Subgroups — #45656253

Order 32576.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22550871	C₂	1	1	yes	maximal, characteristic
2	D₅₀₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#30654286	C₂	1	2	yes	maximal, complemented
4	#30654285	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₅₀₉	C₂	1	1	yes	maximal, characteristic
6	#22550871	—	2	1	no	—
7	#22550871	C₂²	1	1	yes	characteristic
8	#22550871	C₂²	1	2	yes	—
9	#22550871	C₂²	1	1	yes	characteristic
10	#22550870	—	2	1	no	—
11	D₅₀₉⋊C₈	—	2	1	no	—
12	C₂×#16861332	—	2	1	no	—
13	C₂×C₄×D₅₀₉	C₂²	1	1	yes	characteristic
14	#22550845	C₄	1	1	yes	characteristic
15	#22550844	C₄	1	2	yes	—
16	C₈×D₅₀₉	C₂²	1	2	yes	—
17	M₁₆×C₅₀₉	C₄	1	1	yes	characteristic
18	#16861332	—	2	2	no	—
19	#16861332	—	2	1	no	—
20	#16861332	—	2	1	no	—
21	C₂²×D₅₀₉	Q₈	1	1	yes	characteristic
22	C₂×Dic₅₀₉	D₄	1	1	yes	characteristic
23	C₄×D₅₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₅₀₉	D₄	1	1	yes	characteristic
25	C₄×D₅₀₉	Q₈	1	1	yes	characteristic
26	C₄×D₅₀₉	C₂³	1	1	yes	characteristic
27	#16861325	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
29	C₈×C₅₀₉	C₂×C₄	1	2	yes	—
30	C₂²×C₅₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×D₅₀₉	—	4	1	no	—
32	C₂×D₅₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×D₅₀₉	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₅₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
36	Dic₅₀₉	C₄⋊C₄	1	1	yes	characteristic
37	Dic₅₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₅₀₉	—	2	1	no	—
39	C₂×C₅₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	D₅₀₉	—	2	1	no	—
41	D₅₀₉	—	2	1	no	—
42	C₅₀₉	#430	1	1	yes	characteristic, complemented, Sylow
43	#430	—	509	1	no	maximal, Sylow
44	C₂×M₁₆	—	509	1	no	—
45	C₂×M₁₆	—	509	1	no	—
46	C₂×M₁₆	—	509	1	no	—
47	#108	—	509	2	no	—
48	#108	—	509	2	no	—
49	C₂²×C₄	—	509	1	no	—
50	M₁₆	—	1018	1	no	—
51	M₁₆	—	1018	1	no	—
52	M₁₆	—	509	1	no	—
53	M₁₆	—	509	2	no	—
54	M₁₆	—	509	1	no	—
55	M₁₆	—	509	1	no	—
56	M₁₆	—	509	2	no	—
57	M₁₆	#14523266	1	1	yes	characteristic
58	C₂×C₈	—	1018	1	no	—
59	C₂×C₈	—	1018	1	no	—
60	C₂×C₈	—	509	2	no	—
61	C₂³	—	509	1	no	—
62	C₂×C₄	—	509	1	no	—
63	C₂×C₄	—	509	1	no	—
64	C₂×C₄	—	509	1	no	—
65	C₂×C₄	—	509	1	no	—
66	C₂×C₄	—	509	1	no	—
67	C₂×C₄	C₂×#14523266	1	1	yes	characteristic
68	C₈	—	1018	2	no	—
69	C₈	—	1018	1	no	—
70	C₈	—	1018	1	no	—
71	C₈	—	509	2	no	—
72	C₈	C₂×#14523266	1	2	yes	—
73	C₂²	—	2036	1	no	—
74	C₂²	—	509	1	no	—
75	C₂²	—	509	1	no	—
76	C₂²	C₄⋊#14523266	1	1	yes	characteristic
77	C₄	—	509	1	no	—
78	C₄	—	509	1	no	—
79	C₄	C₄⋊#14523266	1	1	yes	characteristic
80	C₄	C₂²×#14523266	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	1018	1	no	—
83	C₂	—	1018	1	no	—
84	C₂	C₂×C₄⋊#14523266	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#14523266
commutator	C₄×C₅₀₉	C₂²×C₄
Frattini	C₄	C₂²×#14523266
Fitting	M₁₆×C₅₀₉	C₄
socle	C₂×C₅₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₅₀₉	Q₈
cosocle	C₄×D₅₀₉	C₂³
hypercenter	M₁₆	#14523266
hypocenter	C₅₀₉	#430
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #30654286	C₂	semidirect
4. #30654285	C₂	semidirect
42. 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×#22550871	2
D₅₀₉⋊M₁₆	2
#30654286	2
#30654285	2
M₁₆×D₅₀₉	2
#430	509