Subgroups — #45686381

Order 33300.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊(C₅⋊D₁₈₅))	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
3	#20269166	—	5	1	no	maximal
4	#20269165	—	5	1	no	maximal
5	C₃⋊(C₅⋊D₁₈₅)	—	6	2	no	—
6	#16814881	—	9	1	no	maximal, Hall
7	C₃⋊D₅₅₅	—	5	1	no	—
8	C₃⋊D₅₅₅	—	5	1	no	—
9	C₃⋊D₅₅₅	—	10	2	no	—
10	C₃×C₅²×C₃₇	—	2	2	no	—
11	C₅⋊D₁₈₅	—	9	1	no	—
12	C₃²×C₅×C₃₇	—	2	2	no	—
13	C₃²×C₅×C₃₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
14	C₃²×C₅×C₃₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
15	#13050652	—	25	1	no	Hall
16	D₅₅₅	—	30	2	no	—
17	D₅₅₅	—	30	2	no	—
18	D₅₅₅	—	30	8	no	—
19	C₅²×C₃₇	#157	1	1	yes	characteristic, complemented, Hall
20	#11744708	—	37	1	no	maximal, Hall
21	#10625192	—	45	1	no	—
22	#10625191	—	45	1	no	—
23	C₃⋊D₁₁₁	—	25	1	no	—
24	C₃×C₅×C₃₇	—	2	8	no	—
25	C₃×C₅×C₃₇	—	2	2	no	—
26	C₃×C₅×C₃₇	—	2	2	no	—
27	C₃⋊(C₃⋊(C₅⋊D₅))	—	37	1	no	—
28	D₁₈₅	—	45	1	no	—
29	D₁₈₅	—	45	1	no	—
30	D₁₈₅	—	90	2	no	—
31	C₃²×C₃₇	#1044	1	1	yes	characteristic, complemented, Hall
32	C₃²×C₅²	#3866	1	1	yes	characteristic, complemented, Hall
33	D₁₁₁	—	150	2	no	—
34	C₅×C₃₇	—	2	2	no	—
35	C₅×C₃₇	#4380	1	1	yes	characteristic, complemented
36	C₅×C₃₇	#4380	1	1	yes	characteristic, complemented
37	#4380	—	185	1	no	—
38	#4380	—	185	1	no	—
39	C₃⋊(C₅⋊D₅)	—	222	2	no	—
40	#3866	—	225	1	no	Hall
41	C₃×C₃₇	—	2	2	no	—
42	#1044	—	333	1	no	Hall
43	C₃⋊D₁₅	—	370	2	no	—
44	C₃⋊D₁₅	—	185	1	no	—
45	C₃⋊D₁₅	—	185	1	no	—
46	C₃×C₅²	—	2	2	no	—
47	D₃₇	—	225	1	no	—
48	C₅⋊D₅	—	333	1	no	—
49	C₃²×C₅	—	2	2	no	—
50	C₃²×C₅	#10625191	1	1	yes	characteristic, complemented
51	C₃²×C₅	#10625192	1	1	yes	characteristic, complemented
52	C₃₇	#11744708	1	1	yes	characteristic, complemented, Sylow
53	#157	—	925	1	no	Hall
54	D₁₅	—	1110	8	no	—
55	D₁₅	—	1110	2	no	—
56	D₁₅	—	1110	2	no	—
57	C₅²	#13050652	1	1	yes	characteristic, complemented, Sylow
58	AGL(1,𝔽₅)	—	1665	1	no	—
59	AGL(1,𝔽₅)	—	1665	1	no	—
60	C₃⋊D₃	—	925	1	no	—
61	C₃×C₅	—	2	8	no	—
62	C₃×C₅	—	2	2	no	—
63	C₃×C₅	—	2	2	no	—
64	D₅	—	3330	2	no	—
65	D₅	—	1665	1	no	—
66	D₅	—	1665	1	no	—
67	C₃²	#16814881	1	1	yes	characteristic, complemented, Sylow
68	D₃	—	5550	2	no	—
69	C₅	—	2	2	no	—
70	C₅	#20269165	1	1	yes	characteristic, complemented
71	C₅	#20269166	1	1	yes	characteristic, complemented
72	C₄	—	8325	1	no	Sylow
73	C₃	—	2	2	no	—
74	C₂	—	8325	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅²×C₃₇	C₄
prime radical, cosocle	C₃⋊(C₃⋊(C₅⋊D₁₈₅))	C₂
LCS	G ▷ C₃²×C₅²×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₃₇	C₄	semidirect
13. C₃²×C₅×C₃₇	AGL(1,𝔽₅)	semidirect
14. C₃²×C₅×C₃₇	AGL(1,𝔽₅)	semidirect
19. C₅²×C₃₇	#157	semidirect
31. C₃²×C₃₇	#1044	semidirect
32. C₃²×C₅²	#3866	semidirect
35. C₅×C₃₇	#4380	semidirect
36. C₅×C₃₇	#4380	semidirect
50. C₃²×C₅	#10625191	semidirect
51. C₃²×C₅	#10625192	semidirect
52. C₃₇	#11744708	semidirect
57. C₅²	#13050652	semidirect
67. C₃²	#16814881	semidirect
70. C₅	#20269165	semidirect
71. C₅	#20269166	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(C₃⋊(C₅⋊D₁₈₅))	2
#20269166	5
#20269165	5
#16814881	9
#11744708	37