Subgroups — #45655525

Order 32576.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₂₀₃₆	C₂	1	1	yes	maximal, characteristic
2	#30654146	C₂	1	2	yes	maximal, complemented
3	C₄⋊#16861325	C₂	1	1	yes	maximal, characteristic, complemented
4	#30654135	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀₃₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₅₀₉	—	2	1	no	—
8	C₂×#16861325	C₂²	1	2	yes	—
9	C₄⋊Dic₅₀₉	C₂²	1	1	yes	characteristic
10	#22550838	—	2	1	no	—
11	C₂×Q₈×C₅₀₉	—	2	1	no	—
12	C₄⋊C₄×C₅₀₉	—	2	1	no	—
13	C₄⋊C₄×C₅₀₉	C₂²	1	2	yes	—
14	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
15	C₂²×D₅₀₉	—	2	1	no	—
16	D₂₀₃₆	—	2	1	no	—
17	D₂₀₃₆	Q₈	1	2	yes	complemented
18	C₂×Dic₅₀₉	—	2	1	no	—
19	C₄×D₅₀₉	—	4	2	no	—
20	#16861325	—	2	2	no	—
21	Q₈×C₅₀₉	—	4	2	no	—
22	C₂×C₄×C₅₀₉	—	2	2	no	—
23	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
26	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
27	C₂×D₅₀₉	—	2	2	no	—
28	C₂×D₅₀₉	—	4	1	no	—
29	C₂×D₅₀₉	—	4	1	no	—
30	C₄×C₅₀₉	—	2	1	no	—
31	C₄×C₅₀₉	—	4	2	no	—
32	C₄×C₅₀₉	SD₁₆	1	2	yes	—
33	C₄×C₅₀₉	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₅₀₉	#41	1	1	yes	characteristic
35	Dic₅₀₉	—	4	1	no	—
36	C₂×C₅₀₉	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₅₀₉	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₅₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₅₀₉	—	4	2	no	—
40	C₅₀₉	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	509	1	no	maximal, Sylow
42	C₄⋊Q₈	D₅₀₉	1	1	yes	characteristic, complemented
43	C₄×D₄	—	509	1	no	—
44	#106	—	509	2	no	—
45	C₄⋊C₈	—	509	1	no	—
46	#102	—	509	2	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	509	1	no	—
49	C₂²×C₄	—	1018	1	no	—
50	C₂×C₈	—	509	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	509	1	no	—
53	C₄⋊C₄	C₂×D₅₀₉	1	2	yes	—
54	C₂²⋊C₄	—	1018	1	no	—
55	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
56	C₂³	—	1018	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	1018	1	no	—
59	D₄	—	509	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2036	2	no	—
62	C₂×C₄	—	1018	1	no	—
63	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
66	C₈	—	1018	2	no	—
67	C₂²	—	2036	1	no	—
68	C₂²	—	2036	1	no	—
69	C₂²	—	1018	2	no	—
70	C₂²	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	2036	1	no	—
74	C₄	#22550848	1	2	yes	—
75	C₄	Q₈×D₅₀₉	1	1	yes	characteristic
76	C₄	#22550862	1	1	yes	characteristic
77	C₂	—	2036	2	no	—
78	C₂	#30654191	1	1	yes	characteristic
79	C₂	C₂×#22550848	1	1	yes	characteristic
80	C₂	#30654241	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₀₉
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₄⋊Q₈×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂×D₄
prime radical	C₂×D₂₀₃₆	C₂²
hypercenter	C₄⋊Q₈	D₅₀₉
hypocenter	C₅₀₉	#476
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #30654146	C₂	semidirect
3. C₄⋊#16861325	C₂	semidirect
5. C₄⋊Q₈×C₅₀₉	C₂	semidirect
17. D₂₀₃₆	Q₈	semidirect
40. C₅₀₉	#476	semidirect
42. C₄⋊Q₈	D₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₂₀₃₆	2
#30654146	2
C₄⋊#16861325	2
#30654135	2
C₄⋊Q₈×C₅₀₉	2
#476	509