Subgroups — #28476366

Order 15104.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422168	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837155	—	2	2	no	—
4	C₂×C₃₂×C₅₉	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₅₉	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
7	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
8	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₅₉	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
11	C₃₂×C₅₉	—	2	1	no	—
12	C₃₂×C₅₉	C₈	1	2	yes	complemented
13	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
15	C₄²×C₅₉	D₈	1	1	yes	characteristic
16	C₂×C₅₉⋊C₈	—	8	2	no	—
17	C₁₆×C₅₉	—	2	1	no	—
18	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₅₉	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
21	C₂×C₄×C₅₉	D₁₆	1	1	yes	characteristic
22	C₂×C₄×C₅₉	Q₃₂	1	1	yes	characteristic
23	C₈×C₅₉	—	2	1	no	—
24	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₅₉	#108	1	1	yes	characteristic
26	C₅₉⋊C₈	—	16	2	no	—
27	#7460	—	59	1	no	maximal, Sylow
28	C₂²×C₅₉	#366	1	1	yes	characteristic
29	C₄×C₅₉	—	2	1	no	—
30	C₄×C₅₉	#335	1	1	yes	characteristic
31	C₄×C₅₉	#368	1	1	yes	characteristic
32	C₄×C₅₉	D₃₂	1	1	yes	characteristic
33	C₄×C₅₉	Q₆₄	1	1	yes	characteristic
34	C₄×C₃₂	D₅₉	1	1	yes	characteristic
35	#1371	—	59	2	no	—
36	C₂×C₅₉	#1371	1	1	yes	characteristic
37	C₂×C₅₉	#1423	1	1	yes	characteristic
38	C₂×C₅₉	#1425	1	1	yes	characteristic
39	C₂×C₃₂	Dic₅₉	1	1	yes	characteristic
40	C₂×C₃₂	Dic₅₉	1	1	yes	characteristic
41	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
42	#335	—	118	2	no	—
43	C₅₉	#7460	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₁₆	Dic₁₁₈	1	1	yes	characteristic
45	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
46	C₄⋊C₈	—	236	2	no	—
47	C₄×C₈	D₂₃₆	1	1	yes	characteristic
48	C₃₂	—	2	1	no	—
49	C₃₂	C₅₉⋊C₈	1	2	yes	complemented
50	C₂×C₈	—	472	2	no	—
51	C₂×C₈	Dic₂₃₆	1	1	yes	characteristic
52	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
53	C₄²	D₄₇₂	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	C₂×C₅₉⋊C₈	1	1	yes	characteristic
56	C₁₆	#11745664	1	1	yes	characteristic
57	C₂×C₄	#14273661	1	1	yes	characteristic
58	C₂×C₄	D₉₄₄	1	1	yes	characteristic
59	C₂×C₄	Dic₄₇₂	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	—	944	2	no	—
62	C₈	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
63	C₈	#14273663	1	1	yes	characteristic
64	C₂²	#16837228	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	#16837155	1	1	yes	characteristic
67	C₄	#16837232	1	1	yes	characteristic
68	C₄	D₁₈₈₈	1	1	yes	characteristic
69	C₄	Dic₉₄₄	1	1	yes	characteristic
70	C₂	#21422168	1	1	yes	characteristic
71	C₂	#21422296	1	1	yes	characteristic
72	C₂	#21422300	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₉₄₄
commutator	C₁₆×C₅₉	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₅₉
Fitting	C₄×C₃₂×C₅₉	C₂
socle, prime radical	C₂²×C₅₉	#366
cosocle	C₄×C₁₆×C₅₉	C₂²
hypercenter	C₄×C₃₂	D₅₉
hypocenter	C₅₉	#7460
LCS	G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₅₉	C₈	semidirect
43. C₅₉	#7460	semidirect
49. C₃₂	C₅₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#21422168	2
C₄×C₃₂×C₅₉	2
#7460	59