Subgroups — #21422078

Order 7552.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#16837178	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#446	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉⋊C₁₆	—	2	1	no	—
5	C₂×C₅₉⋊C₁₆	—	2	1	no	—
6	C₄×Q₈×C₅₉	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₅₉	—	2	1	no	—
8	C₄⋊C₈×C₅₉	C₄	1	1	yes	characteristic
9	#97×C₅₉	—	2	1	no	—
10	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₅₉	C₈	1	1	yes	characteristic
12	C₂×C₈×C₅₉	—	2	1	no	—
13	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
14	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₅₉	—	2	1	no	—
16	C₄⋊C₄×C₅₉	C₈	1	1	yes	characteristic
17	C₄²×C₅₉	—	2	1	no	—
18	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₅₉⋊C₁₆	—	2	2	no	—
20	C₅₉⋊C₁₆	—	4	1	no	—
21	Q₈×C₅₉	—	4	1	no	—
22	C₂×C₄×C₅₉	—	2	1	no	—
23	C₂×C₄×C₅₉	—	2	1	no	—
24	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅₉	M₁₆	1	1	yes	characteristic
27	C₈×C₅₉	—	2	1	no	—
28	C₈×C₅₉	—	4	1	no	—
29	C₈×C₅₉	SD₁₆	1	1	yes	characteristic
30	C₈×C₅₉	Q₁₆	1	1	yes	characteristic
31	C₂²×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₅₉	—	2	1	no	—
33	C₄×C₅₉	—	2	1	no	—
34	C₄×C₅₉	—	4	1	no	—
35	C₄×C₅₉	—	4	1	no	—
36	C₄×C₅₉	#103	1	1	yes	characteristic
37	C₄×C₅₉	#104	1	1	yes	characteristic
38	#1338	—	59	1	no	maximal, Sylow
39	C₂×C₅₉	Q₈⋊C₈	1	1	yes	characteristic
40	C₂×C₅₉	#349	1	1	yes	characteristic
41	C₂×C₅₉	#350	1	1	yes	characteristic
42	#446	D₅₉	1	1	yes	characteristic
43	C₄⋊C₁₆	—	59	1	no	—
44	#346	—	59	1	no	—
45	C₅₉	#1338	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₈	Dic₅₉	1	1	yes	characteristic
47	C₂×C₁₆	—	118	1	no	—
48	C₂×C₁₆	—	118	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄⋊C₈	Dic₅₉	1	1	yes	characteristic
51	#97	—	2	1	no	—
52	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
53	C₂×Q₈	C₅₉⋊C₈	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
56	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	C₅₉⋊C₈	1	1	yes	characteristic
59	C₄²	—	2	1	no	—
60	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
61	C₁₆	—	118	2	no	—
62	C₁₆	—	236	1	no	—
63	Q₈	—	4	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
67	C₂×C₄	#11745664	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	4	1	no	—
71	C₈	#11745671	1	1	yes	characteristic
72	C₈	#11745672	1	1	yes	characteristic
73	C₂²	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	#14273651	1	1	yes	characteristic
79	C₄	#14273654	1	1	yes	characteristic
80	C₂	Q₈⋊(C₅₉⋊C₈)	1	1	yes	characteristic
81	C₂	#16837184	1	1	yes	characteristic
82	C₂	#16837186	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₅₉
commutator	C₂×C₄×C₅₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₅₉
Fitting	C₅₉×#446	C₂
socle	C₂²×C₅₉	C₂²⋊C₈
hypercenter	#446	D₅₉
hypocenter	C₅₉	#1338
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #446

Complements

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

N	H ≅ G/N	type
45. C₅₉	#1338	semidirect