Subgroups — #21422166

Order 7552.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₅₉⋊C₁₆)	C₂	1	2	yes	maximal
2	C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉⋊C₁₆	—	2	2	no	—
4	C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
5	C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
6	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₅₉	—	2	1	no	—
8	C₂×C₈×C₅₉	C₈	1	1	yes	characteristic
9	C₂×C₈×C₅₉	C₈	1	1	yes	characteristic
10	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
11	C₂×C₈×C₅₉	Q₈	1	1	yes	characteristic
12	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
13	C₅₉⋊C₁₆	—	4	2	no	—
14	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₅₉	M₁₆	1	1	yes	characteristic
17	C₈×C₅₉	—	2	2	no	—
18	C₈×C₅₉	—	2	1	no	—
19	C₈×C₅₉	—	2	1	no	—
20	C₈×C₅₉	C₁₆	1	2	yes	complemented
21	C₈×C₅₉	D₈	1	1	yes	characteristic
22	C₈×C₅₉	Q₁₆	1	1	yes	characteristic
23	C₂²×C₅₉	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₅₉	—	2	1	no	—
25	C₄×C₅₉	#107	1	1	yes	characteristic
26	C₄×C₅₉	#108	1	1	yes	characteristic
27	C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
28	C₄×C₅₉	M₃₂	1	1	yes	characteristic
29	#1370	—	59	1	no	maximal, Sylow
30	C₂×C₅₉	#335	1	1	yes	characteristic
31	C₂×C₅₉	C₄⋊C₁₆	1	1	yes	characteristic
32	C₂×C₅₉	#364	1	1	yes	characteristic
33	C₄⋊C₁₆	—	59	2	no	—
34	C₈²	D₅₉	1	1	yes	characteristic
35	C₅₉	#1370	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₁₆	—	118	2	no	—
37	C₄×C₈	Dic₅₉	1	1	yes	characteristic
38	C₄×C₈	Dic₅₉	1	1	yes	characteristic
39	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	C₅₉⋊C₈	1	1	yes	characteristic
42	C₂×C₈	C₅₉⋊C₈	1	1	yes	characteristic
43	C₂×C₈	Dic₁₁₈	1	1	yes	characteristic
44	C₂×C₈	D₂₃₆	1	1	yes	characteristic
45	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
46	C₁₆	—	236	2	no	—
47	C₂×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
48	C₂×C₄	#11745664	1	1	yes	characteristic
49	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	2	1	no	—
52	C₈	—	2	1	no	—
53	C₈	C₅₉⋊C₁₆	1	2	yes	complemented
54	C₈	D₄₇₂	1	1	yes	characteristic
55	C₈	Dic₂₃₆	1	1	yes	characteristic
56	C₂²	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	#14273661	1	1	yes	characteristic
59	C₄	#14273663	1	1	yes	characteristic
60	C₄	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
61	C₄	#14273667	1	1	yes	characteristic
62	C₂	#16837155	1	1	yes	characteristic
63	C₂	C₄⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic
64	C₂	#16837223	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₂₃₆
commutator	C₄×C₅₉	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₅₉
Fitting	C₈²×C₅₉	C₂
socle	C₂²×C₅₉	C₄⋊C₈
hypercenter	C₈²	D₅₉
hypocenter	C₅₉	#1370
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₅₉	C₁₆	semidirect
35. C₅₉	#1370	semidirect
53. C₈	C₅₉⋊C₁₆	semidirect