Subgroups — #21422398

Order 7552.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11745659	C₂	1	3	yes	maximal
2	#16837123	C₂	1	3	yes	maximal, complemented
3	C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11745659	C₄	1	6	yes	—
5	C₄²×D₅₉	C₂²	1	1	yes	characteristic
6	#14273637	C₄	1	6	yes	—
7	C₄×C₅₉⋊C₈	C₂²	1	3	yes	—
8	C₄×C₈×C₅₉	C₂²	1	3	yes	—
9	C₂×C₄×D₅₉	C₂×C₄	1	3	yes	—
10	C₂×C₈×C₅₉	C₂×C₄	1	6	yes	—
11	C₄²×C₅₉	C₂³	1	1	yes	characteristic
12	C₄×Dic₅₉	C₂×C₄	1	3	yes	—
13	C₂×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
14	#11745659	—	2	12	no	—
15	C₂²×D₅₉	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
17	C₂×Dic₅₉	C₄²	1	3	yes	—
18	C₄×D₅₉	—	2	6	no	—
19	C₈×C₅₉	M₁₆	1	12	yes	complemented
20	C₅₉⋊C₈	—	2	6	no	—
21	C₂²×C₅₉	C₂×C₄²	1	1	yes	characteristic
22	C₂×D₅₉	—	2	3	no	—
23	C₄×C₅₉	C₂×M₁₆	1	6	yes	—
24	Dic₅₉	—	2	3	no	—
25	#1455	—	59	1	no	maximal, Sylow
26	C₂×C₅₉	C₄×M₁₆	1	3	yes	—
27	D₅₉	—	4	1	no	—
28	C₄×M₁₆	—	59	3	no	—
29	#322	—	59	3	no	—
30	C₈²	D₅₉	1	1	yes	characteristic, complemented
31	C₅₉	#1455	1	1	yes	characteristic, complemented, Sylow
32	C₂×M₁₆	—	59	6	no	—
33	C₂×C₄²	—	59	1	no	—
34	#97	—	59	6	no	—
35	C₄×C₈	—	59	3	no	—
36	C₄×C₈	C₂×D₅₉	1	3	yes	—
37	C₂²×C₄	—	59	3	no	—
38	M₁₆	—	118	12	no	—
39	C₂×C₈	—	59	6	no	—
40	C₂×C₈	C₄×D₅₉	1	6	yes	—
41	C₄²	—	59	3	no	—
42	C₄²	C₂²×D₅₉	1	1	yes	characteristic
43	C₂³	—	59	1	no	—
44	C₂×C₄	—	59	3	no	—
45	C₂×C₄	—	118	6	no	—
46	C₂×C₄	C₂×C₄×D₅₉	1	3	yes	—
47	C₈	—	118	6	no	—
48	C₈	#11745659	1	12	yes	complemented
49	C₂²	—	118	3	no	—
50	C₂²	C₄²×D₅₉	1	1	yes	characteristic
51	C₄	—	118	3	no	—
52	C₄	C₂×#11745659	1	6	yes	—
53	C₂	—	236	1	no	—
54	C₂	C₄×#11745659	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂×C₄²
Fitting	C₈²×C₅₉	C₂
hypercenter	C₈²	D₅₉
hypocenter	C₅₉	#1455
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #16837123	C₂	semidirect
3. C₈²×C₅₉	C₂	semidirect
19. C₈×C₅₉	M₁₆	semidirect
30. C₈²	D₅₉	semidirect
31. C₅₉	#1455	semidirect
48. C₈	#11745659	semidirect