Subgroups — #28476343

Order 15104.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₅₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
2	#21422168	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2252	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉⋊C₃₂	—	2	1	no	—
5	#16837155	—	2	1	no	—
6	C₅₉×#500	—	2	1	no	—
7	C₅₉×#366	—	2	1	no	—
8	C₅₉×#366	C₄	1	2	yes	—
9	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
11	C₅₉⋊C₃₂	—	4	1	no	—
12	C₄⋊Q₈×C₅₉	—	4	1	no	—
13	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
15	#107×C₅₉	—	2	2	no	—
16	#107×C₅₉	—	4	1	no	—
17	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₅₉	—	8	1	no	—
19	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
21	C₄⋊C₄×C₅₉	—	4	2	no	—
22	C₄⋊C₄×C₅₉	—	8	1	no	—
23	C₄²×C₅₉	D₈	1	1	yes	characteristic
24	C₂×C₅₉⋊C₈	—	8	1	no	—
25	C₁₆×C₅₉	—	2	1	no	—
26	C₁₆×C₅₉	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₅₉	Q₁₆	1	1	yes	characteristic
28	Q₈×C₅₉	—	16	2	no	—
29	C₂×C₄×C₅₉	—	8	2	no	—
30	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
31	C₂×C₄×C₅₉	D₁₆	1	1	yes	characteristic
32	C₂×C₄×C₅₉	SD₃₂	1	1	yes	characteristic
33	C₈×C₅₉	—	2	1	no	—
34	C₈×C₅₉	#101	1	1	yes	characteristic
35	C₈×C₅₉	#103	1	1	yes	characteristic
36	C₅₉⋊C₈	—	16	1	no	—
37	#7451	—	59	1	no	maximal, Sylow
38	C₂²×C₅₉	#357	1	1	yes	characteristic
39	C₄×C₅₉	—	2	1	no	—
40	C₄×C₅₉	—	16	2	no	—
41	C₄×C₅₉	#332	1	1	yes	characteristic
42	C₄×C₅₉	#362	1	1	yes	characteristic
43	C₄×C₅₉	SD₆₄	1	1	yes	characteristic
44	C₄×C₅₉	Q₆₄	1	1	yes	characteristic
45	#2252	D₅₉	1	1	yes	characteristic
46	C₄⋊C₃₂	—	59	1	no	—
47	#1371	—	59	1	no	—
48	C₂×C₅₉	#1364	1	1	yes	characteristic
49	C₂×C₅₉	#1416	1	1	yes	characteristic
50	C₂×C₅₉	#1419	1	1	yes	characteristic
51	#500	—	2	1	no	—
52	C₂×C₃₂	—	118	1	no	—
53	#366	—	2	1	no	—
54	#366	Dic₅₉	1	2	yes	—
55	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
56	#335	—	118	1	no	—
57	C₅₉	#7451	1	1	yes	characteristic, complemented, Sylow
58	C₄⋊Q₈	—	4	1	no	—
59	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
60	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
61	#107	—	2	2	no	—
62	#107	—	4	1	no	—
63	C₄⋊C₈	—	236	1	no	—
64	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
65	C₃₂	—	236	1	no	—
66	C₂×Q₈	—	8	1	no	—
67	C₂×C₈	—	472	1	no	—
68	C₂×C₈	#11745670	1	1	yes	characteristic
69	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
70	C₄⋊C₄	—	4	2	no	—
71	C₄⋊C₄	—	8	1	no	—
72	C₄²	#11745669	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₁₆	#11745671	1	1	yes	characteristic
75	C₁₆	#11745672	1	1	yes	characteristic
76	Q₈	—	16	2	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	#14273648	1	1	yes	characteristic
79	C₂×C₄	#14273668	1	1	yes	characteristic
80	C₂×C₄	#14273671	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	944	1	no	—
83	C₈	#14273646	1	1	yes	characteristic
84	C₈	#14273651	1	1	yes	characteristic
85	C₂²	#16837203	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	16	2	no	—
88	C₄	#16837148	1	1	yes	characteristic
89	C₄	#16837218	1	1	yes	characteristic
90	C₄	#16837239	1	1	yes	characteristic
91	C₄	#16837242	1	1	yes	characteristic
92	C₂	#21422151	1	1	yes	characteristic
93	C₂	#21422277	1	1	yes	characteristic
94	C₂	#21422286	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837203
commutator	C₂×C₁₆×C₅₉	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₅₉
Fitting	C₅₉×#2252	C₂
socle, prime radical	C₂²×C₅₉	#357
cosocle	C₄×C₁₆×C₅₉	C₂²
hypercenter	#2252	D₅₉
hypocenter	C₅₉	#7451
LCS	G ▷ C₂×C₁₆×C₅₉ ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

Complements

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

N	H ≅ G/N	type
57. C₅₉	#7451	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄⋊(C₅₉⋊C₃₂)	2
#21422168	2
C₅₉×#2252	2
#7451	59