Subgroups — #35216445

Order 20864.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₆₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₆₃⋊C₁₆)	C₂	1	2	yes	maximal
3	#23781441	C₂	1	2	yes	maximal
4	C₄×C₁₆₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₈×Q₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₆₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₆₃⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₁₆₃⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₁₆₃	C₄	1	1	yes	characteristic
10	C₄⋊C₈×C₁₆₃	C₄	1	2	yes	—
11	C₄⋊C₈×C₁₆₃	C₄	1	1	yes	characteristic
12	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₆₃	C₂²	1	2	yes	—
14	C₁₆₃⋊C₁₆	—	2	1	no	—
15	C₁₆₃⋊C₁₆	—	2	2	no	—
16	C₁₆₃⋊C₁₆	Q₈	1	2	yes	complemented
17	C₂×Q₈×C₁₆₃	C₈	1	1	yes	characteristic
18	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₆₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₆₃	C₈	1	2	yes	—
22	C₄⋊C₄×C₁₆₃	C₈	1	1	yes	characteristic
23	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
24	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
25	Q₈×C₁₆₃	—	2	2	no	—
26	C₂×C₄×C₁₆₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₆₃	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₁₆₃	—	2	1	no	—
32	C₈×C₁₆₃	—	2	2	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	2	no	—
37	C₄×C₁₆₃	—	2	2	no	—
38	C₄×C₁₆₃	—	2	1	no	—
39	C₄×C₁₆₃	M₃₂	1	2	yes	—
40	C₄×C₁₆₃	C₄×Q₈	1	1	yes	characteristic
41	C₄×C₁₆₃	#131	1	1	yes	characteristic
42	C₂×C₁₆₃	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₆₃	C₂×M₃₂	1	1	yes	characteristic
44	C₂×C₁₆₃	#504	1	1	yes	characteristic
45	C₁₆₃	#2183	1	1	yes	characteristic, complemented, Sylow
46	#2183	—	163	1	no	maximal, Sylow
47	C₈×Q₈	D₁₆₃	1	1	yes	characteristic
48	C₄⋊C₁₆	—	163	1	no	—
49	C₄⋊C₁₆	—	163	2	no	—
50	#346	—	163	2	no	—
51	C₄×C₁₆	—	163	1	no	—
52	C₄×Q₈	Dic₁₆₃	1	1	yes	characteristic
53	C₂×C₁₆	—	163	1	no	—
54	C₂×C₁₆	—	163	2	no	—
55	C₂×C₁₆	—	163	1	no	—
56	C₄⋊C₈	Dic₁₆₃	1	2	yes	—
57	C₄⋊C₈	Dic₁₆₃	1	1	yes	characteristic
58	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
59	C₄×C₈	C₂×D₁₆₃	1	2	yes	—
60	C₂×Q₈	C₁₆₃⋊C₈	1	1	yes	characteristic
61	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
62	C₂×C₈	C₂×Dic₁₆₃	1	2	yes	—
63	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
64	C₄⋊C₄	C₁₆₃⋊C₈	1	2	yes	—
65	C₄⋊C₄	C₁₆₃⋊C₈	1	1	yes	characteristic
66	C₄²	C₂×Dic₁₆₃	1	2	yes	—
67	C₄²	C₂×Dic₁₆₃	1	1	yes	characteristic
68	C₁₆	—	326	1	no	—
69	C₁₆	—	326	2	no	—
70	C₁₆	—	163	2	no	—
71	Q₈	—	2	2	no	—
72	C₂×C₄	C₂×C₁₆₃⋊C₈	1	2	yes	—
73	C₂×C₄	C₂×C₁₆₃⋊C₈	1	1	yes	characteristic
74	C₂×C₄	C₂×C₁₆₃⋊C₈	1	2	yes	—
75	C₂×C₄	C₂×C₁₆₃⋊C₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	2	no	—
79	C₈	Q₈×D₁₆₃	1	1	yes	characteristic
80	C₈	#14579700	1	1	yes	characteristic
81	C₂²	C₂²×C₁₆₃⋊C₈	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	#19104675	1	2	yes	—
86	C₄	Q₈×Dic₁₆₃	1	1	yes	characteristic
87	C₄	#19104742	1	1	yes	characteristic
88	C₂	Q₈×C₁₆₃⋊C₈	1	1	yes	characteristic
89	C₂	C₂×#19104675	1	1	yes	characteristic
90	C₂	#23782082	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₆₃
commutator, socle, prime radical	C₂²×C₁₆₃	C₂²×C₈
Fitting	C₈×Q₈×C₁₆₃	C₂
cosocle	C₂×C₈×C₁₆₃	C₂³
hypercenter	C₈×Q₈	D₁₆₃
hypocenter	C₁₆₃	#2183
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
16. C₁₆₃⋊C₁₆	Q₈	semidirect
45. C₁₆₃	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₆₃⋊C₁₆)	2
#23781441	2
C₄×C₁₆₃⋊C₁₆	2
C₈×Q₈×C₁₆₃	2
#2183	163