Subgroups — #28505089

Order 15104.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422174	C₂	1	6	yes	maximal
2	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅₉⋊C₁₆	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₅₉	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₅₉	C₂²	1	3	yes	—
6	C₄³×C₅₉	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	C₈	1	3	yes	—
11	C₂×C₄²×C₅₉	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
13	C₄×C₈×C₅₉	—	2	6	no	—
14	C₄×C₈×C₅₉	—	2	6	no	—
15	C₂²×C₄×C₅₉	C₂×C₈	1	3	yes	—
16	C₂²×C₄×C₅₉	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	—	2	3	no	—
19	C₂×C₈×C₅₉	—	2	3	no	—
20	C₂×C₈×C₅₉	M₁₆	1	6	yes	—
21	C₂×C₈×C₅₉	#41	1	6	yes	—
22	C₄²×C₅₉	—	2	6	no	—
23	C₄²×C₅₉	—	2	3	no	—
24	C₄²×C₅₉	—	2	3	no	—
25	C₄²×C₅₉	C₁₆	1	4	yes	complemented
26	C₅₉⋊C₁₆	—	4	4	no	—
27	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₅₉	—	2	3	no	—
29	C₂×C₄×C₅₉	—	2	3	no	—
30	C₂×C₄×C₅₉	—	2	3	no	—
31	C₂×C₄×C₅₉	—	2	3	no	—
32	C₂×C₄×C₅₉	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₅₉	M₃₂	1	6	yes	—
34	C₂×C₄×C₅₉	#117	1	3	yes	—
35	C₂×C₄×C₅₉	C₂×M₁₆	1	3	yes	—
36	C₈×C₅₉	—	2	6	no	—
37	C₈×C₅₉	#126	1	4	yes	—
38	#12894	—	59	1	no	maximal, Sylow
39	C₂²×C₅₉	#431	1	3	yes	—
40	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
41	C₂²×C₅₉	C₂×M₃₂	1	3	yes	—
42	C₄×C₅₉	—	2	6	no	—
43	C₄×C₅₉	—	2	6	no	—
44	C₄×C₅₉	#383	1	1	yes	characteristic
45	C₄×C₅₉	#432	1	3	yes	—
46	C₄²×C₈	D₅₉	1	1	yes	characteristic
47	#1374	—	59	6	no	—
48	C₂×C₅₉	#1839	1	1	yes	characteristic
49	C₂×C₅₉	#2162	1	3	yes	—
50	C₂×C₅₉	#2163	1	3	yes	—
51	C₂²×C₁₆	—	59	4	no	—
52	C₂×C₄×C₈	Dic₅₉	1	3	yes	—
53	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	—
54	C₄³	Dic₅₉	1	1	yes	characteristic
55	C₅₉	#12894	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	C₂×Dic₅₉	1	3	yes	—
57	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
58	C₂×C₄²	C₅₉⋊C₈	1	3	yes	—
59	C₂×C₄²	C₅₉⋊C₈	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₅₉	1	3	yes	—
61	C₂×C₁₆	—	118	12	no	—
62	C₄×C₈	—	2	6	no	—
63	C₄×C₈	—	2	6	no	—
64	C₂²×C₄	C₂×C₅₉⋊C₈	1	3	yes	—
65	C₂²×C₄	C₂×C₅₉⋊C₈	1	3	yes	—
66	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	#11745664	1	6	yes	—
70	C₂×C₈	#11745685	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	C₅₉⋊C₁₆	1	4	yes	complemented
75	C₁₆	—	236	4	no	—
76	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×C₅₉⋊C₁₆	1	6	yes	—
82	C₂×C₄	#14273667	1	6	yes	—
83	C₂×C₄	#14273686	1	3	yes	—
84	C₂×C₄	C₂×#11745664	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#14273721	1	4	yes	—
87	C₂²	#16837455	1	3	yes	—
88	C₂²	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
89	C₂²	C₂×#14273667	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#16837270	1	1	yes	characteristic
93	C₄	#16837459	1	3	yes	—
94	C₂	#21424239	1	1	yes	characteristic
95	C₂	#21425895	1	3	yes	—
96	C₂	#21425899	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂²×C₁₆
Fitting	C₄²×C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂²×C₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄²×C₈	D₅₉
hypocenter	C₅₉	#12894
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
25. C₄²×C₅₉	C₁₆	semidirect
55. C₅₉	#12894	semidirect
74. C₄²	C₅₉⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#21422174	2
C₄²×C₈×C₅₉	2
#12894	59