Subgroups — #35212581

Order 20864.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782042	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₆₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	#335×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₃₀₄	—	2	1	no	—
5	C₂×D₁₃₀₄	C₄	1	2	yes	—
6	#19104732	—	2	1	no	—
7	C₂×C₁₆₃⋊C₁₆	—	2	1	no	—
8	C₄⋊C₈×C₁₆₃	—	2	1	no	—
9	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
10	C₂×D₆₅₂	—	2	2	no	—
11	C₂×D₆₅₂	—	4	1	no	—
12	C₂×D₆₅₂	—	4	1	no	—
13	D₁₃₀₄	—	4	2	no	—
14	D₁₃₀₄	—	4	1	no	—
15	D₁₃₀₄	—	4	1	no	—
16	C₁₆₃⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₁₆₃	—	4	1	no	—
18	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
20	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
21	C₂²×D₁₆₃	—	4	2	no	—
22	D₆₅₂	—	4	2	no	—
23	D₆₅₂	—	4	2	no	—
24	D₆₅₂	—	8	2	no	—
25	D₆₅₂	—	8	1	no	—
26	D₆₅₂	—	8	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	—	8	1	no	—
32	C₈×C₁₆₃	D₈	1	1	yes	characteristic
33	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
34	C₂²×C₁₆₃	#102	1	1	yes	characteristic
35	C₂×D₁₆₃	—	8	2	no	—
36	C₂×D₁₆₃	—	8	2	no	—
37	C₂×D₁₆₃	—	8	2	no	—
38	C₄×C₁₆₃	—	2	1	no	—
39	C₄×C₁₆₃	#100	1	1	yes	characteristic
40	C₄×C₁₆₃	#102	1	1	yes	characteristic
41	C₄×C₁₆₃	D₁₆	1	1	yes	characteristic
42	C₄×C₁₆₃	SD₃₂	1	1	yes	characteristic
43	C₂×C₁₆₃	#331	1	1	yes	characteristic
44	C₂×C₁₆₃	#357	1	1	yes	characteristic
45	C₂×C₁₆₃	#361	1	1	yes	characteristic
46	D₁₆₃	—	16	2	no	—
47	C₁₆₃	#1361	1	1	yes	characteristic, complemented, Sylow
48	#1361	—	163	1	no	maximal, Sylow
49	#493	—	163	1	no	—
50	C₄⋊C₁₆	—	163	1	no	—
51	#335	D₁₆₃	1	1	yes	characteristic, complemented
52	C₂×D₈	—	326	1	no	—
53	C₂×D₈	—	163	2	no	—
54	#127	—	326	1	no	—
55	C₂×C₁₆	—	326	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
58	C₂×D₄	—	326	2	no	—
59	C₂×D₄	—	652	1	no	—
60	C₂×D₄	—	652	1	no	—
61	D₈	—	652	2	no	—
62	D₈	—	652	1	no	—
63	D₈	—	652	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	C₄×D₁₆₃	1	1	yes	characteristic
66	C₂×C₈	D₆₅₂	1	1	yes	characteristic
67	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
68	C₁₆	—	652	1	no	—
69	C₂³	—	652	2	no	—
70	D₄	—	1304	2	no	—
71	D₄	—	1304	1	no	—
72	D₄	—	1304	1	no	—
73	D₄	—	652	2	no	—
74	D₄	—	652	2	no	—
75	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	#14579684	1	1	yes	characteristic
77	C₂×C₄	#14579686	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	8	1	no	—
80	C₈	D₁₃₀₄	1	1	yes	characteristic
81	C₈	#14579688	1	1	yes	characteristic
82	C₂²	—	1304	2	no	—
83	C₂²	—	1304	2	no	—
84	C₂²	—	1304	2	no	—
85	C₂²	#19104655	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#19104651	1	1	yes	characteristic
88	C₄	#19104657	1	1	yes	characteristic
89	C₄	#19104676	1	1	yes	characteristic
90	C₄	#19104678	1	1	yes	characteristic
91	C₂	—	2608	2	no	—
92	C₂	#23781408	1	1	yes	characteristic
93	C₂	#23781465	1	1	yes	characteristic
94	C₂	#23781477	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19104655
commutator	C₂×C₈×C₁₆₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	#335×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#102
prime radical	#23782042	C₂
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#335	D₁₆₃
hypocenter	C₁₆₃	#1361
LCS	G ▷ C₂×C₈×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₁₆₃⋊C₁₆)	C₂	semidirect
3. #335×C₁₆₃	C₂	semidirect
47. C₁₆₃	#1361	semidirect
51. #335	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782042	2
C₄⋊(C₁₆₃⋊C₁₆)	2
#335×C₁₆₃	2
#1361	163