Subgroups — #28476367

Order 15104.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#21422168	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1371	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉⋊C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₅₉⋊C₃₂	C₄	1	1	yes	characteristic
6	#16837155	—	2	1	no	—
7	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
8	C₅₉×#335	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₈)	—	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	Q₁₆	1	1	yes	characteristic
19	C₄²×C₅₉	D₈	1	1	yes	characteristic
20	C₂×C₅₉⋊C₈	—	8	1	no	—
21	C₁₆×C₅₉	—	2	1	no	—
22	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₅₉	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
25	C₂×C₄×C₅₉	D₁₆	1	1	yes	characteristic
26	C₂×C₄×C₅₉	Q₃₂	1	1	yes	characteristic
27	C₈×C₅₉	—	2	1	no	—
28	C₈×C₅₉	—	16	1	no	—
29	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
30	C₈×C₅₉	#108	1	1	yes	characteristic
31	C₅₉⋊C₈	—	16	1	no	—
32	#7461	—	59	1	no	maximal, Sylow
33	C₂²×C₅₉	#366	1	1	yes	characteristic
34	C₄×C₅₉	—	2	1	no	—
35	C₄×C₅₉	#335	1	1	yes	characteristic
36	C₄×C₅₉	#368	1	1	yes	characteristic
37	C₄×C₅₉	SD₆₄	1	1	yes	characteristic
38	C₄×C₅₉	SD₆₄	1	1	yes	characteristic
39	C₄×C₃₂	—	59	1	no	—
40	#1371	—	59	1	no	—
41	#1371	D₅₉	1	1	yes	characteristic
42	C₂×C₅₉	#1371	1	1	yes	characteristic
43	C₂×C₅₉	#1424	1	1	yes	characteristic
44	C₂×C₅₉	#1425	1	1	yes	characteristic
45	C₂×C₃₂	—	59	1	no	—
46	C₂×C₃₂	—	59	1	no	—
47	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
48	#335	—	2	1	no	—
49	#335	—	118	1	no	—
50	C₅₉	#7461	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₁₆	Dic₁₁₈	1	1	yes	characteristic
52	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
53	C₄⋊C₈	—	4	1	no	—
54	C₄⋊C₈	—	236	1	no	—
55	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
56	C₃₂	—	59	2	no	—
57	C₃₂	—	118	1	no	—
58	C₂×C₈	—	8	1	no	—
59	C₂×C₈	—	472	1	no	—
60	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
61	C₂×C₈	#11745672	1	1	yes	characteristic
62	C₄²	#11745669	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	C₈×D₅₉	1	1	yes	characteristic
65	C₁₆	#11745659	1	1	yes	characteristic
66	C₂×C₄	#14273660	1	1	yes	characteristic
67	C₂×C₄	#14273668	1	1	yes	characteristic
68	C₂×C₄	#14273673	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	16	1	no	—
71	C₈	—	944	1	no	—
72	C₈	Dic₅₉⋊C₈	1	1	yes	characteristic
73	C₈	#14273662	1	1	yes	characteristic
74	C₂²	#16837227	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	#16837154	1	1	yes	characteristic
77	C₄	#16837231	1	1	yes	characteristic
78	C₄	#16837239	1	1	yes	characteristic
79	C₄	#16837240	1	1	yes	characteristic
80	C₂	#21422167	1	1	yes	characteristic
81	C₂	#21422297	1	1	yes	characteristic
82	C₂	#21422299	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273668
commutator	C₁₆×C₅₉	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₅₉
Fitting	C₅₉×#1371	C₂
socle, prime radical	C₂²×C₅₉	#366
cosocle	C₄×C₁₆×C₅₉	C₂²
hypercenter	#1371	D₅₉
hypocenter	C₅₉	#7461
LCS	G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1371

Complements

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

N	H ≅ G/N	type
11. C₅₉⋊C₃₂	C₈	semidirect
50. C₅₉	#7461	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅₉⋊C₃₂	2
#21422168	2
C₅₉×#1371	2
#7461	59