Subgroups — #45655421

Order 32576.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654245	C₂	1	2	yes	maximal, complemented
2	#30654213	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654142	C₂	1	2	yes	maximal, complemented
4	#30654134	C₂	1	1	yes	maximal, characteristic, complemented
5	2₋¹⁺⁴×C₅₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₁₈	C₂²	1	1	yes	characteristic
7	#22550851	—	2	2	no	—
8	#22550849	—	2	2	no	—
9	#22550845	C₂²	1	2	yes	—
10	#22550840	—	2	2	no	—
11	C₄×Dic₅₀₉	C₂²	1	1	yes	characteristic
12	#41×C₅₀₉	—	2	2	no	—
13	#41×C₅₀₉	—	2	2	no	—
14	#41×C₅₀₉	C₂²	1	2	yes	—
15	C₂×Q₈×C₅₀₉	—	2	2	no	—
16	C₂×Q₈×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	Q₈×C₅₀₉	—	2	2	no	—
22	Q₈×C₅₀₉	—	2	2	no	—
23	Q₈×C₅₀₉	D₄	1	2	yes	—
24	D₄×C₅₀₉	—	2	2	no	—
25	D₄×C₅₀₉	—	2	2	no	—
26	D₄×C₅₀₉	D₄	1	2	yes	—
27	C₂×C₄×C₅₀₉	—	2	4	no	—
28	C₂×C₄×C₅₀₉	—	2	2	no	—
29	C₂×C₄×C₅₀₉	D₄	1	2	yes	—
30	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
31	C₂²×C₅₀₉	—	2	2	no	—
32	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅₀₉	—	2	2	no	—
34	C₄×C₅₀₉	—	2	2	no	—
35	C₄×C₅₀₉	C₂×D₄	1	2	yes	—
36	Dic₅₀₉	—	2	2	no	—
37	Dic₅₀₉	—	4	1	no	—
38	C₂×C₅₀₉	—	2	1	no	—
39	C₂×C₅₀₉	—	4	2	no	—
40	C₂×C₅₀₉	#120	1	1	yes	characteristic
41	C₅₀₉	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	509	1	no	maximal, Sylow
43	2₋¹⁺⁴	D₅₀₉	1	1	yes	characteristic
44	#137	—	509	2	no	—
45	C₄⋊Q₈	—	509	1	no	—
46	#104	—	509	2	no	—
47	#101	—	509	1	no	—
48	#41	—	2	2	no	—
49	#41	—	2	2	no	—
50	#41	C₂×D₅₀₉	1	2	yes	—
51	C₂×Q₈	—	2	2	no	—
52	C₂×Q₈	—	509	1	no	—
53	C₂×Q₈	C₂×D₅₀₉	1	1	yes	characteristic
54	Q₁₆	—	1018	2	no	—
55	SD₁₆	—	1018	2	no	—
56	M₁₆	—	509	2	no	—
57	C₄⋊C₄	—	1018	2	no	—
58	C₄²	—	509	1	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	1018	2	no	—
62	Q₈	C₂²⋊D₅₀₉	1	2	yes	—
63	D₄	—	2	2	no	—
64	D₄	—	2	2	no	—
65	D₄	C₂²⋊D₅₀₉	1	2	yes	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	1018	1	no	—
69	C₂×C₄	—	1018	1	no	—
70	C₂×C₄	C₂²⋊D₅₀₉	1	2	yes	—
71	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
72	C₈	—	1018	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	2036	1	no	—
78	C₄	—	1018	2	no	—
79	C₄	C₂×C₂²⋊D₅₀₉	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	#30654184	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30654184
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting, prime radical	2₋¹⁺⁴×C₅₀₉	C₂
socle	C₂×C₅₀₉	#120
hypercenter	2₋¹⁺⁴	D₅₀₉
hypocenter	C₅₀₉	#456
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #30654245	C₂	semidirect
2. #30654213	C₂	semidirect
3. #30654142	C₂	semidirect
4. #30654134	C₂	semidirect
41. C₅₀₉	#456	semidirect

Maximal subgroups

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

subgroup	index
#30654245	2
#30654213	2
#30654142	2
#30654134	2
2₋¹⁺⁴×C₅₀₉	2
#456	509