Subgroups — #21426379

Order 7552.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837804	C₂	1	1	yes	maximal, characteristic
2	#16837229	C₂	1	2	yes	maximal
3	#16837227	C₂	1	2	yes	maximal
4	#16837178	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#500	C₂	1	1	yes	maximal, characteristic
6	#14273719	—	2	1	no	—
7	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
8	#14273661	C₂²	1	1	yes	characteristic
9	#14273661	C₂²	1	1	yes	characteristic
10	#14273659	—	2	1	no	—
11	C₄⋊Q₈×C₅₉	—	2	1	no	—
12	#107×C₅₉	—	2	1	no	—
13	#107×C₅₉	C₂²	1	2	yes	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₉	—	4	1	no	—
16	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
17	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₅₉	—	2	2	no	—
19	C₄⋊C₄×C₅₉	—	4	1	no	—
20	C₄²×C₅₉	D₄	1	1	yes	characteristic
21	C₄⋊Dic₅₉	—	2	1	no	—
22	C₄⋊Dic₅₉	—	2	1	no	—
23	Dic₅₉⋊C₄	—	4	1	no	—
24	Dic₅₉⋊C₄	—	4	1	no	—
25	C₅₉⋊C₁₆	Q₈	1	4	yes	complemented
26	Q₈×C₅₉	—	8	2	no	—
27	C₂×C₄×C₅₉	—	4	2	no	—
28	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
29	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₅₉	—	4	1	no	—
32	C₂×Dic₅₉	—	4	1	no	—
33	C₈×C₅₉	Q₁₆	1	2	yes	—
34	C₈×C₅₉	C₂×Q₈	1	2	yes	—
35	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
36	C₄×C₅₉	—	2	1	no	—
37	C₄×C₅₉	—	2	1	no	—
38	C₄×C₅₉	—	8	2	no	—
39	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
40	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
41	Dic₅₉	—	8	1	no	—
42	Dic₅₉	—	8	1	no	—
43	#2255	—	59	1	no	maximal, Sylow
44	C₂×C₅₉	#500	1	1	yes	characteristic
45	C₂×C₅₉	#509	1	1	yes	characteristic
46	C₂×C₅₉	#510	1	1	yes	characteristic
47	#500	D₅₉	1	1	yes	characteristic
48	#499	—	59	1	no	—
49	#367	—	59	2	no	—
50	#366	—	59	2	no	—
51	#346	—	59	1	no	—
52	C₅₉	#2255	1	1	yes	characteristic, complemented, Sylow
53	C₄⋊Q₈	—	2	1	no	—
54	#125	—	118	1	no	—
55	C₂×C₁₆	—	59	2	no	—
56	#107	—	2	1	no	—
57	#107	—	59	1	no	—
58	#107	—	59	1	no	—
59	#107	C₂×D₅₉	1	2	yes	—
60	#106	—	118	1	no	—
61	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
62	C₂×Q₈	—	4	1	no	—
63	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	118	1	no	—
68	C₄⋊C₄	—	118	1	no	—
69	C₄⋊C₄	—	236	1	no	—
70	C₄⋊C₄	—	236	1	no	—
71	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
72	C₁₆	—	59	4	no	—
73	Q₈	—	8	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	236	1	no	—
76	C₂×C₄	—	236	1	no	—
77	C₂×C₄	#11745669	1	1	yes	characteristic
78	C₂×C₄	#11745669	1	1	yes	characteristic
79	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
80	C₈	#11745672	1	2	yes	—
81	C₈	Q₈×D₅₉	1	2	yes	—
82	C₂²	C₂×#11745669	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	2	no	—
86	C₄	—	472	1	no	—
87	C₄	—	472	1	no	—
88	C₄	#14273725	1	1	yes	characteristic
89	C₄	C₂×#11745672	1	1	yes	characteristic
90	C₂	#16837805	1	1	yes	characteristic
91	C₂	#16837838	1	1	yes	characteristic
92	C₂	#16837843	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	#2255
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
25. C₅₉⋊C₁₆	Q₈	semidirect
52. C₅₉	#2255	semidirect