Subgroups — #45655614

Order 32576.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22550846	C₂	1	4	yes	maximal, complemented
2	#30654212	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#16861325	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₀₃₆	—	2	2	no	—
6	C₂×D₂₀₃₆	C₂²	1	2	yes	—
7	#22550846	—	2	8	no	—
8	C₂×#16861325	C₂²	1	2	yes	—
9	C₂×D₄×C₅₀₉	—	2	2	no	—
10	C₂×D₄×C₅₀₉	C₂²	1	2	yes	—
11	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
12	C₂²×D₅₀₉	—	2	2	no	—
13	D₂₀₃₆	—	2	4	no	—
14	D₂₀₃₆	—	4	4	no	—
15	#16861325	D₄	1	4	yes	complemented
16	C₂³×C₅₀₉	—	2	2	no	—
17	D₄×C₅₀₉	—	2	4	no	—
18	D₄×C₅₀₉	—	4	4	no	—
19	C₂×C₄×C₅₀₉	D₄	1	2	yes	—
20	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
21	C₂²×C₅₀₉	—	4	2	no	—
22	C₂²×C₅₀₉	—	4	4	no	—
23	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
24	C₂×D₅₀₉	—	4	2	no	—
25	C₂×D₅₀₉	—	4	4	no	—
26	C₄×C₅₀₉	D₈	1	4	yes	complemented
27	C₄×C₅₀₉	C₂×D₄	1	2	yes	—
28	C₂×C₅₀₉	—	8	2	no	—
29	C₂×C₅₀₉	#127	1	1	yes	characteristic
30	C₂×C₅₀₉	C₂×D₈	1	2	yes	—
31	D₅₀₉	—	8	2	no	—
32	C₅₀₉	#493	1	1	yes	characteristic, complemented, Sylow
33	#493	—	509	1	no	maximal, Sylow
34	C₂×D₈	—	509	4	no	—
35	#127	—	509	1	no	—
36	#127	D₅₀₉	1	1	yes	characteristic, complemented
37	C₄×C₈	—	509	1	no	—
38	C₂×D₄	—	2	2	no	—
39	C₂×D₄	—	1018	2	no	—
40	C₂×D₄	—	509	2	no	—
41	C₂×D₄	C₂×D₅₀₉	1	2	yes	—
42	D₈	—	1018	8	no	—
43	C₂×C₈	—	509	2	no	—
44	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
45	C₂³	—	2	2	no	—
46	C₂³	—	1018	2	no	—
47	D₄	—	2	4	no	—
48	D₄	—	4	4	no	—
49	D₄	—	2036	4	no	—
50	D₄	—	1018	4	no	—
51	C₂×C₄	C₂²⋊D₅₀₉	1	2	yes	—
52	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
53	C₈	—	509	4	no	—
54	C₂²	—	4	2	no	—
55	C₂²	—	4	4	no	—
56	C₂²	—	2036	2	no	—
57	C₂²	—	2036	4	no	—
58	C₂²	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
59	C₄	#22550846	1	4	yes	complemented
60	C₄	D₄×D₅₀₉	1	2	yes	—
61	C₂	—	8	2	no	—
62	C₂	—	4072	2	no	—
63	C₂	#30654211	1	1	yes	characteristic
64	C₂	C₂×#22550846	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₀₉
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	#127×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂×D₄
hypercenter	#127	D₅₀₉
hypocenter	C₅₀₉	#493
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#22550846	C₂	semidirect
2. #30654212	C₂	semidirect
3. C₄×#16861325	C₂	semidirect
4. #127×C₅₀₉	C₂	semidirect
15. #16861325	D₄	semidirect
26. C₄×C₅₀₉	D₈	semidirect
32. C₅₀₉	#493	semidirect
36. #127	D₅₀₉	semidirect
59. C₄	#22550846	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22550846	2
#30654212	2
C₄×#16861325	2
#127×C₅₀₉	2
#493	509