Subgroups — #45655586

Order 32576.

118 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	C₄×#16861325	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₀₃₆	—	2	1	no	—
6	C₂×D₂₀₃₆	—	2	1	no	—
7	C₂×D₂₀₃₆	C₂²	1	2	yes	—
8	C₂×#16861325	C₂²	1	2	yes	—
9	C₂×Q₈×C₅₀₉	—	2	1	no	—
10	C₄⋊C₄×C₅₀₉	—	2	1	no	—
11	C₄⋊C₄×C₅₀₉	C₂²	1	2	yes	—
12	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
13	C₂²×D₅₀₉	—	2	2	no	—
14	D₂₀₃₆	—	2	4	no	—
15	D₂₀₃₆	—	4	2	no	—
16	D₂₀₃₆	—	4	2	no	—
17	#16861325	—	2	2	no	—
18	Q₈×C₅₀₉	—	4	2	no	—
19	C₂×C₄×C₅₀₉	—	2	2	no	—
20	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
23	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
24	C₂×D₅₀₉	—	4	2	no	—
25	C₂×D₅₀₉	—	4	2	no	—
26	C₂×D₅₀₉	—	4	2	no	—
27	C₄×C₅₀₉	—	4	2	no	—
28	C₄×C₅₀₉	D₈	1	2	yes	—
29	C₄×C₅₀₉	SD₁₆	1	2	yes	—
30	C₄×C₅₀₉	#41	1	2	yes	—
31	C₂×C₅₀₉	#124	1	1	yes	characteristic
32	C₂×C₅₀₉	C₂×D₈	1	1	yes	characteristic
33	C₂×C₅₀₉	C₂×SD₁₆	1	1	yes	characteristic
34	D₅₀₉	—	8	2	no	—
35	C₅₀₉	#486	1	1	yes	characteristic, complemented, Sylow
36	#486	—	509	1	no	maximal, Sylow
37	C₄⋊Q₈	D₅₀₉	1	1	yes	characteristic, complemented
38	#127	—	509	1	no	—
39	#102	—	509	4	no	—
40	C₄×C₈	—	509	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	1018	1	no	—
43	C₂×D₄	—	1018	1	no	—
44	C₂×D₄	—	509	2	no	—
45	C₂×C₈	—	509	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₅₀₉	1	2	yes	—
48	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
49	C₂³	—	1018	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	2036	2	no	—
52	D₄	—	2036	2	no	—
53	D₄	—	1018	4	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
58	C₈	—	1018	2	no	—
59	C₂²	—	2036	2	no	—
60	C₂²	—	2036	2	no	—
61	C₂²	—	2036	2	no	—
62	C₂²	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#22550846	1	2	yes	—
65	C₄	#22550848	1	2	yes	—
66	C₄	#22550862	1	2	yes	—
67	C₂	—	4072	2	no	—
68	C₂	#30654204	1	1	yes	characteristic
69	C₂	C₂×#22550846	1	1	yes	characteristic
70	C₂	C₂×#22550848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₀₉
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₄⋊Q₈×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂×D₄
prime radical	#30654212	C₂
hypercenter	C₄⋊Q₈	D₅₀₉
hypocenter	C₅₀₉	#486
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #30654135	C₂	semidirect
3. C₄×#16861325	C₂	semidirect
4. C₄⋊Q₈×C₅₀₉	C₂	semidirect
35. C₅₀₉	#486	semidirect
37. C₄⋊Q₈	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
C₄×#16861325	2
C₄⋊Q₈×C₅₀₉	2
#486	509