Subgroups — #45655286

Order 32576.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22550845	C₂	1	1	yes	maximal, characteristic
2	C₂×#22550845	C₂	1	1	yes	maximal, characteristic
3	#30654150	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₅₀₉	C₂	1	1	yes	maximal, characteristic
5	#22550845	—	2	1	no	—
6	#22550845	C₄	1	2	yes	—
7	#22550845	C₄	1	2	yes	—
8	#22550845	C₂²	1	2	yes	—
9	C₂×#16861325	—	2	1	no	—
10	C₂×#16861325	C₂²	1	2	yes	—
11	C₂²×C₄×C₅₀₉	C₂²	1	1	yes	characteristic
12	M₁₆×C₅₀₉	—	2	1	no	—
13	M₁₆×C₅₀₉	C₂²	1	2	yes	—
14	C₂×C₈×C₅₀₉	—	2	1	no	—
15	#16861325	—	2	2	no	—
16	#16861325	C₂×C₄	1	4	yes	—
17	C₂³×C₅₀₉	Q₈	1	1	yes	characteristic
18	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅₀₉	Q₈	1	1	yes	characteristic
23	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
24	C₈×C₅₀₉	—	2	2	no	—
25	C₂²×C₅₀₉	—	4	1	no	—
26	C₂²×C₅₀₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₅₀₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₅₀₉	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₅₀₉	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₅₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₄×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₅₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₅₀₉	—	2	1	no	—
34	C₂×C₅₀₉	—	2	1	no	—
35	C₂×C₅₀₉	—	2	1	no	—
36	C₂×C₅₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₅₀₉	#430	1	1	yes	characteristic, complemented, Sylow
38	#430	—	509	1	no	maximal, Sylow
39	C₂×M₁₆	—	509	1	no	—
40	C₂×M₁₆	—	509	1	no	—
41	C₂×M₁₆	D₅₀₉	1	1	yes	characteristic
42	#108	—	509	4	no	—
43	C₂²×C₄	C₂×D₅₀₉	1	1	yes	characteristic
44	M₁₆	—	2	1	no	—
45	M₁₆	—	1018	1	no	—
46	M₁₆	—	509	2	no	—
47	M₁₆	—	509	2	no	—
48	M₁₆	—	509	2	no	—
49	M₁₆	C₂×D₅₀₉	1	2	yes	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	1018	1	no	—
52	C₂×C₈	—	509	2	no	—
53	C₂³	Dic₁₀₁₈	1	1	yes	characteristic
54	C₂×C₄	C₄×D₅₀₉	1	1	yes	characteristic
55	C₂×C₄	C₄×D₅₀₉	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
58	C₂×C₄	Dic₁₀₁₈	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	1018	2	no	—
62	C₈	—	509	4	no	—
63	C₂²	—	4	1	no	—
64	C₂²	#22550840	1	1	yes	characteristic
65	C₂²	#22550840	1	1	yes	characteristic
66	C₂²	C₂×Dic₁₀₁₈	1	1	yes	characteristic
67	C₄	#22550840	1	1	yes	characteristic
68	C₄	#22550840	1	1	yes	characteristic
69	C₄	C₂×C₄×D₅₀₉	1	1	yes	characteristic
70	C₄	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×#22550840	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊D₅₀₉
commutator	C₄×C₅₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₂×M₁₆×C₅₀₉	C₂
socle	C₂×C₅₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₅₀₉	Q₈
cosocle	C₂×C₄×C₅₀₉	C₂³
hypercenter	C₂×M₁₆	D₅₀₉
hypocenter	C₅₀₉	#430
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #30654150	C₂	semidirect
37. C₅₀₉	#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22550845	2
#30654150	2
C₂×M₁₆×C₅₀₉	2
#430	509