Subgroups — #35212516

Order 20864.

94 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	C₄×C₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₃₂₆	C₄	1	1	yes	characteristic
5	C₂×C₁₆₃⋊C₁₆	—	2	1	no	—
6	C₄⋊(C₁₆₃⋊C₈)	C₄	1	1	yes	characteristic
7	Dic₁₆₃⋊C₈	—	2	1	no	—
8	C₈×Dic₁₆₃	—	2	1	no	—
9	C₂×C₁₆×C₁₆₃	—	2	1	no	—
10	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₂₆	C₈	1	1	yes	characteristic
12	C₂×C₁₆₃⋊C₈	—	2	1	no	—
13	C₄⋊Dic₁₆₃	C₈	1	1	yes	characteristic
14	Dic₁₆₃⋊C₄	—	2	1	no	—
15	C₄×Dic₁₆₃	—	2	1	no	—
16	C₁₆₃⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
19	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₆₃	—	2	2	no	—
21	Dic₃₂₆	—	2	1	no	—
22	Dic₃₂₆	C₁₆	1	2	yes	complemented
23	C₂×Dic₁₆₃	—	2	1	no	—
24	C₂×Dic₁₆₃	—	2	1	no	—
25	C₁₆₃⋊C₈	—	4	1	no	—
26	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₆₃	M₁₆	1	1	yes	characteristic
29	C₈×C₁₆₃	—	2	1	no	—
30	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₆₃	Q₁₆	1	1	yes	characteristic
32	C₂²×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₆₃	—	2	1	no	—
34	C₄×C₁₆₃	#103	1	1	yes	characteristic
35	C₄×C₁₆₃	#104	1	1	yes	characteristic
36	C₄×C₁₆₃	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₆₃	M₃₂	1	1	yes	characteristic
38	Dic₁₆₃	—	2	2	no	—
39	Dic₁₆₃	—	4	1	no	—
40	C₂×C₁₆₃	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₆₃	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₆₃	#350	1	1	yes	characteristic
43	C₁₆₃	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
44	Q₈⋊C₁₆	—	163	1	no	maximal, Sylow
45	C₈×Q₈	—	163	1	no	—
46	C₄⋊C₁₆	—	163	1	no	—
47	C₄×C₁₆	D₁₆₃	1	1	yes	characteristic
48	C₄×Q₈	—	163	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	326	1	no	—
51	C₄⋊C₈	—	326	1	no	—
52	C₄⋊C₈	—	163	1	no	—
53	C₄×C₈	—	326	1	no	—
54	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
55	C₂×Q₈	—	163	1	no	—
56	C₂×C₈	—	326	1	no	—
57	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
58	C₂×C₈	D₆₅₂	1	1	yes	characteristic
59	C₄⋊C₄	—	326	1	no	—
60	C₄⋊C₄	—	163	1	no	—
61	C₄²	—	326	1	no	—
62	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	652	1	no	—
65	Q₈	—	326	1	no	—
66	Q₈	—	163	2	no	—
67	C₂×C₄	—	326	1	no	—
68	C₂×C₄	—	326	1	no	—
69	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
70	C₂×C₄	C₈×D₁₆₃	1	1	yes	characteristic
71	C₂×C₄	#14579682	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	652	1	no	—
74	C₈	#14579688	1	1	yes	characteristic
75	C₈	Dic₆₅₂	1	1	yes	characteristic
76	C₂²	(C₂×D₁₆₃)⋊C₈	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	326	2	no	—
79	C₄	—	652	1	no	—
80	C₄	#19104660	1	1	yes	characteristic
81	C₄	#19104663	1	1	yes	characteristic
82	C₄	C₁₆×D₁₆₃	1	1	yes	characteristic
83	C₄	#19104674	1	1	yes	characteristic
84	C₂	#23781397	1	1	yes	characteristic
85	C₂	(C₂×D₁₆₃)⋊C₁₆	1	1	yes	characteristic
86	C₂	#23781448	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₆₅₂
commutator	C₄×C₁₆₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	C₄×C₁₆×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	C₂²⋊C₈
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	C₄×C₁₆	D₁₆₃
hypocenter	C₁₆₃	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
22. Dic₃₂₆	C₁₆	semidirect
43. C₁₆₃	Q₈⋊C₁₆	semidirect

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
C₄×C₁₆×C₁₆₃	2
Q₈⋊C₁₆	163