Subgroups — C₄×#22550851 #45655331

Order 32576.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22550851	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₁₀₁₈	C₂	1	1	yes	maximal, characteristic
3	#30654148	C₂	1	1	yes	maximal, characteristic
4	#30654139	C₂	1	1	yes	maximal, characteristic
5	#30654138	C₂	1	1	yes	maximal, characteristic
6	C₄×#16861325	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀₁₈	C₂²	1	1	yes	characteristic
9	#22550851	C₄	1	4	yes	direct product factor
10	C₂×#16861325	C₂²	1	1	yes	characteristic
11	C₂×#16861325	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₅₀₉	C₂²	1	1	yes	characteristic
13	#22550840	—	2	1	no	—
14	C₄×Dic₅₀₉	—	2	1	no	—
15	C₂×Q₈×C₅₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₅₀₉	—	2	1	no	—
17	C₄⋊C₄×C₅₀₉	C₂²	1	1	yes	characteristic
18	C₄²×C₅₀₉	—	2	1	no	—
19	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
20	Dic₁₀₁₈	—	2	1	no	—
21	Dic₁₀₁₈	C₂×C₄	1	2	yes	—
22	C₂×Dic₅₀₉	—	2	1	no	—
23	C₂×Dic₅₀₉	—	2	1	no	—
24	#16861325	—	2	1	no	—
25	#16861325	C₂×C₄	1	2	yes	—
26	Q₈×C₅₀₉	—	2	1	no	—
27	Q₈×C₅₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₅₀₉	—	2	1	no	—
29	C₂×C₄×C₅₀₉	—	2	1	no	—
30	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
33	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅₀₉	—	2	2	no	—
35	C₄×C₅₀₉	—	2	1	no	—
36	C₄×C₅₀₉	—	4	1	no	—
37	C₄×C₅₀₉	Q₁₆	1	2	yes	complemented
38	C₄×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₅₀₉	#41	1	1	yes	characteristic
40	Dic₅₀₉	—	2	2	no	—
41	Dic₅₀₉	—	4	1	no	—
42	C₂×C₅₀₉	C₄×D₄	1	1	yes	characteristic
43	C₂×C₅₀₉	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₅₀₉	#135	1	1	yes	characteristic
45	C₅₀₉	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	509	1	no	maximal, Sylow
47	C₂×Q₁₆	—	509	1	no	—
48	C₄×Q₈	—	509	1	no	—
49	C₄×Q₈	D₅₀₉	1	1	yes	characteristic
50	#107	—	509	1	no	—
51	#103	—	509	1	no	—
52	#103	—	509	1	no	—
53	C₄×C₈	—	509	1	no	—
54	C₂×Q₈	—	509	1	no	—
55	C₂×Q₈	C₂×D₅₀₉	1	1	yes	characteristic
56	Q₁₆	—	509	4	no	—
57	C₂×C₈	—	509	1	no	—
58	C₂×C₈	—	509	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	1018	1	no	—
61	C₄⋊C₄	—	509	1	no	—
62	C₄⋊C₄	C₂×D₅₀₉	1	1	yes	characteristic
63	C₄²	—	2	1	no	—
64	C₄²	—	1018	1	no	—
65	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
66	Q₈	—	2	1	no	—
67	Q₈	—	1018	1	no	—
68	Q₈	—	509	2	no	—
69	Q₈	C₄×D₅₀₉	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	1018	1	no	—
73	C₂×C₄	—	1018	1	no	—
74	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
75	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
77	C₈	—	1018	1	no	—
78	C₈	—	509	2	no	—
79	C₂²	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	2	no	—
82	C₄	—	4	1	no	—
83	C₄	—	2036	1	no	—
84	C₄	—	1018	2	no	—
85	C₄	#22550851	1	2	yes	direct product factor
86	C₄	C₂×C₄×D₅₀₉	1	1	yes	characteristic
87	C₄	#22550861	1	1	yes	characteristic
88	C₂	C₄×C₂²⋊D₅₀₉	1	1	yes	characteristic
89	C₂	C₂×#22550851	1	1	yes	characteristic
90	C₂	#30654235	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₅₀₉
commutator	C₄×C₅₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₄×Q₈×C₅₀₉	C₂
socle, prime radical	C₂²×C₅₀₉	C₂×D₄
cosocle	C₂×C₄×C₅₀₉	C₂³
hypercenter	C₄×Q₈	D₅₀₉
hypocenter	C₅₀₉	C₄×Q₁₆
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
9. #22550851	C₄	direct (N × H)
37. C₄×C₅₀₉	Q₁₆	semidirect
45. C₅₀₉	C₄×Q₁₆	semidirect
85. C₄	#22550851	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#22550851	2
C₄×Dic₁₀₁₈	2
#30654148	2
#30654139	2
#30654138	2
C₄×#16861325	2
C₄×Q₈×C₅₀₉	2
C₄×Q₁₆	509

Subgroups — C4×#22550851 #45655331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#22550851 #45655331