Subgroups — #35212500

Order 20864.

98 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	#23781418	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆₃⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₁₆₃⋊C₈)	—	2	1	no	—
6	C₂×Q₁₆×C₁₆₃	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₆₃	—	2	1	no	—
8	#107×C₁₆₃	C₄	1	1	yes	characteristic
9	#103×C₁₆₃	—	2	1	no	—
10	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₆₃⋊C₈	—	4	1	no	—
12	C₁₆₃⋊C₁₆	—	2	2	no	—
13	C₂×Q₈×C₁₆₃	—	2	1	no	—
14	Q₁₆×C₁₆₃	—	2	1	no	—
15	Q₁₆×C₁₆₃	C₈	1	2	yes	complemented
16	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
18	C₄⋊C₄×C₁₆₃	—	2	1	no	—
19	C₄⋊C₄×C₁₆₃	—	4	1	no	—
20	C₄²×C₁₆₃	—	4	1	no	—
21	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
22	C₁₆₃⋊C₈	—	8	1	no	—
23	Q₈×C₁₆₃	—	2	2	no	—
24	Q₈×C₁₆₃	—	4	1	no	—
25	C₂×C₄×C₁₆₃	—	4	1	no	—
26	C₂×C₄×C₁₆₃	—	4	1	no	—
27	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₆₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₆₃	—	2	1	no	—
31	C₈×C₁₆₃	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₆₃	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₆₃	#102	1	1	yes	characteristic
34	C₄×C₁₆₃	—	2	1	no	—
35	C₄×C₁₆₃	—	4	2	no	—
36	C₄×C₁₆₃	—	8	1	no	—
37	C₄×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
38	C₄×C₁₆₃	#104	1	1	yes	characteristic
39	C₄×C₁₆₃	SD₃₂	1	1	yes	characteristic
40	C₄×C₁₆₃	Q₃₂	1	1	yes	characteristic
41	C₂×C₁₆₃	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₁₆₃	#358	1	1	yes	characteristic
43	C₂×C₁₆₃	#359	1	1	yes	characteristic
44	C₁₆₃	#1332	1	1	yes	characteristic, complemented, Sylow
45	#1332	—	163	1	no	maximal, Sylow
46	C₄×Q₁₆	D₁₆₃	1	1	yes	characteristic
47	C₄×C₁₆	—	163	1	no	—
48	#335	—	163	1	no	—
49	C₂×Q₁₆	Dic₁₆₃	1	1	yes	characteristic
50	C₄×Q₈	—	2	1	no	—
51	C₂×C₁₆	—	326	1	no	—
52	#107	Dic₁₆₃	1	1	yes	characteristic
53	C₄⋊C₈	—	326	1	no	—
54	#103	—	2	1	no	—
55	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
56	C₂×Q₈	—	2	1	no	—
57	Q₁₆	—	2	1	no	—
58	Q₁₆	C₁₆₃⋊C₈	1	2	yes	complemented
59	C₂×C₈	—	652	1	no	—
60	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
61	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
62	C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₄	—	4	1	no	—
64	C₄²	—	4	1	no	—
65	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
66	C₁₆	—	326	2	no	—
67	Q₈	—	2	2	no	—
68	Q₈	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
72	C₂×C₄	#14579684	1	1	yes	characteristic
73	C₂×C₄	#14579687	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	1304	1	no	—
76	C₈	C₂×C₁₆₃⋊C₈	1	1	yes	characteristic
77	C₈	#14579683	1	1	yes	characteristic
78	C₂²	#19104656	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	4	2	no	—
81	C₄	—	8	1	no	—
82	C₄	C₂²⋊(C₁₆₃⋊C₈)	1	1	yes	characteristic
83	C₄	#19104662	1	1	yes	characteristic
84	C₄	#19104678	1	1	yes	characteristic
85	C₄	#19104681	1	1	yes	characteristic
86	C₂	D₄⋊(C₁₆₃⋊C₈)	1	1	yes	characteristic
87	C₂	#23781469	1	1	yes	characteristic
88	C₂	#23781472	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579684
commutator	C₈×C₁₆₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	C₄×Q₁₆×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	#102
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	C₄×Q₁₆	D₁₆₃
hypocenter	C₁₆₃	#1332
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
15. Q₁₆×C₁₆₃	C₈	semidirect
44. C₁₆₃	#1332	semidirect
58. Q₁₆	C₁₆₃⋊C₈	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
#23781418	2
C₄×Q₁₆×C₁₆₃	2
#1332	163