Subgroups — C₂×C₄⋊#16861325 #45655259

Order 32576.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16861325	C₂	1	2	yes	maximal
2	C₄⋊#16861325	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄²×C₅₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#16861325	—	2	4	no	—
5	C₂×#16861325	C₂²	1	4	yes	—
6	C₂²×C₄×C₅₀₉	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅₀₉	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₅₀₉	C₂²	1	1	yes	characteristic
9	C₄²×C₅₀₉	C₄	1	2	yes	—
10	C₄²×C₅₀₉	C₂²	1	2	yes	—
11	#16861325	—	2	4	no	—
12	C₂³×C₅₀₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₅₀₉	—	2	2	no	—
14	C₂×C₄×C₅₀₉	C₈	1	4	yes	complemented
15	C₂×C₄×C₅₀₉	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅₀₉	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₅₀₉	D₄	1	2	yes	—
19	C₂×C₄×C₅₀₉	Q₈	1	2	yes	—
20	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
21	C₂²×C₅₀₉	C₄⋊C₄	1	2	yes	—
22	C₂²×C₅₀₉	C₂×C₈	1	2	yes	—
23	C₂²×C₅₀₉	M₁₆	1	2	yes	—
24	C₂²×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₅₀₉	—	2	2	no	—
26	C₄×C₅₀₉	C₄⋊C₄	1	2	yes	—
27	C₄×C₅₀₉	C₂×C₈	1	4	yes	complemented
28	C₄×C₅₀₉	C₂×D₄	1	1	yes	characteristic
29	C₄×C₅₀₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₅₀₉	C₄⋊C₈	1	4	yes	complemented
31	C₂×C₅₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₅₀₉	C₂²×C₈	1	1	yes	characteristic
33	C₂×C₅₀₉	C₂×M₁₆	1	1	yes	characteristic
34	C₅₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₄⋊C₈	—	509	1	no	maximal, Sylow
36	C₂²×C₈	—	509	2	no	—
37	C₂×C₄²	D₅₀₉	1	1	yes	characteristic
38	C₄⋊C₈	—	509	4	no	—
39	C₂²×C₄	Dic₅₀₉	1	1	yes	characteristic
40	C₂²×C₄	Dic₅₀₉	1	1	yes	characteristic
41	C₂²×C₄	C₂×D₅₀₉	1	1	yes	characteristic
42	C₂×C₈	—	1018	4	no	—
43	C₂×C₈	—	509	4	no	—
44	C₄²	Dic₅₀₉	1	2	yes	—
45	C₄²	C₂×D₅₀₉	1	2	yes	—
46	C₂³	C₂×Dic₅₀₉	1	1	yes	characteristic
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	#16861325	1	4	yes	complemented
49	C₂×C₄	C₂×Dic₅₀₉	1	2	yes	—
50	C₂×C₄	C₂×Dic₅₀₉	1	1	yes	characteristic
51	C₂×C₄	C₂×Dic₅₀₉	1	2	yes	—
52	C₂×C₄	D₂₀₃₆	1	2	yes	—
53	C₂×C₄	Dic₁₀₁₈	1	2	yes	—
54	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
55	C₈	—	1018	4	no	—
56	C₂²	C₄⋊Dic₅₀₉	1	2	yes	—
57	C₂²	C₂×#16861325	1	2	yes	—
58	C₂²	#22550845	1	2	yes	—
59	C₂²	C₂²×Dic₅₀₉	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	C₄⋊Dic₅₀₉	1	2	yes	—
62	C₄	C₂×#16861325	1	4	yes	complemented
63	C₄	C₂×D₂₀₃₆	1	1	yes	characteristic
64	C₄	C₂×Dic₁₀₁₈	1	1	yes	characteristic
65	C₂	C₄⋊#16861325	1	4	yes	direct product factor
66	C₂	C₂×C₄⋊Dic₅₀₉	1	1	yes	characteristic
67	C₂	C₂²×#16861325	1	1	yes	characteristic
68	C₂	C₂×#22550845	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₀₉
commutator	C₂×C₅₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₂×C₄²×C₅₀₉	C₂
socle, prime radical	C₂³×C₅₀₉	C₂×C₄
cosocle	C₂×C₄×C₅₀₉	C₂³
hypercenter	C₂×C₄²	D₅₀₉
hypocenter	C₅₀₉	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄⋊#16861325	C₂	direct (N × H)
14. C₂×C₄×C₅₀₉	C₈	semidirect
27. C₄×C₅₀₉	C₂×C₈	semidirect
30. C₂×C₅₀₉	C₄⋊C₈	semidirect
34. C₅₀₉	C₂×C₄⋊C₈	semidirect
48. C₂×C₄	#16861325	semidirect
62. C₄	C₂×#16861325	semidirect
65. C₂	C₄⋊#16861325	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16861325	2
C₄⋊#16861325	2
C₂×C₄²×C₅₀₉	2
C₂×C₄⋊C₈	509

Subgroups — C2×C4⋊#16861325 #45655259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊#16861325 #45655259