Subgroups — #28476019

Order 15104.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422230	C₂	1	2	yes	maximal
2	C₅₉×#1311	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉⋊C₃₂	C₄	1	4	yes	—
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	C₈	1	8	yes	complemented
8	C₂×C₁₆×C₅₉	C₈	1	2	yes	—
9	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
10	C₄×C₈×C₅₉	C₈	1	2	yes	—
11	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₉	—	2	2	no	—
13	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
14	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
16	C₁₆×C₅₉	—	2	2	no	—
17	C₁₆×C₅₉	C₂×C₈	1	4	yes	—
18	C₂×C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
19	C₂×C₄×C₅₉	M₃₂	1	2	yes	—
20	C₈×C₅₉	—	4	2	no	—
21	C₈×C₅₉	C₄×C₈	1	2	yes	—
22	C₈×C₅₉	M₃₂	1	2	yes	—
23	#7330	—	59	1	no	maximal, Sylow
24	C₂²×C₅₉	#346	1	1	yes	characteristic
25	C₄×C₅₉	—	2	2	no	—
26	C₄×C₅₉	C₈²	1	1	yes	characteristic
27	C₄×C₅₉	#346	1	1	yes	characteristic
28	#1397	—	59	2	no	—
29	#1311	D₅₉	1	1	yes	characteristic
30	C₂×C₅₉	#1311	1	1	yes	characteristic
31	C₂×C₅₉	#1398	1	2	yes	—
32	C₂×C₃₂	—	59	4	no	—
33	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
34	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
35	C₈²	Dic₅₉	1	1	yes	characteristic
36	C₅₉	#7330	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₁₆	C₅₉⋊C₈	1	2	yes	—
38	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
39	C₄×C₈	C₅₉⋊C₈	1	2	yes	—
40	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
41	C₃₂	—	59	8	no	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
44	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
45	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
46	C₁₆	—	2	2	no	—
47	C₁₆	C₈×D₅₉	1	4	yes	—
48	C₂×C₄	C₄×C₅₉⋊C₈	1	1	yes	characteristic
49	C₂×C₄	#14273667	1	2	yes	—
50	C₈	—	4	2	no	—
51	C₈	C₈×Dic₅₉	1	2	yes	—
52	C₈	#14273667	1	2	yes	—
53	C₂²	#16837178	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	C₈×C₅₉⋊C₈	1	1	yes	characteristic
56	C₄	#16837178	1	1	yes	characteristic
57	C₂	#21422013	1	1	yes	characteristic
58	C₂	#21422232	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅₉
commutator	C₄×C₅₉	C₈²
Frattini	C₄×C₁₆	C₂×D₅₉
Fitting	C₅₉×#1311	C₂
socle, prime radical	C₂²×C₅₉	#346
cosocle	C₄×C₁₆×C₅₉	C₂²
hypercenter	#1311	D₅₉
hypocenter	C₅₉	#7330
LCS	G ▷ C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ #1311

Complements

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

N	H ≅ G/N	type
7. C₅₉⋊C₃₂	C₈	semidirect
36. C₅₉	#7330	semidirect

Maximal subgroups

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

subgroup	index
#21422230	2
C₅₉×#1311	2
#7330	59