Subgroups — #28482456

Order 15104.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424253	C₂	1	6	yes	maximal
2	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273659	—	2	12	no	—
4	#16837272	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₅₉	C₂²	1	3	yes	—
6	C₈²×C₅₉	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₅₉	—	2	12	no	—
8	#14273659	—	4	24	no	—
9	C₂²×C₈×C₅₉	D₄	1	6	yes	—
10	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
11	C₄×C₈×C₅₉	C₂×C₄	1	6	yes	—
12	C₄×C₈×C₅₉	Q₈	1	6	yes	—
13	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
14	C₂²×Dic₅₉	—	4	4	no	—
15	C₂×C₈×C₅₉	C₄⋊C₄	1	12	yes	—
16	C₂×C₈×C₅₉	SD₁₆	1	24	yes	—
17	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
18	C₄²×C₅₉	C₂×Q₈	1	3	yes	—
19	C₄⋊Dic₅₉	—	4	24	no	—
20	C₂³×C₅₉	#127	1	1	yes	characteristic
21	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	3	yes	—
22	C₂×C₄×C₅₉	C₄⋊Q₈	1	3	yes	—
23	C₂×C₄×C₅₉	C₂×SD₁₆	1	12	yes	—
24	C₂×Dic₅₉	—	8	12	no	—
25	C₈×C₅₉	#106	1	24	yes	complemented
26	#8813	—	59	1	no	maximal, Sylow
27	C₂²×C₅₉	#384	1	1	yes	characteristic
28	C₂²×C₅₉	#492	1	6	yes	—
29	C₄×C₅₉	C₂×#106	1	6	yes	—
30	C₄×C₅₉	#498	1	6	yes	—
31	Dic₅₉	—	16	4	no	—
32	#1844	—	59	6	no	—
33	C₂×C₈²	D₅₉	1	1	yes	characteristic
34	C₂×C₅₉	#1705	1	4	yes	—
35	C₂×C₅₉	#1844	1	3	yes	—
36	C₂×#106	—	118	12	no	—
37	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	—
38	#384	—	59	4	no	—
39	C₈²	Dic₅₉	1	4	yes	complemented
40	C₅₉	#8813	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₈	D₂₃₆	1	6	yes	—
42	C₂×C₄⋊C₄	—	118	12	no	—
43	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
44	#106	—	236	24	no	—
45	C₄×C₈	Dic₁₁₈	1	6	yes	—
46	C₄×C₈	C₂×Dic₅₉	1	6	yes	—
47	C₂²×C₄	—	236	4	no	—
48	C₂²×C₄	C₂×D₂₃₆	1	3	yes	—
49	C₂×C₈	#11745660	1	24	yes	—
50	C₂×C₈	C₄⋊Dic₅₉	1	12	yes	—
51	C₄⋊C₄	—	236	24	no	—
52	C₄²	C₂×Dic₁₁₈	1	3	yes	—
53	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
54	C₂³	#14273724	1	1	yes	characteristic
55	C₂×C₄	—	472	12	no	—
56	C₂×C₄	C₂×C₄⋊Dic₅₉	1	3	yes	—
57	C₂×C₄	#14273727	1	3	yes	—
58	C₂×C₄	C₂×#11745660	1	12	yes	—
59	C₈	#14273659	1	24	yes	complemented
60	C₂²	#16837272	1	1	yes	characteristic
61	C₂²	#16837775	1	6	yes	—
62	C₄	—	944	4	no	—
63	C₄	C₂×#14273659	1	6	yes	—
64	C₄	#16837800	1	6	yes	—
65	C₂	#21423683	1	4	yes	—
66	C₂	#21424253	1	3	yes	—

	subgroup	G/H
center	C₂³	#14273724
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₈²×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈²	D₅₉
hypocenter	C₅₉	#8813
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
6. C₈²×C₅₉	C₄	semidirect
25. C₈×C₅₉	#106	semidirect
39. C₈²	Dic₅₉	semidirect
40. C₅₉	#8813	semidirect
59. C₈	#14273659	semidirect

Maximal subgroups

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

subgroup	index
#21424253	2
C₂×C₈²×C₅₉	2
#8813	59