Subgroups — #45656212

Order 32576.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22550871	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#16861332	C₂	1	4	yes	maximal, complemented
4	C₂×#22550860	C₂	1	1	yes	maximal, characteristic
5	#22550871	—	2	2	no	—
6	D₅₀₉⋊C₈	—	2	2	no	—
7	C₂×#16861332	C₂²	1	2	yes	—
8	C₂×#16861332	C₂²	1	2	yes	—
9	#22550860	—	2	2	no	—
10	#22550860	—	2	2	no	—
11	C₂×Dic₁₀₁₈	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊D₅₀₉	C₄	1	2	yes	—
13	C₂²×Dic₅₀₉	C₂²	1	2	yes	—
14	C₂×C₄×D₅₀₉	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₅₀₉	C₄	1	1	yes	characteristic
16	#16861332	—	2	2	no	—
17	#16861332	—	2	2	no	—
18	C₂²×D₅₀₉	C₂×C₄	1	1	yes	characteristic
19	Dic₁₀₁₈	—	2	2	no	—
20	C₂²⋊D₅₀₉	—	2	4	no	—
21	C₂×Dic₅₀₉	—	2	2	no	—
22	C₂×Dic₅₀₉	—	2	2	no	—
23	C₂×Dic₅₀₉	C₂×C₄	1	2	yes	—
24	C₂×Dic₅₀₉	C₂³	1	1	yes	characteristic
25	C₄×D₅₀₉	D₄	1	2	yes	—
26	C₄×D₅₀₉	D₄	1	2	yes	—
27	C₂³×C₅₀₉	C₂×C₄	1	2	yes	—
28	D₄×C₅₀₉	—	2	2	no	—
29	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅₀₉	—	2	2	no	—
31	C₂²×C₅₀₉	—	2	4	no	—
32	C₂²×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×D₅₀₉	—	2	2	no	—
34	C₂×D₅₀₉	C₂²⋊C₄	1	2	yes	—
35	C₄×C₅₀₉	C₂²⋊C₄	1	2	yes	—
36	Dic₅₀₉	—	2	2	no	—
37	Dic₅₀₉	C₂×D₄	1	1	yes	characteristic
38	Dic₅₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₅₀₉	—	4	2	no	—
40	C₂×C₅₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₅₀₉	#131	1	2	yes	—
42	D₅₀₉	—	2	2	no	—
43	C₅₀₉	#408	1	1	yes	characteristic, complemented, Sylow
44	#408	—	509	1	no	maximal, Sylow
45	C₂×#41	—	509	1	no	—
46	C₂×M₁₆	—	509	1	no	—
47	C₂²×C₈	—	509	1	no	—
48	C₂²⋊C₈	—	509	4	no	—
49	#41	—	1018	2	no	—
50	#41	—	1018	2	no	—
51	C₂×Q₈	—	509	1	no	—
52	C₂×D₄	—	509	2	no	—
53	C₂×D₄	#14523266	1	1	yes	characteristic
54	C₂²×C₄	—	509	2	no	—
55	C₂²×C₄	—	509	1	no	—
56	M₁₆	—	1018	2	no	—
57	C₂×C₈	—	1018	2	no	—
58	C₂×C₈	—	509	2	no	—
59	C₂×C₈	—	509	2	no	—
60	C₂³	—	509	1	no	—
61	C₂³	C₂×#14523266	1	2	yes	—
62	Q₈	—	1018	2	no	—
63	D₄	—	2	2	no	—
64	D₄	—	1018	4	no	—
65	C₂×C₄	—	1018	2	no	—
66	C₂×C₄	—	1018	2	no	—
67	C₂×C₄	—	509	2	no	—
68	C₂×C₄	—	509	1	no	—
69	C₂×C₄	—	509	2	no	—
70	C₂×C₄	—	509	2	no	—
71	C₂×C₄	C₂×#14523266	1	1	yes	characteristic
72	C₈	—	1018	2	no	—
73	C₈	—	1018	2	no	—
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	—	1018	2	no	—
77	C₂²	—	509	2	no	—
78	C₂²	C₂²×#14523266	1	1	yes	characteristic
79	C₄	—	1018	2	no	—
80	C₄	—	509	1	no	—
81	C₄	—	509	1	no	—
82	C₄	C₂²⋊#14523266	1	2	yes	—
83	C₂	—	4	2	no	—
84	C₂	—	1018	2	no	—
85	C₂	C₂×C₂²⋊#14523266	1	1	yes	characteristic
86	C₂	#30654302	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#14523266
commutator, socle	C₂²×C₅₀₉	C₂²×C₄
Fitting	C₂×D₄×C₅₀₉	C₄
prime radical	C₂×#22550860	C₂
cosocle	C₂×Dic₅₀₉	C₂³
hypercenter	C₂×D₄	#14523266
hypocenter	C₅₀₉	#408
LCS	G ▷ C₂²×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#22550871	C₂	semidirect
2. C₂×D₅₀₉⋊C₈	C₂	semidirect
3. C₂²⋊#16861332	C₂	semidirect
43. C₅₀₉	#408	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22550871	2
C₂×D₅₀₉⋊C₈	2
C₂²⋊#16861332	2
C₂×#22550860	2
#408	509