Subgroups — C₂×#30654290 #45656295

Order 32576.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654290	C₂	1	4	yes	maximal, direct product factor
2	C₂×#22550864	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×D₅₀₉	C₂	1	1	yes	maximal, characteristic
4	#22550864	C₂²	1	4	yes	complemented
5	C₂×C₄×D₅₀₉	C₄	1	1	yes	characteristic
6	C₂×#16861325	C₂²	1	1	yes	characteristic
7	C₈×D₅₀₉	C₄	1	2	yes	—
8	C₈×D₅₀₉	C₂²	1	2	yes	—
9	C₂×C₈×C₅₀₉	C₄	1	1	yes	characteristic
10	C₂²×D₅₀₉	C₈	1	1	yes	characteristic
11	C₂×Dic₅₀₉	C₈	1	1	yes	characteristic
12	C₄×D₅₀₉	C₈	1	2	yes	—
13	C₄×D₅₀₉	C₂×C₄	1	2	yes	—
14	#16861325	C₂×C₄	1	1	yes	characteristic
15	#16861325	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
17	C₈×C₅₀₉	C₂×C₄	1	2	yes	—
18	C₂²×C₅₀₉	C₂×C₈	1	1	yes	characteristic
19	C₂×D₅₀₉	—	2	2	no	—
20	C₂×D₅₀₉	C₂×C₈	1	2	yes	—
21	C₄×C₅₀₉	C₂×C₈	1	1	yes	characteristic
22	C₄×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
23	Dic₅₀₉	C₂×C₈	1	2	yes	—
24	C₂×C₅₀₉	M₃₂	1	2	yes	complemented
25	C₂×C₅₀₉	C₂²×C₈	1	1	yes	characteristic
26	D₅₀₉	—	2	2	no	—
27	C₅₀₉	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
28	C₂×M₃₂	—	509	1	no	maximal, Sylow
29	C₂²×C₈	—	509	1	no	—
30	M₃₂	—	509	4	no	—
31	C₂×C₁₆	—	509	2	no	—
32	C₂²×C₄	—	509	1	no	—
33	C₂×C₈	—	509	1	no	—
34	C₂×C₈	—	509	2	no	—
35	C₂×C₈	—	509	2	no	—
36	C₂×C₈	#14523266	1	1	yes	characteristic
37	C₁₆	—	509	4	no	—
38	C₂³	—	509	1	no	—
39	C₂×C₄	—	509	1	no	—
40	C₂×C₄	—	509	2	no	—
41	C₂×C₄	—	509	2	no	—
42	C₂×C₄	C₂×#14523266	1	1	yes	characteristic
43	C₈	—	509	1	no	—
44	C₈	—	509	1	no	—
45	C₈	C₂×#14523266	1	2	yes	—
46	C₂²	—	1018	2	no	—
47	C₂²	—	509	2	no	—
48	C₂²	D₅₀₉⋊C₈	1	1	yes	characteristic
49	C₄	—	509	2	no	—
50	C₄	D₅₀₉⋊C₈	1	1	yes	characteristic
51	C₄	C₂²×#14523266	1	1	yes	characteristic
52	C₂	—	1018	2	no	—
53	C₂	#30654290	1	2	yes	direct product factor
54	C₂	C₂×D₅₀₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#14523266
commutator	C₂×C₅₀₉	C₂²×C₈
Frattini	C₄	C₂²×#14523266
Fitting	C₂×C₈×C₅₀₉	C₄
socle	C₂²×C₅₀₉	C₂×C₈
prime radical	C₂²×D₅₀₉	C₈
cosocle	#16861325	C₂³
hypercenter	C₂×C₈	#14523266
hypocenter	C₅₀₉	C₂×M₃₂
LCS	G ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #30654290	C₂	direct (N × H)
2. C₂×#22550864	C₂	semidirect
4. #22550864	C₂²	semidirect
24. C₂×C₅₀₉	M₃₂	semidirect
27. C₅₀₉	C₂×M₃₂	semidirect
53. C₂	#30654290	direct (N × H)

Maximal subgroups

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

subgroup	index
#30654290	2
C₂×#22550864	2
C₂×C₈×D₅₀₉	2
C₂×M₃₂	509

Subgroups — C2×#30654290 #45656295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30654290 #45656295