Subgroups — #28476361

Order 15104.

98 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	C₁₆×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₅₉×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉⋊C₃₂	—	2	1	no	—
5	C₄×C₅₉⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₅₉	—	2	1	no	—
8	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
9	C₂×C₅₉⋊C₁₆	—	2	1	no	—
10	C₄×C₅₉⋊C₈	C₈	1	1	yes	characteristic
11	C₄×C₅₉⋊C₈	C₈	1	1	yes	characteristic
12	C₅₉⋊C₃₂	—	4	1	no	—
13	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₉	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₉	—	4	1	no	—
17	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
19	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
20	C₂×C₅₉⋊C₈	—	2	1	no	—
21	C₂×C₅₉⋊C₈	C₁₆	1	1	yes	characteristic
22	C₂×C₅₉⋊C₈	C₁₆	1	1	yes	characteristic
23	C₁₆×C₅₉	—	2	1	no	—
24	C₁₆×C₅₉	D₈	1	1	yes	characteristic
25	C₁₆×C₅₉	Q₁₆	1	1	yes	characteristic
26	C₅₉⋊C₁₆	—	2	2	no	—
27	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
28	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
29	C₂×C₄×C₅₉	M₃₂	1	1	yes	characteristic
30	C₈×C₅₉	—	2	1	no	—
31	C₈×C₅₉	#107	1	1	yes	characteristic
32	C₈×C₅₉	#108	1	1	yes	characteristic
33	C₅₉⋊C₈	—	2	2	no	—
34	C₅₉⋊C₈	—	2	1	no	—
35	C₅₉⋊C₈	C₃₂	1	2	yes	complemented
36	#7458	—	59	1	no	maximal, Sylow
37	C₂²×C₅₉	C₄⋊C₁₆	1	1	yes	characteristic
38	C₄×C₅₉	—	2	1	no	—
39	C₄×C₅₉	#335	1	1	yes	characteristic
40	C₄×C₅₉	#364	1	1	yes	characteristic
41	C₄×C₅₉	C₂×C₃₂	1	1	yes	characteristic
42	C₄×C₅₉	M₆₄	1	1	yes	characteristic
43	C₄⋊C₃₂	—	59	1	no	—
44	C₄⋊C₃₂	D₅₉	1	1	yes	characteristic
45	C₈×C₁₆	—	59	1	no	—
46	C₂×C₅₉	#1370	1	1	yes	characteristic
47	C₂×C₅₉	C₄⋊C₃₂	1	1	yes	characteristic
48	C₂×C₅₉	#1422	1	1	yes	characteristic
49	C₂×C₃₂	—	2	1	no	—
50	C₂×C₃₂	—	118	1	no	—
51	C₄×C₁₆	—	59	1	no	—
52	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
53	C₈²	—	59	1	no	—
54	C₅₉	#7458	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₁₆	—	118	1	no	—
56	C₂×C₁₆	Dic₁₁₈	1	1	yes	characteristic
57	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
58	C₄×C₈	—	59	1	no	—
59	C₄×C₈	—	59	1	no	—
60	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
61	C₃₂	—	4	1	no	—
62	C₃₂	—	236	1	no	—
63	C₂×C₈	—	59	1	no	—
64	C₂×C₈	—	59	1	no	—
65	C₂×C₈	—	118	1	no	—
66	C₂×C₈	#11745659	1	1	yes	characteristic
67	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
68	C₄²	C₈×D₅₉	1	1	yes	characteristic
69	C₁₆	—	2	1	no	—
70	C₁₆	—	118	2	no	—
71	C₁₆	#11745669	1	1	yes	characteristic
72	C₁₆	#11745672	1	1	yes	characteristic
73	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
74	C₂×C₄	C₁₆×D₅₉	1	1	yes	characteristic
75	C₂×C₄	#14273666	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	59	2	no	—
78	C₈	—	118	2	no	—
79	C₈	—	118	1	no	—
80	C₈	#14273660	1	1	yes	characteristic
81	C₈	#14273662	1	1	yes	characteristic
82	C₂²	Dic₅₉⋊C₁₆	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	#16837154	1	1	yes	characteristic
85	C₄	#16837222	1	1	yes	characteristic
86	C₄	C₃₂×D₅₉	1	1	yes	characteristic
87	C₄	#16837235	1	1	yes	characteristic
88	C₂	#21422165	1	1	yes	characteristic
89	C₂	Dic₅₉⋊C₃₂	1	1	yes	characteristic
90	C₂	#21422293	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₅₉
Fitting	C₅₉×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₅₉	C₄⋊C₁₆
cosocle	C₄×C₁₆×C₅₉	C₂²
hypercenter	C₄⋊C₃₂	D₅₉
hypocenter	C₅₉	#7458
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

Complements

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

N	H ≅ G/N	type
35. C₅₉⋊C₈	C₃₂	semidirect
54. C₅₉	#7458	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
C₁₆×C₅₉⋊C₈	2
C₅₉×C₄⋊C₃₂	2
#7458	59