Subgroups — #28475429

Order 15104.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422166	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
4	C₄×C₁₆×C₅₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₅₉	C₄	1	1	yes	characteristic
6	C₈²×C₅₉	C₂²	1	1	yes	characteristic
7	C₂×C₅₉⋊C₁₆	—	4	2	no	—
8	C₂×C₁₆×C₅₉	—	2	1	no	—
9	C₂×C₁₆×C₅₉	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₅₉	C₈	1	1	yes	characteristic
11	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
13	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
14	C₂×C₈×C₅₉	—	2	1	no	—
15	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
16	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
17	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
18	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
19	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
20	C₁₆×C₅₉	—	2	2	no	—
21	C₁₆×C₅₉	—	2	1	no	—
22	C₁₆×C₅₉	C₁₆	1	2	yes	complemented
23	C₅₉⋊C₁₆	—	8	2	no	—
24	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
26	C₂×C₄×C₅₉	#108	1	1	yes	characteristic
27	C₈×C₅₉	—	2	2	no	—
28	C₈×C₅₉	—	2	1	no	—
29	C₈×C₅₉	—	2	1	no	—
30	C₈×C₅₉	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₅₉	M₃₂	1	1	yes	characteristic
32	C₈×C₅₉	D₁₆	1	1	yes	characteristic
33	C₈×C₅₉	Q₃₂	1	1	yes	characteristic
34	#7100	—	59	1	no	maximal, Sylow
35	C₂²×C₅₉	#335	1	1	yes	characteristic
36	C₄×C₅₉	—	2	1	no	—
37	C₄×C₅₉	C₄⋊C₁₆	1	1	yes	characteristic
38	C₄×C₅₉	#364	1	1	yes	characteristic
39	C₄×C₅₉	#366	1	1	yes	characteristic
40	C₄×C₅₉	#368	1	1	yes	characteristic
41	#1370	—	59	2	no	—
42	C₈×C₁₆	D₅₉	1	1	yes	characteristic
43	C₂×C₅₉	#1370	1	1	yes	characteristic
44	C₂×C₅₉	#1371	1	1	yes	characteristic
45	C₂×C₅₉	#1373	1	1	yes	characteristic
46	C₄⋊C₁₆	—	118	2	no	—
47	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
48	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
49	C₈²	C₂×D₅₉	1	1	yes	characteristic
50	C₅₉	#7100	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	236	2	no	—
53	C₂×C₁₆	C₅₉⋊C₈	1	1	yes	characteristic
54	C₂×C₁₆	C₅₉⋊C₈	1	1	yes	characteristic
55	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
56	C₄×C₈	D₂₃₆	1	1	yes	characteristic
57	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	D₄₇₂	1	1	yes	characteristic
60	C₂×C₈	Dic₂₃₆	1	1	yes	characteristic
61	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
62	C₂×C₈	#11745664	1	1	yes	characteristic
63	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
64	C₁₆	—	2	2	no	—
65	C₁₆	—	2	1	no	—
66	C₁₆	—	472	2	no	—
67	C₁₆	C₅₉⋊C₁₆	1	2	yes	complemented
68	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
69	C₂×C₄	#14273661	1	1	yes	characteristic
70	C₂×C₄	#14273663	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	2	1	no	—
73	C₈	—	2	1	no	—
74	C₈	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
75	C₈	#14273667	1	1	yes	characteristic
76	C₈	D₉₄₄	1	1	yes	characteristic
77	C₈	Dic₄₇₂	1	1	yes	characteristic
78	C₂²	#16837155	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	C₄⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic
81	C₄	#16837223	1	1	yes	characteristic
82	C₄	#16837228	1	1	yes	characteristic
83	C₄	#16837232	1	1	yes	characteristic
84	C₂	#21422166	1	1	yes	characteristic
85	C₂	#21422168	1	1	yes	characteristic
86	C₂	#21422172	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄₇₂
commutator	C₈×C₅₉	C₂×C₁₆
Frattini	C₈²	C₂×D₅₉
Fitting	C₈×C₁₆×C₅₉	C₂
socle, prime radical	C₂²×C₅₉	#335
cosocle	C₈²×C₅₉	C₂²
hypercenter	C₈×C₁₆	D₅₉
hypocenter	C₅₉	#7100
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
22. C₁₆×C₅₉	C₁₆	semidirect
50. C₅₉	#7100	semidirect
67. C₁₆	C₅₉⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#21422166	2
C₈×C₁₆×C₅₉	2
#7100	59