₃×C₅	—	74	6	no	—
64	C₅²	C₃×C₃⋊#3866	1	1	yes	characteristic, direct product factor, Sylow
65	C₄×C₅	—	111	6	no	—
66	C₂×C₃²	—	37	1	no	—
67	C₃×C₅	—	2	6	no	—
68	C₃×C₅	C₃×C₅×#3866	1	6	yes	complemented
69	C₃×C₅	C₅×C₃⋊#3866	1	6	yes	direct product factor
70	C₃×C₄	—	111	1	no	—
71	Dic₃	—	37	1	no	—
72	C₂×C₅	—	37	6	no	—
73	C₃²	C₅²×#3866	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	—	37	1	no	—
75	C₂×C₃	—	37	1	no	—
76	C₂×C₃	—	74	1	no	—
77	C₅	C₃×C₅×C₃⋊#3866	1	6	yes	direct product factor
78	C₄	—	111	1	no	Sylow
79	C₃	—	2	1	no	—
80	C₃	C₃×C₅²×#3866	1	1	yes	characteristic, complemented
81	C₃	C₅²×C₃⋊#3866	1	1	yes	characteristic, direct product factor
82	C₂	—	37	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅²	C₃⋊#3866
commutator, hypocenter	C₃×C₃₇	C₃×C₄×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₄
prime radical	C₃²×C₅²×D₃₇	C₂
cosocle	C₃×D₃₇	C₂×C₃×C₅²
LCS	G ▷ C₃×C₃₇
UCS	C₁ ◁ C₃×C₅²

Complements

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

N	H ≅ G/N	type
2. C₅²×C₃⋊#3866	C₃	direct (N × H)
4. C₃²×C₅²×C₃₇	C₄	semidirect
5. C₃×C₅×C₃⋊#3866	C₅	direct (N × H)
12. C₃×C₅²×C₃₇	Dic₃	semidirect
13. C₃×C₅²×C₃₇	C₃×C₄	semidirect
14. C₅×C₃⋊#3866	C₃×C₅	direct (N × H)
17. C₃²×C₅×C₃₇	C₄×C₅	semidirect
18. C₃×C₃⋊#3866	C₅²	direct (N × H)
22. C₅²×C₃₇	C₃×Dic₃	semidirect
27. C₃×C₅×C₃₇	C₅×Dic₃	semidirect
28. C₃×C₅×C₃₇	C₃×C₄×C₅	semidirect
30. C₃⋊#3866	C₃×C₅²	direct (N × H)
33. C₃²×C₃₇	C₄×C₅²	semidirect
36. C₃²×C₅²	#3866	semidirect
40. C₅×C₃₇	C₃×C₅×Dic₃	semidirect
47. C₃×C₃₇	C₅²×Dic₃	semidirect
48. C₃×C₃₇	C₃×C₄×C₅²	semidirect
52. C₃×C₅²	C₃×#3866	semidirect
53. C₃×C₅²	C₃⋊#3866	direct (N × H)
58. C₃²×C₅	C₅×#3866	semidirect
59. C₃₇	C₃×C₅²×Dic₃	semidirect
64. C₅²	C₃×C₃⋊#3866	direct (N × H)
68. C₃×C₅	C₃×C₅×#3866	semidirect
69. C₃×C₅	C₅×C₃⋊#3866	direct (N × H)
73. C₃²	C₅²×#3866	semidirect
77. C₅	C₃×C₅×C₃⋊#3866	direct (N × H)
80. C₃	C₃×C₅²×#3866	semidirect
81. C₃	C₅²×C₃⋊#3866	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅²×D₃₇	2
C₅²×C₃⋊#3866	3
C₃×C₅²×#3866	3
C₃×C₅×C₃⋊#3866	5
C₃×C₅²×Dic₃	37

Subgroups — C3×C52×C3⋊#3866 #45686406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×C₃⋊#3866 #45686406