Subgroups — #21425613

Order 7552.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837451	C₂	1	3	yes	maximal, complemented
2	#16837365	C₂	1	3	yes	maximal
3	C₅₉×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₅₉	C₂²	1	1	yes	characteristic
5	C₂×#11745664	C₂²	1	3	yes	—
6	#14273663	—	2	6	no	—
7	(C₂×D₅₉)⋊C₈	—	2	6	no	—
8	C₂×M₁₆×C₅₉	C₂²	1	3	yes	—
9	C₂³×D₅₉	Q₈	1	1	yes	characteristic
10	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
11	C₂²×Dic₅₉	Q₈	1	1	yes	characteristic
12	C₂×C₄×D₅₉	—	2	3	no	—
13	C₂×C₄×D₅₉	—	2	3	no	—
14	M₁₆×C₅₉	—	2	6	no	—
15	C₂×C₈×C₅₉	D₄	1	6	yes	—
16	#11745664	—	2	6	no	—
17	C₂×C₅₉⋊C₈	—	2	3	no	—
18	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
19	C₂²×D₅₉	—	2	3	no	—
20	C₂²×D₅₉	—	4	2	no	—
21	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
22	C₂×C₄×C₅₉	#41	1	3	yes	—
23	C₂×Dic₅₉	—	2	3	no	—
24	C₄×D₅₉	—	4	3	no	—
25	C₄×D₅₉	—	4	1	no	—
26	C₈×C₅₉	—	2	6	no	—
27	C₅₉⋊C₈	—	4	3	no	—
28	C₂²×C₅₉	—	4	1	no	—
29	C₂²×C₅₉	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₅₉	—	2	6	no	—
31	C₂×D₅₉	—	4	3	no	—
32	C₂×D₅₉	—	4	1	no	—
33	C₄×C₅₉	#123	1	3	yes	—
34	C₄×C₅₉	#127	1	1	yes	characteristic
35	Dic₅₉	—	4	1	no	—
36	#2091	—	59	1	no	maximal, Sylow
37	C₂×C₅₉	—	2	3	no	—
38	C₂×C₅₉	#399	1	1	yes	characteristic
39	D₅₉	—	4	2	no	—
40	#430	—	59	3	no	—
41	#407	—	59	3	no	—
42	#338	D₅₉	1	1	yes	characteristic, complemented
43	C₅₉	#2091	1	1	yes	characteristic, complemented, Sylow
44	C₂³×C₄	—	59	1	no	—
45	C₂×M₁₆	—	59	3	no	—
46	C₂×M₁₆	C₂×D₅₉	1	3	yes	—
47	#108	—	118	6	no	—
48	C₂²⋊C₈	—	118	6	no	—
49	C₂⁴	—	59	1	no	—
50	C₂²×C₄	—	59	1	no	—
51	C₂²×C₄	—	118	3	no	—
52	C₂²×C₄	—	118	3	no	—
53	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
54	M₁₆	—	2	6	no	—
55	M₁₆	—	118	6	no	—
56	C₂×C₈	—	118	3	no	—
57	C₂×C₈	D₂₃₆	1	6	yes	—
58	C₂³	—	118	3	no	—
59	C₂³	—	236	2	no	—
60	C₂³	Q₈×D₅₉	1	1	yes	characteristic
61	C₂×C₄	—	118	3	no	—
62	C₂×C₄	—	236	3	no	—
63	C₂×C₄	—	236	1	no	—
64	C₂×C₄	C₂×D₂₃₆	1	3	yes	—
65	C₂×C₄	#11745687	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	—	236	3	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	118	6	no	—
70	C₂²	—	236	3	no	—
71	C₂²	—	236	1	no	—
72	C₂²	#14273707	1	3	yes	—
73	C₄	—	236	1	no	—
74	C₄	#14273711	1	3	yes	—
75	C₄	#14273724	1	1	yes	characteristic
76	C₂	—	2	3	no	—
77	C₂	—	236	2	no	—
78	C₂	#16837339	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14273724
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#338	C₂
socle	C₂×C₅₉	#399
hypercenter	#338	D₅₉
hypocenter	C₅₉	#2091
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #16837451	C₂	semidirect
3. C₅₉×#338	C₂	semidirect
42. #338	D₅₉	semidirect
43. C₅₉	#2091	semidirect