Subgroups — #45655521

Order 32576.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654214	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₅₀₉	C₂	1	1	yes	maximal, characteristic
3	#30654147	C₂	1	1	yes	maximal, characteristic
4	#30654146	C₂	1	1	yes	maximal, characteristic
5	#30654144	C₂	1	1	yes	maximal, characteristic
6	#30654139	C₂	1	1	yes	maximal, characteristic
7	#103×C₅₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀₁₈	—	2	1	no	—
9	C₂×#16861325	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₅₀₉	—	2	1	no	—
11	C₄⋊Dic₅₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₅₀₉	C₂²	1	1	yes	characteristic
13	#22550840	—	2	1	no	—
14	C₄×Dic₅₀₉	—	2	1	no	—
15	C₄×Dic₅₀₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₅₀₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₅₀₉	C₂²	1	1	yes	characteristic
19	Dic₁₀₁₈	—	4	2	no	—
20	C₂×Dic₅₀₉	—	2	1	no	—
21	C₂×Dic₅₀₉	—	2	1	no	—
22	C₂×Dic₅₀₉	D₄	1	1	yes	characteristic
23	C₂×Dic₅₀₉	D₄	1	1	yes	characteristic
24	#16861325	—	2	1	no	—
25	Q₈×C₅₀₉	—	2	1	no	—
26	Q₈×C₅₀₉	Q₈	1	2	yes	complemented
27	C₂×C₄×C₅₀₉	—	2	1	no	—
28	C₂×C₄×C₅₀₉	—	2	1	no	—
29	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
30	C₈×C₅₀₉	—	2	1	no	—
31	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅₀₉	—	2	2	no	—
33	C₄×C₅₀₉	—	4	1	no	—
34	C₄×C₅₀₉	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₅₀₉	#41	1	1	yes	characteristic
36	Dic₅₀₉	—	2	1	no	—
37	Dic₅₀₉	—	4	1	no	—
38	Dic₅₀₉	—	4	1	no	—
39	Dic₅₀₉	SD₁₆	1	2	yes	—
40	C₂×C₅₀₉	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₅₀₉	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₅₀₉	#137	1	1	yes	characteristic
43	C₅₀₉	#475	1	1	yes	characteristic, complemented, Sylow
44	#475	—	509	1	no	maximal, Sylow
45	C₄⋊Q₈	—	509	1	no	—
46	C₄×Q₈	—	509	1	no	—
47	#106	—	509	1	no	—
48	#106	—	509	1	no	—
49	C₄⋊C₈	—	509	1	no	—
50	#103	—	509	1	no	—
51	#103	D₅₀₉	1	1	yes	characteristic
52	C₂×Q₈	—	1018	1	no	—
53	C₂×Q₈	C₂×D₅₀₉	1	1	yes	characteristic
54	C₂×C₈	—	509	1	no	—
55	C₂×C₈	C₂×D₅₀₉	1	1	yes	characteristic
56	C₄⋊C₄	—	1018	1	no	—
57	C₄⋊C₄	—	1018	1	no	—
58	C₄⋊C₄	—	509	1	no	—
59	C₄⋊C₄	—	509	1	no	—
60	C₄⋊C₄	C₂×D₅₀₉	1	1	yes	characteristic
61	C₄²	—	1018	1	no	—
62	C₄²	—	509	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	2036	2	no	—
65	Q₈	Dic₁₀₁₈	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	1018	1	no	—
69	C₂×C₄	—	1018	1	no	—
70	C₂×C₄	—	509	1	no	—
71	C₂×C₄	—	509	1	no	—
72	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	1018	1	no	—
75	C₂²	D₄×D₅₀₉	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	2036	1	no	—
79	C₄	—	2036	1	no	—
80	C₄	—	1018	1	no	—
81	C₄	—	509	2	no	—
82	C₄	C₂×Dic₁₀₁₈	1	1	yes	characteristic
83	C₄	#22550860	1	1	yes	characteristic
84	C₂	#30654192	1	1	yes	characteristic
85	C₂	SD₁₆×D₅₀₉	1	1	yes	characteristic
86	C₂	#30654247	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅₀₉
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	#103×C₅₀₉	C₂
socle, prime radical	C₂²×C₅₀₉	C₂×D₄
hypercenter	#103	D₅₀₉
hypocenter	C₅₀₉	#475
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₅₀₉	Q₈	semidirect
43. C₅₀₉	#475	semidirect
65. Q₈	Dic₁₀₁₈	semidirect

Maximal subgroups

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

subgroup	index
#30654214	2
Q₈×Dic₅₀₉	2
#30654147	2
#30654146	2
#30654144	2
#30654139	2
#103×C₅₀₉	2
#475	509