Subgroups — #28476356

Order 15104.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422294	C₂	1	1	yes	maximal, characteristic
2	#21422012	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1422	C₂	1	1	yes	maximal, characteristic
4	#16837236	—	2	1	no	—
5	C₄×C₅₉⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
7	C₅₉×M₆₄	—	2	1	no	—
8	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
9	C₂×C₅₉⋊C₁₆	C₈	1	1	yes	characteristic
10	C₂×C₅₉⋊C₁₆	C₈	1	1	yes	characteristic
11	C₄×C₅₉⋊C₈	—	2	1	no	—
12	C₅₉⋊C₃₂	—	4	1	no	—
13	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₉	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₉	—	4	1	no	—
17	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
19	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
20	C₂×C₅₉⋊C₈	—	2	2	no	—
21	C₁₆×C₅₉	—	2	1	no	—
22	C₁₆×C₅₉	SD₁₆	1	1	yes	characteristic
23	C₁₆×C₅₉	SD₁₆	1	1	yes	characteristic
24	C₅₉⋊C₁₆	—	2	1	no	—
25	C₅₉⋊C₁₆	C₁₆	1	2	yes	—
26	C₂×C₄×C₅₉	—	2	1	no	—
27	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₅₉	#106	1	1	yes	characteristic
29	C₈×C₅₉	#108	1	1	yes	characteristic
30	C₈×C₅₉	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₅₉	M₃₂	1	1	yes	characteristic
32	C₅₉⋊C₈	—	4	2	no	—
33	#7456	—	59	1	no	maximal, Sylow
34	C₂²×C₅₉	#364	1	1	yes	characteristic
35	C₄×C₅₉	—	2	2	no	—
36	C₄×C₅₉	#334	1	1	yes	characteristic
37	C₄×C₅₉	C₄⋊C₁₆	1	1	yes	characteristic
38	#1422	—	59	1	no	—
39	#1422	D₅₉	1	1	yes	characteristic
40	#1311	—	59	1	no	—
41	C₂×C₅₉	—	2	1	no	—
42	C₂×C₅₉	#1367	1	1	yes	characteristic
43	M₆₄	—	2	1	no	—
44	M₆₄	—	118	1	no	—
45	C₄×C₁₆	—	59	1	no	—
46	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
47	C₈²	—	59	1	no	—
48	C₅₉	#7456	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₁₆	—	59	1	no	—
50	C₂×C₁₆	—	59	1	no	—
51	C₂×C₁₆	Dic₁₁₈	1	1	yes	characteristic
52	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
53	C₄×C₈	—	118	1	no	—
54	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
55	C₃₂	—	4	1	no	—
56	C₃₂	—	236	1	no	—
57	C₂×C₈	—	118	2	no	—
58	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
59	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
60	C₄²	#11745659	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	—	59	2	no	—
63	C₁₆	—	118	1	no	—
64	C₁₆	#11745670	1	1	yes	characteristic
65	C₁₆	#11745671	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
68	C₈	—	236	2	no	—
69	C₈	#14273658	1	1	yes	characteristic
70	C₈	#14273662	1	1	yes	characteristic
71	C₈	C₁₆×D₅₉	1	1	yes	characteristic
72	C₈	#14273666	1	1	yes	characteristic
73	C₂²	#16837222	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	#16837152	1	1	yes	characteristic
76	C₄	Dic₅₉⋊C₁₆	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	#21422159	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273658
commutator	C₈×C₅₉	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₅₉
Fitting	C₅₉×#1422	C₂
socle	C₂×C₅₉	#1367
prime radical	C₂²×C₅₉	#364
cosocle	C₄×C₁₆×C₅₉	C₂²
hypercenter	#1422	D₅₉
hypocenter	C₅₉	#7456
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1422

Complements

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

N	H ≅ G/N	type
48. 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
#21422012	2
C₅₉×#1422	2
#7456	59