Subgroups — #28475305

Order 15104.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422014	C₂	1	2	yes	maximal
2	C₅₉×#1311	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅₉⋊C₁₆	C₄	1	4	yes	—
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	—	2	4	no	—
8	C₂×C₁₆×C₅₉	C₈	1	2	yes	—
9	C₂×C₁₆×C₅₉	C₈	1	2	yes	—
10	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
11	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
13	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
14	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
15	C₄²×C₅₉	C₄²	1	1	yes	characteristic
16	C₁₆×C₅₉	—	2	2	no	—
17	C₁₆×C₅₉	C₁₆	1	4	yes	complemented
18	C₅₉⋊C₁₆	—	4	4	no	—
19	C₂×C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
20	C₂×C₄×C₅₉	#97	1	2	yes	—
21	C₈×C₅₉	—	2	4	no	—
22	C₈×C₅₉	C₂×C₁₆	1	2	yes	—
23	C₈×C₅₉	M₃₂	1	2	yes	—
24	#7053	—	59	1	no	maximal, Sylow
25	C₂²×C₅₉	#322	1	1	yes	characteristic
26	C₄×C₅₉	C₄×C₁₆	1	1	yes	characteristic
27	C₄×C₅₉	#346	1	1	yes	characteristic
28	C₄×C₅₉	#347	1	4	yes	—
29	#1312	—	59	2	no	—
30	#1311	D₅₉	1	1	yes	characteristic
31	C₂×C₅₉	#1312	1	1	yes	characteristic
32	C₂×C₅₉	#1313	1	2	yes	—
33	C₄×C₁₆	—	59	4	no	—
34	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
35	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
36	C₈²	C₂×D₅₉	1	1	yes	characteristic
37	C₅₉	#7053	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₁₆	—	118	4	no	—
39	C₂×C₁₆	C₅₉⋊C₈	1	2	yes	—
40	C₂×C₁₆	C₅₉⋊C₈	1	2	yes	—
41	C₄×C₈	C₄×D₅₉	1	2	yes	—
42	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
43	C₂×C₈	#11745659	1	4	yes	—
44	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
45	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
46	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
47	C₁₆	—	2	2	no	—
48	C₁₆	—	236	4	no	—
49	C₁₆	C₅₉⋊C₁₆	1	4	yes	complemented
50	C₂×C₄	C₄×C₅₉⋊C₈	1	1	yes	characteristic
51	C₂×C₄	#14273637	1	2	yes	—
52	C₈	—	2	4	no	—
53	C₈	C₂×C₅₉⋊C₁₆	1	2	yes	—
54	C₈	#14273667	1	2	yes	—
55	C₂²	#16837123	1	1	yes	characteristic
56	C₄	C₄×C₅₉⋊C₁₆	1	1	yes	characteristic
57	C₄	#16837178	1	1	yes	characteristic
58	C₄	#16837179	1	4	yes	—
59	C₂	#21422014	1	1	yes	characteristic
60	C₂	#21422016	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₅₉
commutator	C₄×C₅₉	C₄×C₁₆
Frattini	C₈²	C₂×D₅₉
Fitting	C₅₉×#1311	C₂
socle, prime radical	C₂²×C₅₉	#322
cosocle	C₈²×C₅₉	C₂²
hypercenter	#1311	D₅₉
hypocenter	C₅₉	#7053
LCS	G ▷ C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ #1311

Complements

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

N	H ≅ G/N	type
17. C₁₆×C₅₉	C₁₆	semidirect
37. C₅₉	#7053	semidirect
49. 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
#21422014	2
C₅₉×#1311	2
#7053	59