Subgroups — #44478201

Order 30420.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊(C₅⋊(C₁₃⋊D₁₃)))	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×C₁₃²	C₄	1	1	yes	characteristic, complemented, Hall
3	#20258974	—	5	1	no	maximal, Hall
4	C₃⋊(C₅⋊(C₁₃⋊D₁₃))	—	6	2	no	—
5	#16810004	—	9	1	no	maximal, Hall
6	C₃⋊(C₃⋊(C₁₃⋊D₁₃))	—	5	1	no	—
7	C₃×C₅×C₁₃²	—	2	2	no	—
8	#14556843	—	13	1	no	maximal
9	#14556842	—	13	1	no	maximal
10	C₅⋊(C₁₃⋊D₁₃)	—	9	1	no	—
11	C₃²×C₁₃²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
12	C₃⋊D₁₉₅	—	13	1	no	—
13	C₃⋊D₁₉₅	—	13	1	no	—
14	C₃⋊D₁₉₅	—	26	6	no	—
15	C₃⋊(C₁₃⋊D₁₃)	—	30	2	no	—
16	C₅×C₁₃²	#157	1	1	yes	characteristic, complemented, Hall
17	#10621729	—	45	1	no	Hall
18	C₃²×C₅×C₁₃	—	2	6	no	—
19	C₃²×C₅×C₁₃	#261	1	1	yes	characteristic, complemented
20	C₃²×C₅×C₁₃	#261	1	1	yes	characteristic, complemented
21	C₃×C₁₃²	—	2	2	no	—
22	#90866	—	65	1	no	—
23	#90866	—	65	1	no	—
24	D₁₉₅	—	78	2	no	—
25	D₁₉₅	—	78	2	no	—
26	D₁₉₅	—	78	24	no	—
27	C₁₃⋊D₁₃	—	45	1	no	—
28	#63122	—	117	1	no	—
29	#63121	—	117	1	no	—
30	C₃⋊D₃₉	—	65	1	no	—
31	C₃⋊D₃₉	—	65	1	no	—
32	C₃⋊D₃₉	—	130	6	no	—
33	C₃×C₅×C₁₃	—	2	24	no	—
34	C₃×C₅×C₁₃	—	2	2	no	—
35	C₃×C₅×C₁₃	—	2	2	no	—
36	#4380	—	169	1	no	Hall
37	C₁₃²	#4380	1	1	yes	characteristic, complemented, Sylow
38	D₆₅	—	117	1	no	—
39	D₆₅	—	117	1	no	—
40	D₆₅	—	234	6	no	—
41	C₃²×C₁₃	—	2	6	no	—
42	C₃²×C₁₃	#63121	1	1	yes	characteristic, complemented
43	C₃²×C₁₃	#63122	1	1	yes	characteristic, complemented
44	C₃⋊D₁₅	—	169	1	no	—
45	D₃₉	—	390	2	no	—
46	D₃₉	—	390	2	no	—
47	D₃₉	—	390	24	no	—
48	C₅×C₁₃	—	2	6	no	—
49	C₅×C₁₃	#90866	1	1	yes	characteristic, complemented
50	C₅×C₁₃	#90866	1	1	yes	characteristic, complemented
51	#261	—	585	1	no	—
52	#261	—	585	1	no	—
53	C₃²×C₅	#10621729	1	1	yes	characteristic, complemented, Hall
54	C₃×C₁₃	—	2	24	no	—
55	C₃×C₁₃	—	2	2	no	—
56	C₃×C₁₃	—	2	2	no	—
57	#157	—	845	1	no	Hall
58	D₁₅	—	1014	2	no	—
59	D₁₃	—	585	1	no	—
60	D₁₃	—	585	1	no	—
61	D₁₃	—	1170	6	no	—
62	AGL(1,𝔽₅)	—	1521	1	no	Hall
63	C₃⋊D₃	—	845	1	no	—
64	C₃×C₅	—	2	2	no	—
65	C₁₃	—	2	6	no	—
66	C₁₃	#14556842	1	1	yes	characteristic, complemented
67	C₁₃	#14556843	1	1	yes	characteristic, complemented
68	D₅	—	1521	1	no	—
69	C₃²	#16810004	1	1	yes	characteristic, complemented, Sylow
70	D₃	—	5070	2	no	—
71	C₅	#20258974	1	1	yes	characteristic, complemented, Sylow
72	C₄	—	7605	1	no	Sylow
73	C₃	—	2	2	no	—
74	C₂	—	7605	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅×C₁₃²	C₄
prime radical, cosocle	C₃⋊(C₃⋊(C₅⋊(C₁₃⋊D₁₃)))	C₂
LCS	G ▷ C₃²×C₅×C₁₃²
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅×C₁₃²	C₄	semidirect
11. C₃²×C₁₃²	AGL(1,𝔽₅)	semidirect
16. C₅×C₁₃²	#157	semidirect
19. C₃²×C₅×C₁₃	#261	semidirect
20. C₃²×C₅×C₁₃	#261	semidirect
37. C₁₃²	#4380	semidirect
42. C₃²×C₁₃	#63121	semidirect
43. C₃²×C₁₃	#63122	semidirect
49. C₅×C₁₃	#90866	semidirect
50. C₅×C₁₃	#90866	semidirect
53. C₃²×C₅	#10621729	semidirect
66. C₁₃	#14556842	semidirect
67. C₁₃	#14556843	semidirect
69. C₃²	#16810004	semidirect
71. C₅	#20258974	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⋊(C₁₃⋊D₁₃)))	2
#20258974	5
#16810004	9
#14556843	13
#14556842	13