Subgroups — #44477648

Order 30420.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊((C₁₃⋊C₉)⋊D₁₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×#16810004	C₃	1	1	yes	maximal, characteristic
3	C₅×C₁₃×C₁₃⋊C₉	C₄	1	1	yes	characteristic, complemented, Hall
4	#20258775	—	5	1	no	maximal, Hall
5	C₃×C₅⋊(C₁₃⋊D₁₃)	C₂×C₃	1	1	yes	characteristic
6	#16810004	C₉	1	1	yes	characteristic, complemented, Hall
7	(C₁₃⋊C₉)⋊D₁₃	—	5	1	no	—
8	C₃×C₅×C₁₃²	C₃×C₄	1	1	yes	characteristic
9	C₉×#63122	—	13	1	no	maximal
10	#63121⋊C₉	—	13	1	no	maximal
11	C₃×#10621729	—	5	1	no	—
12	C₅⋊(C₁₃⋊D₁₃)	C₂×C₉	1	1	yes	characteristic
13	C₁₃×C₁₃⋊C₉	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
14	C₉×D₆₅	—	13	1	no	—
15	C₅⋊(D₁₃⋊C₉)	—	13	1	no	—
16	C₃×C₁₃⋊D₁₃	—	5	1	no	—
17	C₅×C₁₃²	C₄×C₉	1	1	yes	characteristic, complemented, Hall
18	C₃×#63121	—	13	1	no	—
19	C₃×#63122	—	39	1	no	—
20	#10621729	—	5	1	no	Hall
21	C₅×C₉×C₁₃	—	13	1	no	—
22	C₅×C₁₃⋊C₉	#261	1	1	yes	characteristic, complemented
23	C₃×C₁₃²	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
24	C₉×#261	—	65	1	no	—
25	#261⋊C₉	—	65	1	no	—
26	C₃×D₆₅	—	13	1	no	—
27	C₃×D₆₅	—	39	1	no	—
28	C₃×D₆₅	—	78	2	no	—
29	C₁₃⋊D₁₃	—	5	1	no	—
30	#63122	—	39	1	no	—
31	#63121	—	13	1	no	—
32	C₉×D₁₃	—	65	1	no	—
33	D₁₃⋊C₉	—	65	1	no	—
34	C₃×C₅×C₁₃	—	6	2	no	—
35	C₃×C₅×C₁₃	AGL(1,𝔽₁₃)	1	1	yes	characteristic
36	C₃×C₅×C₁₃	C₃×#261	1	1	yes	characteristic
37	C₉×AGL(1,𝔽₅)	—	169	1	no	Hall
38	C₁₃²	C₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
39	C₃×#261	—	65	1	no	—
40	C₃×#261	—	195	1	no	—
41	D₆₅	—	13	1	no	—
42	D₆₅	—	39	1	no	—
43	D₆₅	—	78	2	no	—
44	C₉×C₁₃	—	13	1	no	—
45	C₁₃⋊C₉	#63122	1	1	yes	characteristic, complemented
46	C₉×D₅	—	169	1	no	—
47	C₃×D₁₃	—	65	1	no	—
48	C₃×D₁₃	—	65	1	no	—
49	C₃×D₁₃	—	390	2	no	—
50	C₅×C₁₃	—	6	2	no	—
51	C₅×C₁₃	#261⋊C₉	1	1	yes	characteristic, complemented
52	C₅×C₁₃	C₉×#261	1	1	yes	characteristic, complemented
53	C₃×AGL(1,𝔽₅)	—	507	1	no	—
54	#261	—	65	1	no	—
55	#261	—	195	1	no	—
56	C₅×C₉	—	13	1	no	Hall
57	C₃×C₁₃	—	6	2	no	—
58	C₃×C₁₃	#63121⋊C₃	1	1	yes	characteristic
59	C₃×C₁₃	C₃×#63122	1	1	yes	characteristic
60	C₄×C₉	—	845	1	no	Hall
61	C₃×D₅	—	507	1	no	—
62	D₁₃	—	65	1	no	—
63	D₁₃	—	65	1	no	—
64	D₁₃	—	390	2	no	—
65	AGL(1,𝔽₅)	—	507	1	no	Hall
66	C₂×C₉	—	845	1	no	—
67	C₃×C₅	#14523151	1	1	yes	characteristic
68	C₁₃	—	6	2	no	—
69	C₁₃	#63121⋊C₉	1	1	yes	characteristic, complemented
70	C₁₃	C₉×#63122	1	1	yes	characteristic, complemented
71	C₃×C₄	—	2535	1	no	—
72	D₅	—	507	1	no	—
73	C₉	—	13	1	no	Sylow
74	C₂×C₃	—	845	1	no	—
75	C₅	#20258775	1	1	yes	characteristic, complemented, Sylow
76	C₄	—	2535	1	no	Sylow
77	C₃	#23778466	1	1	yes	characteristic
78	C₂	—	845	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#23778466
commutator, hypocenter	C₅×C₁₃²	C₄×C₉
Fitting, socle	C₃×C₅×C₁₃²	C₃×C₄
prime radical, cosocle	C₃×C₅⋊(C₁₃⋊D₁₃)	C₂×C₃
LCS	G ▷ C₅×C₁₃²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
3. C₅×C₁₃×C₁₃⋊C₉	C₄	semidirect
6. #16810004	C₉	semidirect
13. C₁₃×C₁₃⋊C₉	AGL(1,𝔽₅)	semidirect
17. C₅×C₁₃²	C₄×C₉	semidirect
22. C₅×C₁₃⋊C₉	#261	semidirect
38. C₁₃²	C₉×AGL(1,𝔽₅)	semidirect
45. C₁₃⋊C₉	#63122	semidirect
51. C₅×C₁₃	#261⋊C₉	semidirect
52. C₅×C₁₃	C₉×#261	semidirect
69. C₁₃	#63121⋊C₉	semidirect
70. C₁₃	C₉×#63122	semidirect
75. C₅	#20258775	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊((C₁₃⋊C₉)⋊D₁₃)	2
C₃×#16810004	3
#20258775	5
C₉×#63122	13
#63121⋊C₉	13