Subgroups — #28509727

Order 15104.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21426389	C₂	1	1	yes	maximal, characteristic
2	#21422307	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422306	C₂	1	2	yes	maximal, complemented
4	C₆₄×D₅₉	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₉×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837818	C₄	1	1	yes	characteristic
7	#16837236	C₄	1	1	yes	characteristic
8	#16837235	C₄	1	2	yes	—
9	C₃₂×D₅₉	C₂²	1	2	yes	—
10	C₅₉⋊C₆₄	C₂²	1	2	yes	—
11	C₂×C₃₂×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×C₆₄	C₂²	1	1	yes	characteristic
13	C₅₉×C₆₄	C₂²	1	1	yes	characteristic
14	#14273733	C₈	1	1	yes	characteristic
15	#14273667	C₈	1	1	yes	characteristic
16	#14273666	C₈	1	2	yes	—
17	C₁₆×D₅₉	C₂×C₄	1	2	yes	—
18	C₅₉⋊C₃₂	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₃₂×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₃₂×C₅₉	C₂³	1	1	yes	characteristic
22	#11745685	C₁₆	1	1	yes	characteristic
23	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
24	#11745664	C₁₆	1	1	yes	characteristic
25	#11745659	C₁₆	1	2	yes	—
26	C₈×D₅₉	C₂×C₈	1	2	yes	—
27	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
28	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
29	C₅₉⋊C₁₆	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
31	C₂²⋊D₅₉	C₃₂	1	2	yes	—
32	D₂₃₆	C₃₂	1	1	yes	characteristic
33	C₄×D₅₉	C₂×C₁₆	1	2	yes	—
34	Dic₁₁₈	C₃₂	1	1	yes	characteristic
35	C₈×C₅₉	C₂×C₁₆	1	1	yes	characteristic
36	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
37	C₅₉⋊C₈	C₂×C₁₆	1	2	yes	—
38	#13737	—	59	1	no	maximal, Sylow
39	C₂²×C₅₉	C₂×C₃₂	1	1	yes	characteristic
40	C₂×D₅₉	C₂×C₃₂	1	2	yes	—
41	C₄×C₅₉	C₂×C₃₂	1	1	yes	characteristic
42	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
43	Dic₅₉	C₂×C₃₂	1	2	yes	—
44	#2258	—	59	1	no	—
45	M₁₂₈	—	59	1	no	—
46	M₁₂₈	—	59	2	no	—
47	C₂×C₆₄	—	59	2	no	—
48	C₂×C₆₄	D₅₉	1	1	yes	characteristic, complemented
49	C₂×C₅₉	—	2	1	no	—
50	C₂×C₅₉	C₂²×C₃₂	1	1	yes	characteristic
51	D₅₉	—	2	2	no	—
52	#504	—	59	1	no	—
53	M₆₄	—	59	1	no	—
54	M₆₄	—	59	2	no	—
55	C₂×C₃₂	—	59	2	no	—
56	C₂×C₃₂	C₂×D₅₉	1	1	yes	characteristic
57	C₆₄	—	59	2	no	—
58	C₆₄	C₂×D₅₉	1	1	yes	characteristic
59	C₆₄	C₂×D₅₉	1	1	yes	characteristic
60	C₅₉	#13737	1	1	yes	characteristic, complemented, Sylow
61	#131	—	59	1	no	—
62	M₃₂	—	59	1	no	—
63	M₃₂	—	59	2	no	—
64	C₂×C₁₆	—	59	2	no	—
65	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
66	C₃₂	—	59	2	no	—
67	C₃₂	C₄×D₅₉	1	1	yes	characteristic
68	C₃₂	C₂²×D₅₉	1	1	yes	characteristic
69	#41	—	59	1	no	—
70	M₁₆	—	59	1	no	—
71	M₁₆	—	59	2	no	—
72	C₂×C₈	—	59	2	no	—
73	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
74	C₁₆	—	59	2	no	—
75	C₁₆	C₈×D₅₉	1	1	yes	characteristic
76	C₁₆	C₂×C₄×D₅₉	1	1	yes	characteristic
77	Q₈	—	59	1	no	—
78	D₄	—	59	2	no	—
79	D₄	—	59	1	no	—
80	C₂×C₄	—	59	2	no	—
81	C₂×C₄	C₁₆×D₅₉	1	1	yes	characteristic
82	C₈	—	59	2	no	—
83	C₈	C₁₆×D₅₉	1	1	yes	characteristic
84	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
85	C₂²	—	59	2	no	—
86	C₂²	C₃₂×D₅₉	1	1	yes	characteristic
87	C₄	—	59	2	no	—
88	C₄	C₃₂×D₅₉	1	1	yes	characteristic
89	C₄	C₂×C₁₆×D₅₉	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	118	2	no	—
92	C₂	C₂×C₃₂×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₆₄	C₂×D₅₉
commutator, socle	C₂×C₅₉	C₂²×C₃₂
Frattini	C₃₂	C₂²×D₅₉
Fitting	C₂×C₅₉×C₆₄	C₂
prime radical	#11745685	C₁₆
cosocle	C₃₂×C₅₉	C₂³
hypercenter	C₂×C₆₄	D₅₉
hypocenter	C₅₉	#13737
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₆₄ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #21422307	C₂	semidirect
3. #21422306	C₂	semidirect
4. C₆₄×D₅₉	C₂	semidirect
5. C₂×C₅₉×C₆₄	C₂	semidirect
48. C₂×C₆₄	D₅₉	semidirect
60. C₅₉	#13737	semidirect

Maximal subgroups

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

subgroup	index
#21426389	2
#21422307	2
#21422306	2
C₆₄×D₅₉	2
C₂×C₅₉×C₆₄	2
#13737	59