Subgroups — #45655615

Order 32576.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₀₇₂	C₂	1	4	yes	maximal, complemented
2	#30654212	C₂	1	2	yes	maximal, complemented
3	C₄×C₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₀₃₆	—	2	4	no	—
5	C₂×D₂₀₃₆	C₂²	1	4	yes	—
6	D₄₀₇₂	—	2	8	no	—
7	C₂×C₈×C₅₀₉	C₂²	1	2	yes	—
8	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
9	C₂²×D₅₀₉	—	2	4	no	—
10	D₂₀₃₆	—	2	8	no	—
11	D₂₀₃₆	—	4	8	no	—
12	C₂×C₄×C₅₀₉	D₄	1	2	yes	—
13	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
14	C₈×C₅₀₉	D₄	1	4	yes	complemented
15	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
16	C₂×D₅₀₉	—	4	4	no	—
17	C₂×D₅₀₉	—	4	8	no	—
18	C₄×C₅₀₉	D₈	1	4	yes	complemented
19	C₄×C₅₀₉	C₂×D₄	1	2	yes	—
20	C₂×C₅₀₉	#127	1	1	yes	characteristic
21	C₂×C₅₀₉	C₂×D₈	1	2	yes	—
22	D₅₀₉	—	8	4	no	—
23	C₅₀₉	#493	1	1	yes	characteristic, complemented, Sylow
24	#493	—	509	1	no	maximal, Sylow
25	C₂×D₈	—	509	4	no	—
26	#127	—	509	2	no	—
27	C₄×C₈	D₅₀₉	1	1	yes	characteristic, complemented
28	C₂×D₄	—	1018	4	no	—
29	C₂×D₄	—	509	4	no	—
30	D₈	—	1018	8	no	—
31	C₂×C₈	C₂×D₅₀₉	1	2	yes	—
32	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
33	C₂³	—	1018	4	no	—
34	D₄	—	2036	8	no	—
35	D₄	—	1018	8	no	—
36	C₂×C₄	D₂₀₃₆	1	2	yes	—
37	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
38	C₈	D₂₀₃₆	1	4	yes	complemented
39	C₂²	—	2036	4	no	—
40	C₂²	—	2036	8	no	—
41	C₂²	C₂×D₂₀₃₆	1	1	yes	characteristic
42	C₄	D₄₀₇₂	1	4	yes	complemented
43	C₄	C₂×D₂₀₃₆	1	2	yes	—
44	C₂	—	4072	4	no	—
45	C₂	#30654212	1	1	yes	characteristic
46	C₂	C₂×D₄₀₇₂	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₀₃₆
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₄×C₈×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂×D₄
hypercenter	C₄×C₈	D₅₀₉
hypocenter	C₅₀₉	#493
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₄₀₇₂	C₂	semidirect
2. #30654212	C₂	semidirect
3. C₄×C₈×C₅₀₉	C₂	semidirect
14. C₈×C₅₀₉	D₄	semidirect
18. C₄×C₅₀₉	D₈	semidirect
23. C₅₀₉	#493	semidirect
27. C₄×C₈	D₅₀₉	semidirect
38. C₈	D₂₀₃₆	semidirect
42. C₄	D₄₀₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₀₇₂	2
#30654212	2
C₄×C₈×C₅₀₉	2
#493	509