Subgroups — #45655149

Order 32576.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22550838	C₂	1	3	yes	maximal
2	#30654122	C₂	1	3	yes	maximal, complemented
3	C₂²×Q₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₅₀₉	C₂²	1	1	yes	characteristic
5	C₂²×Dic₅₀₉	C₂²	1	3	yes	—
6	#22550838	—	2	6	no	—
7	C₂×Q₈×C₅₀₉	—	2	6	no	—
8	C₂²×C₄×C₅₀₉	C₂²	1	3	yes	—
9	C₂²×D₅₀₉	—	2	3	no	—
10	C₂²×D₅₀₉	—	2	3	no	—
11	C₂²×D₅₀₉	Q₈	1	2	yes	complemented
12	C₂×Dic₅₀₉	—	2	3	no	—
13	C₂×Dic₅₀₉	—	2	6	no	—
14	C₂³×C₅₀₉	C₂³	1	1	yes	characteristic
15	Q₈×C₅₀₉	—	2	8	no	—
16	C₂×C₄×C₅₀₉	—	2	6	no	—
17	C₂×C₄×C₅₀₉	D₄	1	6	yes	—
18	C₂²×C₅₀₉	C₂×D₄	1	3	yes	—
19	C₂²×C₅₀₉	C₂×Q₈	1	1	yes	characteristic
20	C₂²×C₅₀₉	#41	1	3	yes	—
21	C₂×D₅₀₉	—	2	6	no	—
22	C₂×D₅₀₉	—	4	1	no	—
23	C₂×D₅₀₉	—	4	3	no	—
24	C₄×C₅₀₉	—	2	6	no	—
25	Dic₅₀₉	—	4	3	no	—
26	C₂×C₅₀₉	#120	1	1	yes	characteristic
27	C₂×C₅₀₉	C₂²⋊Q₈	1	3	yes	—
28	C₂×C₅₀₉	#124	1	3	yes	—
29	D₅₀₉	—	4	2	no	—
30	C₅₀₉	#393	1	1	yes	characteristic, complemented, Sylow
31	#393	—	509	1	no	maximal, Sylow
32	C₂²×Q₈	D₅₀₉	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₄	—	509	3	no	—
34	#95	—	509	3	no	—
35	C₂⁴	—	509	1	no	—
36	C₂×Q₈	—	2	6	no	—
37	C₂²×C₄	—	509	3	no	—
38	C₂²×C₄	C₂×D₅₀₉	1	3	yes	—
39	C₂²⋊C₄	—	1018	6	no	—
40	C₂³	—	1018	3	no	—
41	C₂³	—	1018	3	no	—
42	C₂³	—	509	2	no	—
43	C₂³	C₂²×D₅₀₉	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	1018	3	no	—
47	C₂×C₄	—	1018	6	no	—
48	C₂×C₄	C₂²⋊D₅₀₉	1	6	yes	—
49	C₂²	—	2036	1	no	—
50	C₂²	—	2036	3	no	—
51	C₂²	—	1018	6	no	—
52	C₂²	C₂×C₂²⋊D₅₀₉	1	3	yes	—
53	C₂²	Q₈×D₅₀₉	1	1	yes	characteristic
54	C₂²	#22550862	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	2036	3	no	—
57	C₂	—	2036	2	no	—
58	C₂	#30654184	1	1	yes	characteristic
59	C₂	#30654191	1	3	yes	—
60	C₂	#30654204	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₅₀₉
commutator, socle, cosocle	C₂³×C₅₀₉	C₂³
Fitting	C₂²×Q₈×C₅₀₉	C₂
prime radical	C₂³×D₅₀₉	C₂²
hypercenter	C₂²×Q₈	D₅₀₉
hypocenter	C₅₀₉	#393
LCS	G ▷ C₂³×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. #30654122	C₂	semidirect
3. C₂²×Q₈×C₅₀₉	C₂	semidirect
11. C₂²×D₅₀₉	Q₈	semidirect
30. C₅₀₉	#393	semidirect
32. C₂²×Q₈	D₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22550838	2
#30654122	2
C₂²×Q₈×C₅₀₉	2
#393	509