Subgroups — #28476357

Order 15104.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422294	C₂	1	2	yes	maximal
2	C₅₉×#1311	C₂	1	1	yes	maximal, characteristic
3	#16837236	—	2	2	no	—
4	C₄×C₁₆×C₅₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
6	C₈²×C₅₉	C₄	1	1	yes	characteristic
7	C₅₉⋊C₃₂	—	4	2	no	—
8	C₂×C₁₆×C₅₉	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₅₉	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₅₉	Q₈	1	1	yes	characteristic
12	C₄×C₈×C₅₉	—	2	1	no	—
13	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₉	—	2	2	no	—
15	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
18	C₁₆×C₅₉	—	2	1	no	—
19	C₁₆×C₅₉	—	2	1	no	—
20	C₁₆×C₅₉	C₁₆	1	2	yes	—
21	C₁₆×C₅₉	SD₁₆	1	2	yes	—
22	C₂×C₄×C₅₉	—	2	1	no	—
23	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₅₉	—	4	2	no	—
25	C₈×C₅₉	#106	1	1	yes	characteristic
26	C₈×C₅₉	#108	1	1	yes	characteristic
27	C₈×C₅₉	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₅₉	M₃₂	1	1	yes	characteristic
29	#7456	—	59	1	no	maximal, Sylow
30	C₂²×C₅₉	#364	1	1	yes	characteristic
31	C₄×C₅₉	—	2	2	no	—
32	C₄×C₅₉	#334	1	1	yes	characteristic
33	C₄×C₅₉	C₄⋊C₁₆	1	1	yes	characteristic
34	#1422	—	59	2	no	—
35	#1311	D₅₉	1	1	yes	characteristic
36	C₂×C₅₉	—	2	1	no	—
37	C₂×C₅₉	#1367	1	1	yes	characteristic
38	M₆₄	—	118	2	no	—
39	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
40	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
41	C₈²	Dic₅₉	1	1	yes	characteristic
42	C₅₉	#7456	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₁₆	C₅₉⋊C₈	1	1	yes	characteristic
44	C₂×C₁₆	C₅₉⋊C₈	1	1	yes	characteristic
45	C₂×C₁₆	Dic₁₁₈	1	1	yes	characteristic
46	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
49	C₃₂	—	236	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
52	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
53	C₄²	#11745664	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	2	1	no	—
56	C₁₆	C₅₉⋊C₁₆	1	2	yes	—
57	C₁₆	#11745660	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
60	C₈	—	4	2	no	—
61	C₈	#14273659	1	1	yes	characteristic
62	C₈	#14273663	1	1	yes	characteristic
63	C₈	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
64	C₈	#14273667	1	1	yes	characteristic
65	C₂²	#16837223	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	#16837153	1	1	yes	characteristic
68	C₄	C₄⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#21422160	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273659
commutator	C₈×C₅₉	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₅₉
Fitting	C₅₉×#1311	C₂
socle	C₂×C₅₉	#1367
prime radical	C₂²×C₅₉	#364
cosocle	C₄×C₁₆×C₅₉	C₂²
hypercenter	#1311	D₅₉
hypocenter	C₅₉	#7456
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1311

Complements

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

N	H ≅ G/N	type
42. C₅₉	#7456	semidirect

Maximal subgroups

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

subgroup	index
#21422294	2
C₅₉×#1311	2
#7456	59