Subgroups — C₄⋊Q₈×D₅₀₉ #45655779

Order 32576.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₅₀₉	C₂	1	2	yes	maximal
2	#30654215	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654214	C₂	1	4	yes	maximal, complemented
4	#30654213	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×D₅₀₉	C₂	1	4	yes	maximal
6	C₄²×D₅₀₉	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₁₀₁₈	C₂²	1	2	yes	—
9	C₂×Dic₁₀₁₈	C₂²	1	4	yes	—
10	Q₈×D₅₀₉	—	2	8	no	—
11	C₂×C₄×D₅₀₉	C₂²	1	4	yes	—
12	C₂×C₄×D₅₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₅₀₉	C₂²	1	2	yes	—
14	C₄⋊Dic₅₀₉	C₂²	1	4	yes	—
15	#22550840	C₂²	1	8	yes	—
16	C₄×Dic₅₀₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₅₀₉	C₂²	1	2	yes	—
18	C₂×Q₈×C₅₀₉	C₂²	1	2	yes	—
19	C₄⋊C₄×C₅₀₉	C₂²	1	4	yes	—
20	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
21	C₂²×D₅₀₉	C₂³	1	1	yes	characteristic
22	Dic₁₀₁₈	—	2	4	no	—
23	Dic₁₀₁₈	—	2	8	no	—
24	C₂×Dic₅₀₉	C₂³	1	4	yes	—
25	C₂×Dic₅₀₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₅₀₉	C₂³	1	2	yes	—
27	C₄×D₅₀₉	—	2	8	no	—
28	C₄×D₅₀₉	D₄	1	4	yes	—
29	C₄×D₅₀₉	Q₈	1	8	yes	complemented
30	Q₈×C₅₀₉	—	2	4	no	—
31	C₂×C₄×C₅₀₉	C₂³	1	4	yes	—
32	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₅₀₉	C₂³	1	2	yes	—
34	C₂²×C₅₀₉	C₂⁴	1	1	yes	characteristic
35	C₂×D₅₀₉	C₂×D₄	1	2	yes	—
36	C₂×D₅₀₉	C₂×Q₈	1	4	yes	—
37	C₄×C₅₀₉	—	2	4	no	—
38	C₄×C₅₀₉	C₂×D₄	1	2	yes	—
39	C₄×C₅₀₉	C₂×Q₈	1	4	yes	complemented
40	Dic₅₀₉	—	2	4	no	—
41	Dic₅₀₉	C₂×D₄	1	2	yes	—
42	Dic₅₀₉	C₂×Q₈	1	4	yes	complemented
43	C₂×C₅₀₉	C₂²×D₄	1	1	yes	characteristic
44	C₂×C₅₀₉	C₂²×Q₈	1	2	yes	—
45	D₅₀₉	C₄⋊Q₈	1	4	yes	direct product factor
46	C₅₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄⋊Q₈	—	509	1	no	maximal, Sylow
48	C₂²×Q₈	—	509	2	no	—
49	C₄⋊Q₈	—	509	2	no	—
50	C₄⋊Q₈	—	509	4	no	—
51	C₄⋊Q₈	—	509	1	no	—
52	C₄⋊Q₈	D₅₀₉	1	1	yes	characteristic, direct product factor
53	C₂×C₄⋊C₄	—	509	4	no	—
54	C₂×C₄²	—	509	1	no	—
55	C₂×Q₈	—	1018	8	no	—
56	C₂×Q₈	—	509	2	no	—
57	C₂×Q₈	—	509	4	no	—
58	C₂×Q₈	C₂×D₅₀₉	1	2	yes	—
59	C₂²×C₄	—	509	4	no	—
60	C₂²×C₄	—	509	2	no	—
61	C₂²×C₄	—	509	1	no	—
62	C₄⋊C₄	—	509	8	no	—
63	C₄⋊C₄	—	509	4	no	—
64	C₄⋊C₄	C₂×D₅₀₉	1	4	yes	—
65	C₄²	—	509	1	no	—
66	C₄²	—	509	2	no	—
67	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
68	C₂³	—	509	1	no	—
69	Q₈	—	2	4	no	—
70	Q₈	—	1018	8	no	—
71	Q₈	—	1018	4	no	—
72	C₂×C₄	—	1018	8	no	—
73	C₂×C₄	—	509	4	no	—
74	C₂×C₄	—	509	2	no	—
75	C₂×C₄	—	509	1	no	—
76	C₂×C₄	—	509	4	no	—
77	C₂×C₄	—	509	8	no	—
78	C₂×C₄	C₂²×D₅₀₉	1	4	yes	—
79	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₅₀₉	1	2	yes	—
81	C₂²	—	509	2	no	—
82	C₂²	—	509	4	no	—
83	C₂²	C₂³×D₅₀₉	1	1	yes	characteristic
84	C₄	—	2	4	no	—
85	C₄	—	1018	4	no	—
86	C₄	—	509	2	no	—
87	C₄	—	509	4	no	—
88	C₄	D₄×D₅₀₉	1	2	yes	—
89	C₄	Q₈×D₅₀₉	1	4	yes	complemented
90	C₂	—	509	4	no	—
91	C₂	C₂×D₄×D₅₀₉	1	1	yes	characteristic
92	C₂	C₂×Q₈×D₅₀₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₀₉
commutator, socle, cosocle	C₂²×C₅₀₉	C₂⁴
Fitting	C₄⋊Q₈×C₅₀₉	C₂
prime radical	C₂²×D₅₀₉	C₂³
hypercenter	C₄⋊Q₈	D₅₀₉
hypocenter	C₅₀₉	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #30654215	C₂	semidirect
3. #30654214	C₂	semidirect
4. #30654213	C₂	semidirect
7. C₄⋊Q₈×C₅₀₉	C₂	semidirect
29. C₄×D₅₀₉	Q₈	semidirect
39. C₄×C₅₀₉	C₂×Q₈	semidirect
42. Dic₅₀₉	C₂×Q₈	semidirect
45. D₅₀₉	C₄⋊Q₈	direct (N × H)
46. C₅₀₉	C₂×C₄⋊Q₈	semidirect
52. C₄⋊Q₈	D₅₀₉	direct (N × H)
89. 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
C₂×Q₈×D₅₀₉	2
#30654215	2
#30654214	2
#30654213	2
C₄⋊C₄×D₅₀₉	2
C₄²×D₅₀₉	2
C₄⋊Q₈×C₅₀₉	2
C₂×C₄⋊Q₈	509

Subgroups — C4⋊Q8×D509 #45655779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊Q₈×D₅₀₉ #45655779