Subgroups — #45656287

Order 32576.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654282	C₂	1	2	yes	maximal
2	C₄⋊#16861332	C₂	1	1	yes	maximal, characteristic
3	C₄×#16861332	C₂	1	1	yes	maximal, characteristic
4	C₄×#16861332	C₂	1	2	yes	maximal
5	#30654180	C₂	1	1	yes	maximal, characteristic
6	C₂×#16861332	C₂²	1	2	yes	—
7	C₂×#16861332	C₂²	1	1	yes	characteristic
8	C₂×#16861332	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀₁₈	C₄	1	1	yes	characteristic
10	#22550840	C₄	1	2	yes	—
11	C₄×Dic₅₀₉	C₂²	1	1	yes	characteristic
12	C₄×Dic₅₀₉	C₂²	1	2	yes	—
13	C₄⋊C₄×C₅₀₉	C₄	1	1	yes	characteristic
14	#16861332	—	2	2	no	—
15	#16861332	—	2	1	no	—
16	#16861332	Q₈	1	2	yes	complemented
17	Dic₁₀₁₈	C₈	1	4	yes	complemented
18	C₂×Dic₅₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₅₀₉	C₂×C₄	1	2	yes	—
20	C₂×Dic₅₀₉	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₅₀₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₅₀₉	—	2	2	no	—
25	C₄×C₅₀₉	C₂×C₈	1	2	yes	—
26	Dic₅₀₉	—	2	1	no	—
27	Dic₅₀₉	C₂×C₈	1	4	yes	—
28	Dic₅₀₉	C₂×Q₈	1	1	yes	characteristic
29	Dic₅₀₉	#41	1	1	yes	characteristic
30	C₂×C₅₀₉	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₅₀₉	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₅₀₉	#131	1	1	yes	characteristic
33	C₅₀₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	509	1	no	maximal, Sylow
35	C₄×Q₈	—	509	1	no	—
36	C₄⋊C₈	—	509	1	no	—
37	C₄⋊C₈	—	509	2	no	—
38	C₄×C₈	—	509	1	no	—
39	C₄×C₈	—	509	2	no	—
40	C₂×Q₈	—	509	1	no	—
41	C₂×C₈	—	509	2	no	—
42	C₂×C₈	—	509	1	no	—
43	C₂×C₈	—	509	1	no	—
44	C₄⋊C₄	—	509	2	no	—
45	C₄⋊C₄	#14523266	1	1	yes	characteristic
46	C₄²	—	509	1	no	—
47	C₄²	—	509	2	no	—
48	Q₈	—	509	4	no	—
49	C₂×C₄	—	509	1	no	—
50	C₂×C₄	—	509	1	no	—
51	C₂×C₄	—	509	2	no	—
52	C₂×C₄	C₂×#14523266	1	2	yes	—
53	C₂×C₄	C₂×#14523266	1	1	yes	characteristic
54	C₈	—	1018	2	no	—
55	C₈	—	1018	1	no	—
56	C₈	—	509	2	no	—
57	C₂²	C₂²×#14523266	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	1018	1	no	—
60	C₄	—	509	1	no	—
61	C₄	—	509	1	no	—
62	C₄	—	509	4	no	—
63	C₄	D₅₀₉⋊C₈	1	2	yes	—
64	C₂	Q₈×#14523266	1	1	yes	characteristic
65	C₂	C₂×D₅₀₉⋊C₈	1	1	yes	characteristic
66	C₂	#30654302	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#14523266
commutator	C₂×C₅₀₉	C₂²×C₈
Fitting	C₄⋊C₄×C₅₀₉	C₄
socle, prime radical	C₂²×C₅₀₉	C₂²×C₄
cosocle	C₂×Dic₅₀₉	C₂³
hypercenter	C₄⋊C₄	#14523266
hypocenter	C₅₀₉	C₈×Q₈
LCS	G ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
16. #16861332	Q₈	semidirect
17. Dic₁₀₁₈	C₈	semidirect
33. C₅₀₉	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
#30654282	2
C₄⋊#16861332	2
C₄×#16861332	2
#30654180	2
C₈×Q₈	509