Subgroups — #21422239

Order 7552.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837818	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#14273733	C₄	1	1	yes	characteristic
5	#14273667	C₄	1	1	yes	characteristic
6	#14273666	—	2	1	no	—
7	C₁₆×D₅₉	—	2	1	no	—
8	C₅₉⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
10	C₃₂×C₅₉	—	2	1	no	—
11	#11745685	C₈	1	1	yes	characteristic
12	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
13	#11745664	C₈	1	1	yes	characteristic
14	#11745659	—	2	1	no	—
15	C₈×D₅₉	—	2	1	no	—
16	C₁₆×C₅₉	D₄	1	1	yes	characteristic
17	C₁₆×C₅₉	D₄	1	1	yes	characteristic
18	C₅₉⋊C₁₆	—	2	1	no	—
19	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
20	C₂²⋊D₅₉	—	2	1	no	—
21	D₂₃₆	C₁₆	1	1	yes	characteristic
22	C₄×D₅₉	—	2	1	no	—
23	Dic₁₁₈	C₁₆	1	1	yes	characteristic
24	C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₅₉	M₁₆	1	1	yes	characteristic
26	C₅₉⋊C₈	—	2	1	no	—
27	C₂²×C₅₉	M₃₂	1	1	yes	characteristic
28	C₂×D₅₉	—	2	1	no	—
29	C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
31	Dic₅₉	—	2	1	no	—
32	#1401	—	59	1	no	maximal, Sylow
33	C₂×C₅₉	—	2	1	no	—
34	C₂×C₅₉	C₂²⋊C₁₆	1	1	yes	characteristic
35	D₅₉	—	4	1	no	—
36	#504	—	59	1	no	—
37	M₆₄	D₅₉	1	1	yes	characteristic, complemented
38	C₂×C₃₂	—	59	1	no	—
39	C₅₉	#1401	1	1	yes	characteristic, complemented, Sylow
40	#131	—	59	1	no	—
41	M₃₂	—	59	1	no	—
42	M₃₂	—	118	1	no	—
43	C₂×C₁₆	—	118	1	no	—
44	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	118	1	no	—
47	#41	—	59	1	no	—
48	M₁₆	—	59	1	no	—
49	M₁₆	—	118	1	no	—
50	C₂×C₈	—	118	1	no	—
51	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
52	C₁₆	—	118	1	no	—
53	C₁₆	D₂₃₆	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
55	Q₈	—	59	1	no	—
56	D₄	—	59	1	no	—
57	D₄	—	118	1	no	—
58	C₂×C₄	—	118	1	no	—
59	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
60	C₈	—	118	1	no	—
61	C₈	#11745659	1	1	yes	characteristic
62	C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
63	C₂²	—	118	1	no	—
64	C₂²	#14273666	1	1	yes	characteristic
65	C₄	—	118	1	no	—
66	C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
67	C₄	C₁₆×D₅₉	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	236	1	no	—
70	C₂	(C₂×D₅₉)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₅₉
Fitting	C₅₉×M₆₄	C₂
socle	C₂×C₅₉	C₂²⋊C₁₆
hypercenter	M₆₄	D₅₉
hypocenter	C₅₉	#1401
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₉⋊C₃₂	C₂	semidirect
3. C₅₉×M₆₄	C₂	semidirect
37. M₆₄	D₅₉	semidirect
39. C₅₉	#1401	semidirect