Subgroups — #28476358

Order 15104.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422232	C₂	1	1	yes	maximal, characteristic
2	#21422164	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1368	C₂	1	1	yes	maximal, characteristic
4	#16837236	C₄	1	1	yes	characteristic
5	#16837236	C₄	1	1	yes	characteristic
6	#16837223	—	2	1	no	—
7	C₅₉×#346	C₂²	1	1	yes	characteristic
8	C₅₉×#334	—	2	1	no	—
9	#14273667	—	4	1	no	—
10	C₅₉⋊C₃₂	—	2	1	no	—
11	C₅₉⋊C₃₂	C₈	1	2	yes	complemented
12	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₉	Q₈	1	1	yes	characteristic
14	C₄⋊C₈×C₅₉	—	4	1	no	—
15	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
16	C₂×C₈×C₅₉	—	8	1	no	—
17	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
19	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
20	C₁₆×C₅₉	—	2	1	no	—
21	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
22	C₁₆×C₅₉	M₁₆	1	1	yes	characteristic
23	C₅₉⋊C₁₆	—	8	1	no	—
24	C₂×C₄×C₅₉	—	2	1	no	—
25	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
26	C₈×C₅₉	—	2	1	no	—
27	C₈×C₅₉	—	16	1	no	—
28	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
29	C₈×C₅₉	#108	1	1	yes	characteristic
30	#7457	—	59	1	no	maximal, Sylow
31	C₂²×C₅₉	#365	1	1	yes	characteristic
32	C₄×C₅₉	—	4	1	no	—
33	C₄×C₅₉	#334	1	1	yes	characteristic
34	C₄×C₅₉	#365	1	1	yes	characteristic
35	#1398	—	59	1	no	—
36	#1369	—	59	1	no	—
37	#1368	D₅₉	1	1	yes	characteristic
38	C₂×C₅₉	—	2	1	no	—
39	C₂×C₅₉	#1368	1	1	yes	characteristic
40	M₆₄	—	59	1	no	—
41	M₆₄	—	59	1	no	—
42	#364	—	118	1	no	—
43	#346	C₂×D₅₉	1	1	yes	characteristic
44	#334	—	2	1	no	—
45	C₅₉	#7457	1	1	yes	characteristic, complemented, Sylow
46	M₃₂	—	236	1	no	—
47	C₂×C₁₆	Dic₁₁₈	1	1	yes	characteristic
48	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
49	C₄⋊C₈	—	4	1	no	—
50	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
51	C₃₂	—	59	2	no	—
52	C₃₂	—	118	1	no	—
53	C₂×C₈	—	8	1	no	—
54	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
55	C₂×C₈	#11745671	1	1	yes	characteristic
56	C₄²	#11745670	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₁₆	—	472	1	no	—
59	C₁₆	C₈×D₅₉	1	1	yes	characteristic
60	C₁₆	#11745659	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#14273658	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	16	1	no	—
65	C₈	Dic₅₉⋊C₈	1	1	yes	characteristic
66	C₈	#14273662	1	1	yes	characteristic
67	C₂²	#16837224	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	#16837152	1	1	yes	characteristic
70	C₄	#16837226	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#21422161	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21422161
commutator	C₁₆×C₅₉	C₂×C₈
Frattini	#346	C₂×D₅₉
Fitting	C₅₉×#1368	C₂
socle	C₂×C₅₉	#1368
prime radical	C₂²×C₅₉	#365
cosocle	C₅₉×#346	C₂²
hypercenter	#1368	D₅₉
hypocenter	C₅₉	#7457
LCS	G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #1368

Complements

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

N	H ≅ G/N	type
11. C₅₉⋊C₃₂	C₈	semidirect
45. C₅₉	#7457	semidirect

Maximal subgroups

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

subgroup	index
#21422232	2
#21422164	2
C₅₉×#1368	2
#7457	59