Subgroups — #21426369

Order 7552.

128 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	#16837227	C₂	1	4	yes	maximal
3	C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#500	C₂	1	1	yes	maximal, characteristic
5	#14273727	—	2	1	no	—
6	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
7	#14273661	—	2	1	no	—
8	#14273661	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₅₉	—	2	1	no	—
10	#107×C₅₉	—	2	1	no	—
11	#107×C₅₉	C₂²	1	2	yes	—
12	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₅₉	—	4	1	no	—
14	C₂×Dic₁₁₈	—	4	1	no	—
15	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
16	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₅₉	—	2	2	no	—
18	C₄⋊C₄×C₅₉	—	4	1	no	—
19	C₄²×C₅₉	D₄	1	1	yes	characteristic
20	C₄⋊Dic₅₉	—	2	2	no	—
21	C₄⋊Dic₅₉	—	4	1	no	—
22	C₅₉⋊C₁₆	Q₈	1	4	yes	complemented
23	Q₈×C₅₉	—	8	2	no	—
24	C₂×C₄×C₅₉	—	4	2	no	—
25	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
26	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
27	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₅₉	—	4	2	no	—
29	Dic₁₁₈	—	8	2	no	—
30	C₈×C₅₉	Q₁₆	1	2	yes	—
31	C₈×C₅₉	C₂×Q₈	1	2	yes	—
32	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
33	C₄×C₅₉	—	8	2	no	—
34	C₄×C₅₉	D₁₆	1	2	yes	—
35	C₄×C₅₉	Q₃₂	1	2	yes	—
36	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₅₉	—	8	2	no	—
39	#2252	—	59	1	no	maximal, Sylow
40	C₂×C₅₉	#500	1	1	yes	characteristic
41	C₂×C₅₉	C₂×D₁₆	1	1	yes	characteristic
42	C₂×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
43	#500	—	59	1	no	—
44	#500	D₅₉	1	1	yes	characteristic
45	#366	—	59	4	no	—
46	C₄×C₁₆	—	59	1	no	—
47	C₅₉	#2252	1	1	yes	characteristic, complemented, Sylow
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	118	1	no	—
50	C₂×C₁₆	—	59	2	no	—
51	#107	—	2	1	no	—
52	#107	—	59	2	no	—
53	#107	—	118	1	no	—
54	#107	C₂×D₅₉	1	2	yes	—
55	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	236	1	no	—
58	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	118	2	no	—
63	C₄⋊C₄	—	236	1	no	—
64	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
65	C₁₆	—	59	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	472	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	236	2	no	—
70	C₂×C₄	#11745669	1	1	yes	characteristic
71	C₂×C₄	#11745669	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
73	C₈	#11745672	1	2	yes	—
74	C₈	Q₈×D₅₉	1	2	yes	—
75	C₂²	C₂×#11745669	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	472	2	no	—
78	C₄	#14273668	1	2	yes	—
79	C₄	#14273673	1	2	yes	—
80	C₄	#14273725	1	1	yes	characteristic
81	C₄	C₂×#11745672	1	1	yes	characteristic
82	C₂	#16837805	1	1	yes	characteristic
83	C₂	C₂×#14273668	1	1	yes	characteristic
84	C₂	C₂×#14273673	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#500	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#500	D₅₉
hypocenter	C₅₉	#2252
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
22. C₅₉⋊C₁₆	Q₈	semidirect
47. C₅₉	#2252	semidirect