Subgroups — #35216856

Order 20864.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782097	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₃₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₆₀₈	C₂	1	2	yes	maximal, complemented
4	#23781504	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19104757	—	2	1	no	—
7	C₂×Dic₆₅₂	—	2	1	no	—
8	Dic₁₃₀₄	—	2	1	no	—
9	Dic₁₃₀₄	C₂²	1	2	yes	—
10	Dic₁₃₀₄	C₂²	1	1	yes	characteristic
11	#19104680	—	2	1	no	—
12	D₂₆₀₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₆₃	C₂²	1	2	yes	—
15	#14579699	—	4	1	no	—
16	C₂×Dic₃₂₆	—	4	1	no	—
17	Dic₆₅₂	—	2	1	no	—
18	Dic₆₅₂	—	2	2	no	—
19	Dic₆₅₂	—	4	1	no	—
20	#14579688	—	4	1	no	—
21	D₁₃₀₄	—	2	1	no	—
22	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
23	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
24	C₁₆×C₁₆₃	C₂³	1	1	yes	characteristic
25	Dic₃₂₆	—	4	1	no	—
26	Dic₃₂₆	—	4	2	no	—
27	Dic₃₂₆	—	8	1	no	—
28	D₆₅₂	—	4	1	no	—
29	C₂²⋊D₁₆₃	—	8	1	no	—
30	C₂×Dic₁₆₃	—	8	1	no	—
31	C₄×D₁₆₃	—	8	1	no	—
32	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
33	C₈×C₁₆₃	D₈	1	1	yes	characteristic
34	C₈×C₁₆₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₆₃	D₁₆	1	1	yes	characteristic
36	C₂×D₁₆₃	—	8	1	no	—
37	C₄×C₁₆₃	D₁₆	1	1	yes	characteristic
38	C₄×C₁₆₃	C₂×D₈	1	1	yes	characteristic
39	Dic₁₆₃	—	8	1	no	—
40	Dic₁₆₃	—	8	2	no	—
41	C₂×C₁₆₃	—	2	1	no	—
42	C₂×C₁₆₃	C₂×D₁₆	1	1	yes	characteristic
43	D₁₆₃	—	16	1	no	—
44	C₁₆₃	#2264	1	1	yes	characteristic, complemented, Sylow
45	#2264	—	163	1	no	maximal, Sylow
46	#508	—	163	1	no	—
47	C₂×Q₃₂	—	163	1	no	—
48	Q₆₄	—	163	2	no	—
49	SD₆₄	—	163	2	no	—
50	M₆₄	D₁₆₃	1	1	yes	characteristic, complemented
51	#135	—	326	1	no	—
52	C₂×Q₁₆	—	326	1	no	—
53	Q₃₂	—	326	1	no	—
54	Q₃₂	—	163	2	no	—
55	Q₃₂	—	163	1	no	—
56	SD₃₂	—	326	1	no	—
57	D₁₆	—	163	1	no	—
58	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
59	C₃₂	C₂×D₁₆₃	1	2	yes	—
60	#41	—	652	1	no	—
61	C₂×Q₈	—	652	1	no	—
62	Q₁₆	—	326	1	no	—
63	Q₁₆	—	326	2	no	—
64	Q₁₆	—	652	1	no	—
65	SD₁₆	—	652	1	no	—
66	D₈	—	326	1	no	—
67	C₂×C₈	D₆₅₂	1	1	yes	characteristic
68	C₁₆	D₆₅₂	1	1	yes	characteristic
69	C₁₆	C₂²×D₁₆₃	1	1	yes	characteristic
70	Q₈	—	1304	1	no	—
71	Q₈	—	652	1	no	—
72	Q₈	—	652	2	no	—
73	D₄	—	1304	1	no	—
74	D₄	—	652	1	no	—
75	C₂×C₄	—	1304	1	no	—
76	C₂×C₄	—	1304	1	no	—
77	C₂×C₄	D₁₃₀₄	1	1	yes	characteristic
78	C₈	D₁₃₀₄	1	1	yes	characteristic
79	C₈	C₂×D₆₅₂	1	1	yes	characteristic
80	C₂²	—	1304	1	no	—
81	C₂²	D₂₆₀₈	1	1	yes	characteristic
82	C₄	—	1304	1	no	—
83	C₄	—	1304	2	no	—
84	C₄	D₂₆₀₈	1	1	yes	characteristic
85	C₄	C₂×D₁₃₀₄	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2608	1	no	—
88	C₂	C₂×D₂₆₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₆₀₈
commutator, cosocle	C₁₆×C₁₆₃	C₂³
Frattini	C₁₆	C₂²×D₁₆₃
Fitting	M₆₄×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₂×D₁₆
prime radical	#23782097	C₂
hypercenter	M₆₄	D₁₆₃
hypocenter	C₁₆₃	#2264
LCS	G ▷ C₁₆×C₁₆₃ ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₃₀₄	C₂	semidirect
3. Dic₂₆₀₈	C₂	semidirect
4. #23781504	C₂	semidirect
5. M₆₄×C₁₆₃	C₂	semidirect
44. C₁₆₃	#2264	semidirect
50. M₆₄	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782097	2
C₂×Dic₁₃₀₄	2
Dic₂₆₀₈	2
#23781504	2
M₆₄×C₁₆₃	2
#2264	163