Subgroups — #28505091

Order 15104.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422174	C₂	1	6	yes	maximal
2	C₅₉×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅₉⋊C₁₆	C₂²	1	4	yes	—
4	C₂×#97×C₅₉	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₅₉	C₂²	1	3	yes	—
6	C₅₉×#376	C₄	1	1	yes	characteristic
7	C₂×C₅₉⋊C₁₆	—	2	12	no	—
8	C₂²×C₈×C₅₉	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
11	#97×C₅₉	—	2	6	no	—
12	C₄×C₈×C₅₉	—	2	6	no	—
13	#95×C₅₉	C₈	1	4	yes	—
14	C₂²×C₄×C₅₉	C₂×C₈	1	6	yes	—
15	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₉	—	2	3	no	—
17	C₂×C₈×C₅₉	—	2	3	no	—
18	C₂×C₈×C₅₉	M₁₆	1	6	yes	—
19	C₂×C₈×C₅₉	#41	1	6	yes	—
20	C₄²×C₅₉	—	2	3	no	—
21	C₄²×C₅₉	—	2	3	no	—
22	C₅₉⋊C₁₆	—	4	4	no	—
23	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₉	—	2	6	no	—
25	C₂×C₄×C₅₉	—	2	12	no	—
26	C₂×C₄×C₅₉	#117	1	3	yes	—
27	C₂×C₄×C₅₉	C₂×M₁₆	1	3	yes	—
28	C₈×C₅₉	—	2	6	no	—
29	C₈×C₅₉	#126	1	4	yes	—
30	#12895	—	59	1	no	maximal, Sylow
31	C₂²×C₅₉	#431	1	3	yes	—
32	C₂²×C₅₉	#504	1	4	yes	—
33	C₄×C₅₉	—	4	6	no	—
34	C₄×C₅₉	#383	1	1	yes	characteristic
35	C₄×C₅₉	#432	1	3	yes	—
36	#1726	D₅₉	1	1	yes	characteristic
37	#1374	—	59	6	no	—
38	C₂×C₅₉	#1839	1	1	yes	characteristic
39	C₂×C₅₉	#2164	1	6	yes	—
40	C₂²×C₁₆	—	59	4	no	—
41	C₂×#97	Dic₅₉	1	3	yes	—
42	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	—
43	#376	Dic₅₉	1	1	yes	characteristic
44	C₅₉	#12895	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₈	C₂×Dic₅₉	1	3	yes	—
46	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
47	C₂×C₄²	C₂×Dic₅₉	1	3	yes	—
48	C₂×C₁₆	—	118	12	no	—
49	#97	—	2	6	no	—
50	C₄×C₈	—	2	6	no	—
51	#95	C₅₉⋊C₈	1	4	yes	—
52	C₂²×C₄	C₂×C₅₉⋊C₈	1	6	yes	—
53	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
54	C₂×C₈	—	2	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	#11745664	1	6	yes	—
57	C₂×C₈	#11745685	1	6	yes	—
58	C₄²	—	2	3	no	—
59	C₄²	—	2	3	no	—
60	C₁₆	—	236	4	no	—
61	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	#14273686	1	3	yes	—
65	C₂×C₄	C₂×#11745664	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	#14273721	1	4	yes	—
68	C₂²	#16837455	1	3	yes	—
69	C₂²	#16837819	1	4	yes	—
70	C₄	—	4	6	no	—
71	C₄	#16837270	1	1	yes	characteristic
72	C₄	#16837459	1	3	yes	—
73	C₂	#21424239	1	1	yes	characteristic
74	C₂	#21425903	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator, socle, prime radical	C₂³×C₅₉	C₂²×C₈
Fitting	C₅₉×#1726	C₂
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1726	D₅₉
hypocenter	C₅₉	#12895
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
44. C₅₉	#12895	semidirect

Maximal subgroups

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

subgroup	index
#21422174	2
C₅₉×#1726	2
#12895	59