Subgroups — C₁₆×Dic₃₂₆ #35216439

Order 20864.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₃₂₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₆₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	Dic₁₆₃⋊C₁₆	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₆₃	C₂	1	2	yes	maximal
5	C₄×C₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₃₂₆	C₄	1	1	yes	characteristic
7	C₂×C₁₆₃⋊C₁₆	C₂²	1	2	yes	—
8	C₄⋊(C₁₆₃⋊C₈)	C₄	1	1	yes	characteristic
9	Dic₁₆₃⋊C₈	C₄	1	2	yes	—
10	C₈×Dic₁₆₃	C₂²	1	2	yes	—
11	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₂₆	C₈	1	1	yes	characteristic
15	C₂×C₁₆₃⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₆₃	C₈	1	1	yes	characteristic
17	Dic₁₆₃⋊C₄	C₈	1	2	yes	—
18	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
19	C₁₆₃⋊C₁₆	—	2	2	no	—
20	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
22	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₁₆₃	—	2	1	no	—
24	C₁₆×C₁₆₃	Q₈	1	2	yes	complemented
25	Dic₃₂₆	C₁₆	1	4	yes	direct product factor
26	C₂×Dic₁₆₃	C₂×C₈	1	2	yes	—
27	C₂×Dic₁₆₃	C₂×C₈	1	2	yes	—
28	C₁₆₃⋊C₈	—	2	2	no	—
29	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₆₃	—	2	1	no	—
33	C₈×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₁₆₃	#41	1	1	yes	characteristic
35	C₂²×C₁₆₃	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₆₃	—	2	1	no	—
37	C₄×C₁₆₃	C₂×C₁₆	1	2	yes	—
38	C₄×C₁₆₃	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₁₆₃	#131	1	1	yes	characteristic
40	Dic₁₆₃	—	2	2	no	—
41	Dic₁₆₃	C₂×C₁₆	1	4	yes	—
42	C₂×C₁₆₃	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₆₃	C₂²×C₁₆	1	1	yes	characteristic
44	C₂×C₁₆₃	#504	1	1	yes	characteristic
45	C₁₆₃	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
46	Q₈×C₁₆	—	163	1	no	maximal, Sylow
47	C₈×Q₈	—	163	1	no	—
48	C₄⋊C₁₆	—	163	2	no	—
49	C₄⋊C₁₆	—	163	1	no	—
50	C₄×C₁₆	—	163	2	no	—
51	C₄×C₁₆	D₁₆₃	1	1	yes	characteristic
52	C₄×Q₈	—	163	1	no	—
53	C₂×C₁₆	—	163	2	no	—
54	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
55	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
56	C₄⋊C₈	—	163	2	no	—
57	C₄⋊C₈	—	163	1	no	—
58	C₄×C₈	—	163	2	no	—
59	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
60	C₂×Q₈	—	163	1	no	—
61	C₂×C₈	—	163	2	no	—
62	C₂×C₈	C₄×D₁₆₃	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
64	C₄⋊C₄	—	163	2	no	—
65	C₄⋊C₄	—	163	1	no	—
66	C₄²	—	163	2	no	—
67	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	—	326	2	no	—
70	C₁₆	Dic₃₂₆	1	2	yes	direct product factor
71	Q₈	—	163	4	no	—
72	C₂×C₄	—	163	2	no	—
73	C₂×C₄	—	163	2	no	—
74	C₂×C₄	C₈×D₁₆₃	1	1	yes	characteristic
75	C₂×C₄	C₈×D₁₆₃	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	326	2	no	—
79	C₈	C₂×Dic₃₂₆	1	1	yes	characteristic
80	C₈	#14579699	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₁₆₃	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	326	2	no	—
84	C₄	—	163	4	no	—
85	C₄	C₁₆×D₁₆₃	1	2	yes	—
86	C₄	C₄×Dic₃₂₆	1	1	yes	characteristic
87	C₄	#19104741	1	1	yes	characteristic
88	C₂	C₈×Dic₃₂₆	1	1	yes	characteristic
89	C₂	C₂×C₁₆×D₁₆₃	1	1	yes	characteristic
90	C₂	#23782081	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₁₆₃
commutator	C₂×C₁₆₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₁₆₃
Fitting	C₄×C₁₆×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	C₂²×C₈
cosocle	C₂×C₈×C₁₆₃	C₂³
hypercenter	C₄×C₁₆	D₁₆₃
hypocenter	C₁₆₃	Q₈×C₁₆
LCS	G ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
24. C₁₆×C₁₆₃	Q₈	semidirect
25. Dic₃₂₆	C₁₆	direct (N × H)
45. C₁₆₃	Q₈×C₁₆	semidirect
70. C₁₆	Dic₃₂₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×Dic₃₂₆	2
C₄⋊(C₁₆₃⋊C₁₆)	2
Dic₁₆₃⋊C₁₆	2
C₁₆×Dic₁₆₃	2
C₄×C₁₆×C₁₆₃	2
Q₈×C₁₆	163

Subgroups — C16×Dic326 #35216439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×Dic₃₂₆ #35216439