Subgroups — #28509728

Order 15104.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422307	C₂	1	3	yes	maximal, complemented
2	C₂×C₅₉⋊C₆₄	C₂	1	3	yes	maximal, complemented
3	C₅₉×#2258	C₂	1	1	yes	maximal, characteristic
4	C₅₉⋊C₆₄	C₂²	1	1	yes	characteristic
5	C₅₉⋊C₆₄	C₂²	1	3	yes	—
6	C₅₉×#504	C₄	1	1	yes	characteristic
7	C₅₉×M₆₄	C₄	1	3	yes	—
8	C₂×C₃₂×C₅₉	C₂²	1	3	yes	—
9	#131×C₅₉	C₈	1	1	yes	characteristic
10	M₃₂×C₅₉	C₈	1	3	yes	—
11	C₂×C₁₆×C₅₉	C₂×C₄	1	3	yes	—
12	C₃₂×C₅₉	C₂×C₄	1	3	yes	—
13	C₃₂×C₅₉	C₂³	1	1	yes	characteristic
14	#41×C₅₉	C₁₆	1	1	yes	characteristic
15	M₁₆×C₅₉	C₁₆	1	3	yes	—
16	C₂×C₈×C₅₉	C₂×C₈	1	3	yes	—
17	C₁₆×C₅₉	C₂×C₈	1	3	yes	—
18	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
19	Q₈×C₅₉	C₃₂	1	1	yes	characteristic
20	D₄×C₅₉	C₃₂	1	3	yes	—
21	C₂×C₄×C₅₉	C₂×C₁₆	1	3	yes	—
22	C₈×C₅₉	C₂×C₁₆	1	3	yes	—
23	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
24	#13737	—	59	1	no	maximal, Sylow
25	C₂²×C₅₉	C₂×C₃₂	1	3	yes	—
26	C₄×C₅₉	C₂×C₃₂	1	3	yes	—
27	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
28	#2258	D₅₉	1	1	yes	characteristic
29	M₁₂₈	—	59	3	no	—
30	C₂×C₆₄	—	59	3	no	—
31	C₂×C₅₉	—	2	3	no	—
32	C₂×C₅₉	C₂²×C₃₂	1	1	yes	characteristic
33	#504	Dic₅₉	1	1	yes	characteristic
34	M₆₄	Dic₅₉	1	3	yes	—
35	C₂×C₃₂	C₂×D₅₉	1	3	yes	—
36	C₆₄	—	59	1	no	—
37	C₆₄	—	59	3	no	—
38	C₅₉	#13737	1	1	yes	characteristic, complemented, Sylow
39	#131	C₅₉⋊C₈	1	1	yes	characteristic
40	M₃₂	C₅₉⋊C₈	1	3	yes	—
41	C₂×C₁₆	C₂×Dic₅₉	1	3	yes	—
42	C₃₂	C₂×Dic₅₉	1	3	yes	—
43	C₃₂	C₂²×D₅₉	1	1	yes	characteristic
44	#41	C₅₉⋊C₁₆	1	1	yes	characteristic
45	M₁₆	C₅₉⋊C₁₆	1	3	yes	—
46	C₂×C₈	C₂×C₅₉⋊C₈	1	3	yes	—
47	C₁₆	C₂×C₅₉⋊C₈	1	3	yes	—
48	C₁₆	C₂²×Dic₅₉	1	1	yes	characteristic
49	Q₈	C₅₉⋊C₃₂	1	1	yes	characteristic
50	D₄	C₅₉⋊C₃₂	1	3	yes	—
51	C₂×C₄	C₂×C₅₉⋊C₁₆	1	3	yes	—
52	C₈	C₂×C₅₉⋊C₁₆	1	3	yes	—
53	C₈	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
54	C₂²	C₂×C₅₉⋊C₃₂	1	3	yes	—
55	C₄	C₂×C₅₉⋊C₃₂	1	3	yes	—
56	C₄	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₅₉⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₅₉
commutator, socle	C₂×C₅₉	C₂²×C₃₂
Fitting	C₅₉×#2258	C₂
prime radical	#41×C₅₉	C₁₆
cosocle	C₃₂×C₅₉	C₂³
hypercenter	#2258	D₅₉
hypocenter	C₅₉	#13737
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₃₂ ◁ #2258

Complements

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

N	H ≅ G/N	type
1. #21422307	C₂	semidirect
2. C₂×C₅₉⋊C₆₄	C₂	semidirect
38. C₅₉	#13737	semidirect

Maximal subgroups

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

subgroup	index
#21422307	2
C₂×C₅₉⋊C₆₄	2
C₅₉×#2258	2
#13737	59