Subgroups — #45655604

Order 32576.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654212	C₂	1	1	yes	maximal, characteristic
2	#30654135	C₂	1	4	yes	maximal, complemented
3	#30654126	C₂	1	1	yes	maximal, characteristic, complemented
4	#125×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₀₃₆	—	2	2	no	—
6	C₂×D₂₀₃₆	C₂²	1	2	yes	—
7	C₂×#16861325	C₂²	1	2	yes	—
8	C₄⋊C₄×C₅₀₉	—	2	2	no	—
9	C₄⋊C₄×C₅₀₉	C₂²	1	2	yes	—
10	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
11	C₂²×D₅₀₉	—	2	2	no	—
12	D₂₀₃₆	—	2	4	no	—
13	D₂₀₃₆	—	4	4	no	—
14	#16861325	—	2	2	no	—
15	C₂×C₄×C₅₀₉	—	2	2	no	—
16	C₂×C₄×C₅₀₉	D₄	1	2	yes	—
17	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
18	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
19	C₂×D₅₀₉	—	4	2	no	—
20	C₂×D₅₀₉	—	4	4	no	—
21	C₄×C₅₀₉	—	2	2	no	—
22	C₄×C₅₀₉	—	4	2	no	—
23	C₄×C₅₀₉	#41	1	2	yes	—
24	C₂×C₅₀₉	#124	1	1	yes	characteristic
25	C₂×C₅₀₉	AGL(1,ℤ/8)	1	2	yes	—
26	D₅₀₉	—	8	2	no	—
27	C₅₀₉	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	509	1	no	maximal, Sylow
29	#127	—	509	1	no	—
30	#125	D₅₀₉	1	1	yes	characteristic, complemented
31	#102	—	509	4	no	—
32	#97	—	509	1	no	—
33	C₂×D₄	—	1018	2	no	—
34	C₂×D₄	—	509	2	no	—
35	C₂×C₈	—	509	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	C₂×D₅₀₉	1	2	yes	—
38	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
39	C₂³	—	1018	2	no	—
40	D₄	—	2036	4	no	—
41	D₄	—	1018	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂²⋊D₅₀₉	1	2	yes	—
44	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
45	C₈	—	1018	2	no	—
46	C₂²	—	2036	4	no	—
47	C₂²	—	2036	2	no	—
48	C₂²	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#22550862	1	2	yes	—
52	C₂	—	4072	2	no	—
53	C₂	#30654204	1	1	yes	characteristic
54	C₂	#30654240	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₀₉
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	#125×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂×D₄
prime radical	#30654212	C₂
hypercenter	#125	D₅₀₉
hypocenter	C₅₀₉	#490
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #30654135	C₂	semidirect
3. #30654126	C₂	semidirect
4. #125×C₅₀₉	C₂	semidirect
27. C₅₀₉	#490	semidirect
30. #125	D₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
#30654212	2
#30654135	2
#30654126	2
#125×C₅₀₉	2
#490	509