Subgroups — #21423713

Order 7552.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837807	C₂	1	1	yes	maximal, characteristic
2	#16837805	C₂	1	2	yes	maximal
3	#16837781	C₂	1	2	yes	maximal
4	C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745672	—	2	4	no	—
7	#14273727	C₂²	1	2	yes	—
8	#14273661	—	2	2	no	—
9	#14273660	—	2	4	no	—
10	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
13	#107×C₅₉	—	2	2	no	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₉	—	2	2	no	—
16	C₂×Dic₁₁₈	—	2	2	no	—
17	C₂×C₈×C₅₉	D₄	1	2	yes	—
18	C₄⋊C₄×C₅₉	—	2	2	no	—
19	C₄⋊C₄×C₅₉	—	2	2	no	—
20	C₄²×C₅₉	C₂³	1	1	yes	characteristic
21	#11745672	—	4	8	no	—
22	C₄⋊Dic₅₉	—	2	2	no	—
23	C₄⋊Dic₅₉	—	2	2	no	—
24	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
25	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
26	Q₈×C₅₉	—	4	4	no	—
27	C₂×C₄×C₅₉	—	4	2	no	—
28	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₅₉	—	4	2	no	—
32	Dic₁₁₈	—	4	4	no	—
33	C₈×C₅₉	Q₁₆	1	4	yes	complemented
34	C₅₉⋊C₈	D₈	1	4	yes	—
35	C₅₉⋊C₈	Q₁₆	1	4	yes	complemented
36	C₂²×C₅₉	#127	1	1	yes	characteristic
37	C₄×C₅₉	—	8	2	no	—
38	C₄×C₅₉	C₂×D₈	1	2	yes	—
39	C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
40	C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
41	Dic₅₉	—	8	2	no	—
42	#1713	—	59	1	no	maximal, Sylow
43	C₂×C₅₉	#493	1	1	yes	characteristic
44	C₂×C₅₉	#494	1	1	yes	characteristic
45	C₂×C₅₉	#494	1	1	yes	characteristic
46	#500	—	59	2	no	—
47	#500	—	59	1	no	—
48	#500	D₅₉	1	1	yes	characteristic
49	#494	—	59	2	no	—
50	C₈²	—	59	1	no	—
51	C₅₉	#1713	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₁₆	—	118	4	no	—
53	C₄⋊Q₈	—	59	2	no	—
54	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
55	#107	—	2	2	no	—
56	#107	—	118	4	no	—
57	#107	—	118	2	no	—
58	C₄×C₈	—	59	1	no	—
59	C₄×C₈	—	59	1	no	—
60	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	118	2	no	—
63	Q₁₆	—	236	8	no	—
64	C₂×C₈	—	59	2	no	—
65	C₂×C₈	—	59	2	no	—
66	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	118	2	no	—
70	C₄⋊C₄	—	118	2	no	—
71	C₄²	C₂²×D₅₉	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	236	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	236	2	no	—
76	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
77	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
79	C₈	—	59	4	no	—
80	C₈	—	59	4	no	—
81	C₈	#11745672	1	4	yes	complemented
82	C₂²	#14273723	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	472	2	no	—
85	C₄	D₈×D₅₉	1	2	yes	—
86	C₄	Q₁₆×D₅₉	1	2	yes	—
87	C₄	C₂×#11745672	1	2	yes	—
88	C₂	#16837777	1	1	yes	characteristic
89	C₂	#16837780	1	1	yes	characteristic
90	C₂	#16837781	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14273723
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#500	C₂
socle	C₂²×C₅₉	#127
hypercenter	#500	D₅₉
hypocenter	C₅₉	#1713
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
33. C₈×C₅₉	Q₁₆	semidirect
35. C₅₉⋊C₈	Q₁₆	semidirect
51. C₅₉	#1713	semidirect
81. C₈	#11745672	semidirect