Subgroups — #45655394

Order 32576.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22550851	C₂	1	2	yes	maximal, complemented
2	#30654194	C₂	1	1	yes	maximal, characteristic
3	#30654139	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#16861325	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₁₈	C₂²	1	1	yes	characteristic
7	#22550851	—	2	4	no	—
8	C₂×#16861325	C₂²	1	2	yes	—
9	C₄⋊Dic₅₀₉	C₂²	1	1	yes	characteristic
10	#22550840	—	2	1	no	—
11	C₂²⋊Dic₅₀₉	—	2	1	no	—
12	C₂×Q₈×C₅₀₉	—	2	4	no	—
13	C₂×Q₈×C₅₀₉	—	2	1	no	—
14	C₂×Q₈×C₅₀₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₀₉	—	2	1	no	—
16	C₂²×C₄×C₅₀₉	C₂²	1	1	yes	characteristic
17	Dic₁₀₁₈	—	2	2	no	—
18	C₂×Dic₅₀₉	—	2	1	no	—
19	C₂×Dic₅₀₉	—	2	1	no	—
20	#16861325	—	2	2	no	—
21	C₂³×C₅₀₉	D₄	1	1	yes	characteristic
22	Q₈×C₅₀₉	—	2	4	no	—
23	Q₈×C₅₀₉	—	2	2	no	—
24	Q₈×C₅₀₉	D₄	1	4	yes	—
25	C₂×C₄×C₅₀₉	—	2	4	no	—
26	C₂×C₄×C₅₀₉	—	2	2	no	—
27	C₂×C₄×C₅₀₉	—	2	2	no	—
28	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
30	C₂²×C₅₀₉	—	2	1	no	—
31	C₂²×C₅₀₉	—	2	1	no	—
32	C₂²×C₅₀₉	Q₁₆	1	2	yes	complemented
33	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅₀₉	—	2	4	no	—
35	C₄×C₅₀₉	—	2	1	no	—
36	C₄×C₅₀₉	C₂×D₄	1	2	yes	—
37	Dic₅₀₉	—	4	1	no	—
38	Dic₅₀₉	—	4	1	no	—
39	C₂×C₅₀₉	—	2	2	no	—
40	C₂×C₅₀₉	#120	1	1	yes	characteristic
41	C₂×C₅₀₉	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×C₅₀₉	#137	1	1	yes	characteristic
43	C₅₀₉	#451	1	1	yes	characteristic, complemented, Sylow
44	#451	—	509	1	no	maximal, Sylow
45	C₂²×Q₈	D₅₀₉	1	1	yes	characteristic
46	C₂×Q₁₆	—	509	2	no	—
47	C₂²⋊Q₈	—	509	1	no	—
48	#103	—	509	2	no	—
49	C₂²⋊C₈	—	509	1	no	—
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	—	2	1	no	—
52	C₂×Q₈	—	509	1	no	—
53	C₂×Q₈	C₂×D₅₀₉	1	2	yes	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	C₂×D₅₀₉	1	1	yes	characteristic
56	Q₁₆	—	1018	4	no	—
57	C₂×C₈	—	509	2	no	—
58	C₄⋊C₄	—	1018	1	no	—
59	C₄⋊C₄	—	509	1	no	—
60	C₂²⋊C₄	—	1018	1	no	—
61	C₂³	C₂²⋊D₅₀₉	1	1	yes	characteristic
62	Q₈	—	2	4	no	—
63	Q₈	—	2	2	no	—
64	Q₈	—	1018	2	no	—
65	Q₈	C₂²⋊D₅₀₉	1	4	yes	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	1018	1	no	—
70	C₂×C₄	—	1018	1	no	—
71	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
73	C₈	—	1018	2	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	#22550851	1	2	yes	complemented
77	C₂²	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	2	1	no	—
80	C₄	—	2036	1	no	—
81	C₄	—	2036	1	no	—
82	C₄	C₂×C₂²⋊D₅₀₉	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	#30654184	1	1	yes	characteristic
85	C₂	C₂×#22550851	1	1	yes	characteristic
86	C₂	#30654246	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₀₉
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₂²×Q₈×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂×D₄
prime radical	C₂³×C₅₀₉	D₄
hypercenter	C₂²×Q₈	D₅₀₉
hypocenter	C₅₀₉	#451
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#22550851	C₂	semidirect
3. #30654139	C₂	semidirect
32. C₂²×C₅₀₉	Q₁₆	semidirect
43. C₅₀₉	#451	semidirect
76. C₂²	#22550851	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22550851	2
#30654194	2
#30654139	2
C₂²⋊#16861325	2
C₂²×Q₈×C₅₀₉	2
#451	509